
























































































































































































































































































































































































































Glass_lX-i^--.) -5 J 

Book_ •-Ijk-3_ 

l?^7 


COPYRIGHT DEPOSIT 






























NYSTROM’S 

POCKET-BOOK 



OF 


MECHANICS 


AND 


ENGINEERING. 


REVISED AND CORRECTED BY 


WILLIAM DENNIS MARKS, Ph. B. f C. E. [YALE s. S. S.], 


Vhiiney Professor of Dynamical Engineering, University of Penn¬ 
sylvania; Honorary Life-Member of the Franklin Institute; 
Member of the American Philosophical Society, Phila¬ 
delphia; Member of the Electric Club, N. Y. ; and 
Consulting Mechanical and Civil Engineer. 



NINETEENTH EDITION. REVISED AND GREATLY ENLARGEI 


with 


ORIGINAL MATTER 



PHILADELPHIAT 


J. B. LIPPINCOTT COMPANY. 

LONDON: 10 HENRIETTA STREET, CO VENT GARDEN. „ 


1887 


A 























\ 












Entered according to Act of Congress, in the year 1872, by 
JOHN W. NYSTROM, 


In the office of the Librarian of Congress, at Washington. 


Copyright, 1883, by JOHN W. NYSTROM. 




Copyright, 1887, by J. B. LIFfINCOTT COMPANY, 




\ft 




















Preface. 


3 


/ fcA- 11 

PREFACE TO THE NINETEENTH EDITION...-* 

: . 6 / ?? 

The very large number of this Pocket-Book sold in past years 

seems to prove its utility in its present shape. The Reviser, there¬ 
fore,. has principally confined himself to corrections of obvious 
errors in the English and the formulae. 

He cannot hope to have found and corrected every error, and 
will esteem it as the greatest possible favor if any person will call 
his attention to oversights. 

An elementary article on dynamic electricity has been added by 
him, as also one on the expansion of steam. f 

In the form of notes, the Reviser has at times stated his differ- 
ent opinion, and also referred the reader to the literature of topics 
which required more space than can be given to them in a pocket- 
book. 

A single branch of mechanical engineering may require half a 
lifetime of study and experience before the engineer is competent 
to undertake designing in it. 

The writer cannot too strongly call the attention of engineers to 
the deplorable results of over-confidence and superficial reasoning. 
It would be far better that our laws should hold him a criminal 
who, through ignorance or dishonest pretension to knowledge, 
makes blunders resulting in the loss of life and property, than 
that the present loose ideas as to the dignity and responsibility of 
an engineer’s position and work should prevail. 

A pocket-book serves as a useful memorandum for the expe¬ 
rienced engineer, and should guide the inexperienced engineer to 
higher and wider fields of thought and research. 

Modestly remember, in engineering, that the possibility of error 
or oversight in our premises is so great that the result of unverified 
mathematical investigation remains only a “presumption” until 
verified by experiment. It makes no difference what the intention 
or wish of the designer may be, matter and force blindly and inev¬ 
itably follow the laws of nature with certainty and precision. The 
designer must learn these laws, and all of them that affect the partic- j 
ular case he has, before he can effect a safe and general solution. 

WILLIAM DENNIS MARKS. 

University of Pennsylvania, 

Philadelphia, 1887. 











PREFACE. 


Every Engineer should make his own Pocket-Book, as lie pr< 
Ceeds in study and practice, to suit his particular business. Tli 
present work has been accumulated in that way during the author 
professional career. It was originally not intended for publics 
tion, but grew too large for the pocket in form of manuscrip 
which circumstance, combined with repeated requests to publish i 
first placed it before the public in the year 1854. 

The author claims to have given a goodly share of origins 
matter, and has spent much labor and money in experiments o 
subjects requiring elucidation. 

The authors consulted are distinguished experimenters, such a 
Dalton, on air and heat; Regnault, on steam; Kopp, on the ex 
pansion of water; Morin, on friction and strength of materials 
Joule, on the mechanical equivalent of heat; the Franklin Inst: 
tute, on the strength of iron and copper at different temperatures 
the Boyal Technological Institute, Stockholm, on dynamics; an 
various others of equal authority; but these savans are not it 
sponsible for the formulas and tables which are herein deduce 
from their experiments. 

The solution of mathematical formulas leads to powerful pre¬ 
sumptions in the revelation of physical laws, which could never b 
attained or realized from mere observation of facts in experiment 
and practice. All observation and contemplation which involve 
mind, involves theory, which is the foundation of our practic 
and prograss. 

A knowledge of algebra is not necessary for the use of the for¬ 
mulas, and it is satisfactory to know that most engineers who ar 
not versed in mathematics have acquired the very important habit 
of inserting numerical values for the corresponding letters, which 
they prefer to cumbrous written rules, which are impracticable i: 
extensive problems. If all the formulas herein were explained i 
words, the book would exceed in volume Webster’s unabbrevi 
ated Dictionary, and the matter would be only so much the mor 
complicated. The algebraical formulas herein are solved into al 
their functions, ready to receive what is given and refund what i. 
required. They not only tell what is to be done, but at a glanc 
impress the mind with the complete operation. 

JOHN W. NYSTROM. 


1010 Spruce street, Philadelphia. 










Index. 


5 


INDEX. 


PAGE 

Abbreviations of metric term. 60 
Accelerated motion, formulas . . 436 
“ circular motion . . . 447 
Acceleratrix of gravity . . . 435,554 
Acids, binary compounds .... 628 
“ for soldering or tinning . 631 
“ tests for metalsin solution. 637 

Acoustics, music.622 

Acre. 38 

Acres into hectares. 53 

Actual horse-power.547 

Addition in algebra. 19 

Adhesion on rails.157 

Adiabatic curve.554 

Adulteration of metals.461 

Aerostatics and aerodynamics . . 486 

Age, moon’s.666 

Air and heat.512, 519 

blowing machines.590 

composition of.628 

fans, ventilators.592 

for furnaces and cupolas . . 585 

moisture in.488 

properties of. 488, 591 

pumps.540 

resistance of.438 

respiration of.516 

velocity of.486 

warminsr and ventilation . . 516 
“ weight and volume of . .489,519 

Alcohol in liq ids.632 

Algebra.17-19 

Alligation . 33 

Alloys.464 


659 

657 

671 

671 

638 


Almanac, astronomical 

“ for the 19th century 
Alphabets for headings . . . 

“ deaf and dumb . 
Amalgams, gold and silver . 

American wire-gauge.360 

Amp6re, electric intensity . . 554,644 

Amplitude.670 

Amsler’s plauimeter.551 

Analytical geometry.176 

Ancient measures. 58 

Aneroid barometer . . .... 495 

Angles by a two-foot rule .... 44 

Animal strength.396 

Annuity .. 37 

Annular double cylinder .... 549 
“ expansion engine . . . 547 
Anti-friction curve, Shield’s . 75,180 

Apothecary weights. 39 

Apparent time and latitude . , . 664 

Arabic notation. 17 

Area of circles. 94-107 

“ circular figures.146 

“ foot-valves in air-pumps, 540 

“ inland lakes.492 


PAGE 

Area of plane figures. 86 

“ solids . 89 

“ spheres or balls.353 

“ steam-ports .542 

Arithmetical progression . . . 30,31 
“ “ high or¬ 
der . 32 

Arithmetics. 17 

Artificial cold.507 

“ horizon.664 

Artillery, heavy.523 

Asphaltum for street pavement . 630 
Assaying gold and silver .... 634 

Astronomical almanac.659 

“ signs. 18 

Astronomy.656 

Atmosphere .485 

“ columns.585 

“ height of.493 

“ refraction of ... . 663 

Atomic formulas into weights . . 628 

“ weights ..626 

Audibility of sound.621 

Avoirdupois weight . . •. 39 

Axes, number on steamboats . . 587 

Axles and shafts.293 

Azimuth . 554,670 

Balance, differential.352 

Balls and shells, piling of ... . 32 

“ capacity and weight of . . 353 

Bar iron, weight of. 356, 358 

Barometer.487 

Barometrical observations . 493-501 
Barrel, cask, volume of ... . 92, 631 

Battery of steam-boilers.554 

Ban me scale.466 

Beam, to cut the strongest from 

a log.319 

Beams, elements of iron.338 

“ working, Fig. 89. 83 

Bear or burden on animals . . . 396 

Bells, ringing.622 

Belting.398 

Belting friction.399 

Bend of pipes, flow of water in . 474 

Billiard problem.454 

Binary arithmetics.554 

“ compound . 627, 628 

Birmingham wire gauge .... 361 
“ “ gold and silver. 361 

Blast furnaces and engines . . . 593 

“ warm and cold.593 

Blasting with dynamite.629 

Blowing machines . . . , , 590,592 
M off salt water ,,,,,, 580 

Board measure 576 

Bodies in collision , ..454 

Boilers, inspection, U. S .569 

“ steam. 554,560-571 





















































































6 


Index. 


page 

Boiling-point, salt water .... 581 
“ temperature of . . 507 
Boiling water, barometer .... 499 

Bollman’s truss-bridge.333 

Bolts and nuts . 360, 363, 347, 348, 349 

“ copper, weight of.359 

Bowstring bridge.325 

Bramah’s press.467 

Brass tubes, seamless.364 

Breadth of Belts.408 

Breast wheels, water.478 

Bricks.631 

Bridges. 322-333 

“ Bollman’s truss.333 

“ catenary.328 

“ cost of.327 

“ stone.830 

“ suspension.326 

“ truss, many panels . . . 324 

“ truss, queen-rods .... 323 

“ Warren’s.334 

Brown mortar.630 

Buckets, number on steamboats . 587 

Bushel. 38, 40, 631 

Butter and cheese from milk . . 632 
California rule for silver and 

gold.635 

Calorics, units of heat .... 59, 522 

Calorimeter. 561,572 

Calculus, formulae of . . . 61,66,101 

Candle, sperm.484 

Cannons, heavy artillery .... 523 
Capacity and weight of sub¬ 
stances . 371, 460 

Capacity, cask. 92 

“ English measures of • 64 

“ measure of. 39 

“ solids. 90 

“ spheres.353 

Carat, diamond. 43 

Castings, shrinkage of.365 

“ weight of, by patterns . 365 

Cast-iron cylinders and pipes . . 355 

“ girders.318 

“ pillars, strength of . . 313 

“ produce of.593 

Cateuaria. 305, 309, 328 

Cement and concrete mortar . . 630 

“ for cast-iron.630 

Centimetres and inches. 48 

Centre of gravity.456 

gyration. 446, 449 

“ percussion.456 

“ pressure, hydraulic . . 467 

Centrifugal force.450 

“ propeller.618 

Chain, surveying . .45, 83 

Chain-line catenaria.328 

Chains for railroads.321 

“ rope and wire.315 

“ strength of.315 

Chapman’s rule for areas .... 146 

Charcoal from wood.465 

Charge in blast furnaces .... 593 


PAGE 

Charge of powder.523 

Charts of long measures . . 46, 47 

Chemical formulas. 627, 628 

Chemistry.626 

Cheval, puissance de.58, 60 

Chimneys, height of ,IP of . 561,574 

Chinese gunpowder.523 

Chlorination of gold.638 

Chlorine gas, to make.638 

Chronology. 378, 658, 659 

Circle. 67, 68, 78 

Circular motion.394 

“ saw.396 

Circumference of circles . . . 94-107 

“ ellipse .... 84 

“ ropes . . 423, 429 

Climate, mean temperature . 490, 502 

Clock, sidereal. 658,662 

Coal, consumption of ... 537, 575 

“ properties of.576 

Coefficients of vessels.606 

Cog-wheels.307 

Cohesive strength.814 

Coins, foreign. 42 

Cold, artificial.507 

Collision of bodies.454 

Columb, electric quantity . . . . 554 
Colors, selection of water .... 871 

“ spectrum.484 

“ tempering steel.404 

Columns, Phoenix.344 

“ water and mercury. 476, 485 

Combination. 29 

Combustion, properties of. . . . 577 
Command the engineer ?. . . . 581 

Compass, points of the,.168 

Composition air and water . . . 628 
“ nails and rivets . . 365 

Compound engines.552 

“ interest. 36 

f ‘ pendulum .452 

Compounds, binary, chemical . . 627 
Compression and expansion of 

air.. . 517-519 

Compression, strength of ... . 312 
Concave and convex mirrors. 639, 640 
Concrete, cements, mortar . . . 680 
Condenser, surface, fresh-water . 543 
Condensing water, quantity of.540, 541 
Conducting power for heat . . . 511 
Conductivity of electricity .... 647 

Cone pulleys.404 

“ surface of. 89 

“ volume of. 90 

Conic sections.177-181 

Construction, geometrical . . . 68-92 

“ of ships,.606 

“ of teet'n for wheels. 367 

Consumption of fuel .... 637,561 

“ gas .484 

“ water in cities . 474 

Contracted vein.471 

Conversion of acres and hectares. 53 
“ cub. feet aud dm 3 . o-i 
























































































Index. 


7 


PAGE 


Conversion of cub. ins. and cm 3 . 

50 

“ 

cub. yards and m 3 . 

51 

U 

Eng. and Fr. mea- 


{ 

sures.46-59 

« 

Eng. and Fr. tons . 

56 

i< 

feet and metres . . 

48 

u 

foot-pounds and 



kilogram meters. 

58 

u 

foot-tons and ton- 



metres. 

59 

u 

gallons and litres . 

51 

a 

grain Troy and 



grammes .... 

57 

« 

heat and calories . 

59 

u 

horse-power and 



force de cheval. 

58 

“ 

inches and centi- 


| 

metres. 

48 

44 

oz. and grammes . 

57 

a 

pounds per sq. ft. 



and kg. per ra 2 . 

54 

u 

pounds per in 2 and 



atmos. pressure. 

55 

u 

pounds per in 2 and 



cm 2 . 

55 

u 

pounds and kilo- 



grammes .... 

56 


44 

sea miles and kilo- 



metres. 

49 

<4 

sidereal and solar 



time. 

663 

44 

statute miles and 



kilometres . . . 

49 

44 

sq. inches and cm 2 . 

50 

44 

sq. yards and m 2 . 

52 

44 

sq. miles and km 2 . 

53 

44 

yards and metres . 

52 


Copper bolts, weight of.359 

“ ropes.429 

“ strength of, at high heat. 3G4 

Cord of wood .... *.576 

Correction for It. A. and decli¬ 
nation .G62 

Cosine, logarithmic.199 

“ natural .245 

Cotton ropes.425 

Counting seconds.620 

Couplings, price of.364 

Cranes, hoisting.302 

Crank and pin.294 

Cream and cheese.632 

Creation of words . ..658 

Crushing strength.312,630 

Cube and cube roots.112 

Cubic feet into dm 3 . 54 

“ inches into cm 3 . 50 

“ incites, water, iron, lead . 553 

“ yards into m 3 . 51 

Cupola.585 

Currency of different nations . . 43 

Curvature of the earth.1G4 

Curves, inclination of tracks . . 151 

“ railroad.147-153 

Cut-off steam, expansion .... 594 


PAGE 

Cut-off valve.558 

Cut or embankment on slopes . . 153 

Cycle of the sun.658 

Cycloid. 77, 178, 439 

Cyma, to construct. 72 


Dates, civil and astronomical . G58 
Day and night, length of ... . 670 
Deaf and dumb alphabet .... 671 
Decimals and vulgar fractions . . 44 

of an hour, degree . . . 244 


Decimeter. 61 

Declination, the sun’s.660 


Decomposition of light, spectrum. 484 
Definition of mechanical terms.386,554 
Degree, length of, in parallel . . 161 
“ of the earth's circle ... 38 


Departure. 159 , 163 

Dew-point.488 

Diagrams, indicator. 550-553 

Diameter of shafts.319, 420 

“ of the earth.648 

Diamond. 43 

Difference, mean. 22 

“ of longitude.651 

Differential balance.352 

Dimensions of the earth .... 648 

Dip of horizon.163 

Dipper dredge. 897 

Discount or rebate.. 28 

Displacement of vessels.598 

“ scale.607 

Distances by sea.652 

inaccessible, to find . . 294 

in Europe.653 

(land) in U. S.655 

of objects at sea . . . 163 

(sea) in U. S.654 

spherical.172 

to sun and moon . . . 656 


which sound travels . 621 


Distillation temperature .... 507 
Divergency of tlie parallel . . . 165 

Division in algebra. 20 

Doctors on food.632 

Dodecahedron. 85 

Donkey-pump, price of.557 

Double cylinder engines .... 549 

“ fellowship. 29 

“ riveted joints.568 

Drain, motion of water in ,. . . 469 

Dredging machines.397 

Driving a nail into a piece of 
wood.582 

Dynamic work of food.632 

Dynamics. 297, 386 

“ electro.644 

of matter. 442, 445 

Dynamite blasting ....... 629 

Dynamometer, Prony’s friction . 546 
Dyne, electro-motive force . . . 554 

Earth, dimensions of.648 

Eccentrics.558 


Eclipse of Jupiter’s satellites . . 665 















































































8 


Index. 


page 

Economy of expansion of steam . 594 

Effect, dynamic.387 

“ of evaporation of water . 562 
“ of steam-engines .... 549 
“ of waterfall, natural . . . 480 


Elasticity of materials . 317, 320, 335 

Elbow, to cut out.583 

Elect ricity, posit ive and negative. 644 

Electro-dynamics. 554, 644 

Elementary substances.626 

Elements . ..76,179 


ami functions .... 386 
“ of Jupiter’s satellites . 666 

“ of mechanics.297 

Elevation of external rail . . . . 151 
Ellipse, circumference of .... 84 

“ construction of. 74 

“ formulas lor.179 

“ isometric. 74 

Ellipsoid. 88 

Elliptic mirror.640 

“ railway curves.151 

“ stern of vessels.606 

Embankment and excavation . . 154 

Energy.554 

Engineer’s command.552 

Engines, steam, of different kinds 549 
English terms abbreviated ... 60 

Epact of the year and month . , 656 

Epoch.387 

Equilateral hyperbola.181 

Equivalent, chemical.626 

Equation of time.660 

Erg. 554 


Estimate price of engines . . . . 557 

Evaporation in open air, seas . . 492 

“ natural effect of . . 562 

“ reduced to 212° . . 575 

Events before and after Christ . 659 


Evolute of a circle. 77 

Evolution .143 


Excavation and embankment. . 154 


Expansion and compression of 

air.517-519 

“ linear, of cast-iron . 510 

“ of bodies by heat . 508 

“ of steam, economy 

of.594 

“ of water .... 508,524 

Explosions, dynamite, nitro¬ 
glycerine . . . 629 
“ gunpowder .... 523 

“ steam-boilers . . . 5S6 


Eyes, long- and near-sighted . . 639 

Knees of the moon.668 

Falling bodies, table of.440 

“ machine.439 

Falls, water, height of.501 

Fan, ventilator.592 

Farad, electric capacity.555 

Fathom.38, 46 

Feed-pumps for boilers.540 


Feet and meters, conversion of . 48 

“ per sec. = miles per hour . . 486 


PAGE 

Fellowship.28, 29 

Felt covering for steam-pipes . . 578 
Fertilizing value of manures . . 633 
Fifth and fourth powers .... 145 
Fineness of silver and gold . . . 636 
Fire, assay of silver and gold . . 634 

“ engines.469 

“ grate, horse-power of . . . 560 
“ precaution against .... 5S7 

Fixed stars.666 

Flagging.349 

Flags of all nations—front plate. 

Floors with iron joists.340 

Flour-mills.396 

Flow of water in bends.474 

“ “ pipes.470 

“ “ rivers.472 

Flues for steam-boilers . . . 561, 579 

Fly-wheels. 446, 447, 449 

Focus of optical lenses.642 

Food, what the doctors say about 632 
Foot-pounds and foot-tons . . 389,391 
“ pounds and kilogrammeters 58 
“ tons and metre-tons .... 59 

“ valves in air-pumps .... 540 

Force by a screw-jack.311 

“ de eheval.442 

“ definition of.386 

“ of inertia . .-. 60 

“ of temperature.509 

“ of wind.486 

“ pump.540 

Foreign coins, money. 42 

“ measures and weights. 62-65 
Forging by steam-hammer . . . 333 

Forms.89,90 

Formulas for electro-dynamics . 645 
Fourth and fifth powers .... 145 
Fractions, vulgar and decimal . 44 

Freezing mixture.507 

French metric system. 45 

Fresh-water condenser . . . 540,543 

Friction axles.433 

“ belting.399 

“ curve, anti. Shield’s . . 180 

“ in machinery.434 

Frogs, railway ......... 167 

Fuel, consumption of.537 

“ properties of different . . . 576 

Fulcrum. 298-303 

Funicular machines .309 

Furnaces, blast.593 

Fusion, temperature of.507 

Gallon and litre. 51 

“ English Imperial ... 64 

“ the U. S. standard . . 38 

Galvanic current, galvanometer 555 

Gas, motion of, in pipes.484 

Ganges, American wire.360 

“ Birmingham.361 

“ new English standard . 646 

“ railway.158 

“ sheet, nail, rivets . . . 361 
Gay-Lussac’s scale.466 








































































Index. 


9 


PAGE 

Gearing, construction of teeth . 807 

Geography.G48 

Geometrical constructions . . . 68-92 
“ progression .... 84 

“ scale in music . . . 628 

Geometry, analytical.176 

“ plane. 67 

Giant powder.629 

Gitfard’s injector.689 

Girder, box.848 

“ cast-iron.318 

“ compound plate.843 

“ Warren’s.384 

Glass, window.434 

Glossary of bridges.330 

Glues.031 

Gold and silver, value of . . . 635, 636 

“ amalgams.638 

“ imitation metal.464 

Golden number.658 

Government inspector’s tables . 569 

Governors.451 

Grain troy and gramme .... 57 

Grapple-dredge.397 

Gravitation .435 

Gravity, centre of. 446, 449 

“ specific.460 

Guitar, to divide the bridge . . . 622 
Gunpowder, force and work of . 528 

Guns, heavy artillery.523 

Gyration, centre and radius of, 446,449 

Half-trunk engine.5-49 

Hammers, steam.333 

Hardness of substances.465 

Harmonic intonation, music . . 623 
Hay and other stock food .... 632 

Heat and calories. 59, 503 

“ in air, gases.512-519 

“ latent.506 

“ lost by radiation.578 

“ specific.513,521 

“ u nits of.522 

Heating and ventilation .... 516 

“ houses by steam .... 516 

“ surface in boilers . .537,561 
Height, measure by barometer . 493 

“ of chimneys.561 

“ of cities.502 

“ of columns of air, water, 

mercury.476 

“ of mountains and volca¬ 
noes .495 

“ of natural and artificial 

works ..502 

“ of snow-line.485 

“ of the atmosphere . . . 493 

“ of waterfalls.501 

Helix of screws.83,84 

Hemp ropes, tarred.425 

“ “ white.424 

Hexahedron. 85 

High water, time of.668 

Hodgkinsou’s pillar, strength . . 313 

Horizon, artificial.664 


yAGE 

Horizontal range in gunnery . . 439 
“ tubular boilers . . . 573 

Horse-mills.396 

Horse-power and puissance de 

ebeval. 58 

“ actual, ofengines.517,547 

“ chimneys.574 

“ indicated. . . .543,550 

M leather belts. . . . 407 

** locomotives .... 157 

“ of small engines . . 556 

“ per motive force . . 418 

“ ropes ....... 407 

“ steam-boilers . .562,573 

“ steam-engines . . . 544 

“ steamship perform¬ 
ance . 610-613 

“ wrought-iron shafts 420 

Humidity of air .. 488 : 

“ of steam.572 

Hydraulics.468 

Hydraulic mortar and concrete . 640 

“ press.. 467 

“ pumping water . . . 473 

“ radius in rivers . . . 475 

“ ram.475 

Hydrodynamics.. . 476 

Hydrometer.466 

Hydrostatic paradox.467 

Hydrostatics.466 

Hygrometry. 488 

Hyperbola.75,181 

Hyperbolic logarithms . . . . 290 

“ mirrors.640 


Ice, expansion of.508 

Icosahedron. 85 

Impact of bodies.454 

Imponderable matter. 297 

Inches and decimals of a foot. . 44 

“ “ centimetres. 48 

“ eighths into decimals 44 
Inclination of tracks in curves . 151 
Inclined plane . . . 804,310,439,445 

Income on investment. 27 

Incrustation in boilers . . . 543,580 

Index of refraction.641 

Indicator diagrams. 550-553 

Inertia, definition of.442 

“ of reciprocating parts . 539 
Injection, water, velocity of. . . 541 

Injector.589 

Inspection of boilers, U. S. . . 569, 587 

Interest, compound. 36 

law of the States ... 26 

“ simple.23-25 

Interpolation.166 

Intonation, musical.623 

Introduction. 17 

Investment, income on. 27 

Iron and cast-steel ropes .... 426 
“ acid tests for quality . . . . 673 

“ beams.335 

“ blast furnaces.593 



















































































10 


Index. 


page 

Iron bridges. 333 

“ columns.344 

“ flat bar, weight of.356 

“ floors.340 

“ girders, box.343 

“ pyrites, sulphurets .... 638 

“ railroad.348 

“ I'olled, round, and square, 

weight of.354 

“ roofs.332 

“ strength at high temperature 364 

Irrigation, volume of water for . 491 

Isometric perspective. 75 

Isothermal line.554 

Joints, double riveted.568 

“ single riveted.567 

Joists for flooring.337 

Joule’s equivalent of heat . . . 522 

Jouvat’s turbine.480 

Julian period.658 

Jupiter’s satellites. 665,668 

Kilometres and miles .... 49 

Kinetic energy.554 

Kinetics.555 

Knot, sea-mile. 38 

“ tying. 431 

Ladd <:r-(i red ge.397 

Lakes, areas of inland.492 

Land surveying.160 

Lap and lead on slide-valves . . 558 

Latent heat.506 

Lateral strength.316, 317 

Latitude aud apparent time . . 664 
“ “ longitude of places 650 

Law of gravity.435 

“ interest in all the States 26 

Leap year.657 

Leather belts.406 

Legal horse-power of boilers . . 563 
Length, measure of . . 38,44, 45,47, 61 
“ of night and day . . . . 670 
“ of one degree in parallel 161 

“ of vessels.608 

Lenses, optical.642 

Letters for headings.671 

Level, apparent and true .... 164 
Levers, mechanical . . . 299, 304, 306 

Light and colors.484 

“ velocity of.633 

Lime mortar.630 

Linear expansion of cast iron . . 510 

Liquid, measure of . .. 39 

Llama, Peru.396 

Lloyd’s rule for boilers, British . 570 

Load od roofs.331 

Locomotive indicator cards . . . 552 

“ traction of.157 

Logarithms, common .... 182-284 

“ hyperbolic.290 

“ trigonometrical . . 199 

Log-line, length of ...... . 38 

London bridge, high-water . . . 668 
Longitude, difference in time . . 651 
“ to find.665 


PAGE 

Lunar cycle.658 

Magnifying opera-glasses . . 643 
power of lenses . 642 
“ telescopes .... 643 

Manilla ropes.424 

Man-power.387 

Mantissa of logarithms.182 

Manual labor.396 

Manures, fertilizing, value of . . 663 

Mariner’s compass.162 

“ date.658 

Mass, definition of.442 

Mathematics. 17 

Mean pressure indicator cards . 551 

“ “ steam.534 

“ proportion aud difference . 22 

“ time.658 

Measures, ancient. 58 

“ by triaugulation . . 294-296 

“ foreign.59, 63 

“ of length . 61 

“ on sloping ground . . 161 

“ U. S.38, 39 

Mechanics, elements of.297 

Meniscus, optical.642 

Meridian passage.667 

“ to find.667 

Meta-centrum of vessels .... 614 
Metals, comparative value of . . 620 

“ hardness of.451 

Metre and feet. 48 

“ and yards. 52 

Metric system.45-60 

“ terms abbreviated .... 60 

Metrology.65, 66 

Micro-farad.555 

Microscope, magnifying power . 639 

Miles and kilometres. 49 

“ per hour = feet per second 489 
“ statute and nautical ... 38 

Milk, cream and cheese.632 

Mills, flour, saw, rolling .... 396 

Minerals, hardness of.465 

Mirrors, concave and convex .639, 640 
Miscellaneous temperatures . . . 511 

Moment, static. 298, 306 

“ of stability.299 

“ “ meta-centre . 614 

“ of inertia.555 

Momentum, dynamic.442 

** in bodies.454 

Monuments, height of.502 

Moon, elements of.656 

Moon’s age. 656, 668 

“ faces.* ‘ 668 


Moons, number of, to each planet 668 
Morris, Tasker & Co.’s iron tubes. 579 
Mortar, cement, concrete .... 630 

“ piece of ordnance . . . 656 

Motion defined.386 

“ gas in pipes.484 

“ of bodies in collision . . 454 

“ water in pipes.468 

“ water in livers.472 



















































































Index, 


11 


PAGE 

Motive force.386 

Motive per horse-power . . 414-417 
Mountains and volcanoes, height 

of.495 

Multiplication in algebra .... 19 

Musical vibration.622 

Nail, driving in a.582 

Nails and spikes.847 

“ penny.365 

Natural effect of steam.562 

“ effect of waterfalls . . . 480 
“ sines, cosines, tangents . 245 
“ slope of substances . . . 301 

Navigation traverse.159 

New English wire-gauge .... 646 

Night and day, length of .... 671 

Nitro-glycerine.629 

Nominal horse-power of engines. 544 

North by the Polaris.667 

Notation.. . 17 

Nutritive elements in food . . . 633 
Nuts and bolts, weight of. . 347. 349 
“ square and hexagon .... 348 
“ hexagon.347 

Observed results of power . . 396 

Obstruction in livers.473 

Octahedron. 85 

Ohm’s electric resistance . . 555, 644 

Opera-glasses.643 

Optics, mirrors and lenses . . . 639 
Order of conducting power, elec¬ 
tricity .646 

Ordinates for railway curves . . 147 

Ordnance performance.523 

Origin.555 

Oscillation, angle of.452 

“ centre of.452 

“ of pendulum .... 452 

Ounces and grammes. 57 

Overshot water-wheel.477 

“ wheels.479 

Paper, drawing and tracing . . 434 

tt, value of. 78 

Parabolas, to construct ... 75, 180 

Parabolic mirror.639 

“ veiu of water.471 

Paradox, hydrostatic.467 

Parallax, sun’s, in altitude . . . 663 

Parallel, divergency of.165 

Parameter.555 

Pattern-makers’ rule.365 

Pendulum.452 

Penny nails.365 

Perch of stone.631 

Percussion, centre of.456 

Performance of steamships . . . 610 

Periphery of circles.94-107 

“ ellipses. 84 

Permutation.29, 144 

Perspective, isometric. 75 

Peruvian measures of ore .... G34 


u 

tt 

tt 


(C 

if 


PAGE 

Phoenix columns and beams. 335-343 

Pile-driving.422 

Piling of balls and shells .... 32 

Pipes and Hues.579 

“ cast-iron, weight of ... . 355 

motion of gas in.484 

motion of water in ... . 468 
of different metals .... 364 
radiation of heat from . . 578 

“ steam, size of.542 

Pitch of propellers. 619 

of screw helix.83, 84 

of screw thread . . . 362, 363 

of spiral. 77, 83, 84 

“ of standard in wheels . . . 372 
“ of teeth in gearing .... 307 
Plane, inclined . . . 304,310,339,445 

“ sailing traverse.159 

Planetary system, elements of . . 666 

Plate girders.343 

Platinum sheet, weight of. . . . 635 

Plotting out curves. 75 

Plumbing.364 

Points of the compass.162 

Polygons .. 93 

Polyhedrons. 85 

Poueelet’s water-wheel.478 

Population of the world . . 648, 649 

Portland cement.630 

Ports, high water in. 666 

“ steam. 542 

Potential. 555 

Pound avoirdupois.38, 39 

Pounds and kilogrammes .... 56 

per /: 2 and kg. per m' 2 . . 54 

per in- and kg. per cm- . 55 

Powder, charge of, composition . 523 

Power, actual horse. 547 

“ definition of. 386 

for blowing machines . . 590 
for different mills .... 396 
for fans, ventilators . . . 592 
for magnifying lenses . 642 
for pumping water . . . 473 
for punching iron-plates. 585 

for quartz-mill.638 

for steamboats.CIO 

in moving bodies .... 443 

natural effect, waterfall . 480 

nominal. 543, 552 

of locomotives.157 

of man and beasts . . 378, 396 

of steam-boilers.562 

of steam-engines . . 549, 550 

of turbines.480 

of water-wheels.477 

to work machines .... 396 

Powers and roots. 21 

fourth and fifth . . . 144,145 
“ of first nine numbers . . 144 

Precaution against fire on steam¬ 
ers .587 

Press, hydraulic.467 

Pressure, columns water, mer¬ 
cury .. 476, 485 


it 

tt 


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tt 

tt 

a 

it 

it 

tt 

tt 

tt 

tt 

tt 

tt 

tt 

tt 

tt 

tt 

it 

it 









































































12 


Index. 


page 

Price and weight of engines . . 557 

“ of boiler tubes.579 

“ of copper and brass tubes. 364 
“ of couplings for plumbing. 364 

“ of donkey pumps.557 

“ of gold and silver .... 636 
“ of hemp and wire ropes . . 315 
“ of railroad iron ...... 321 

“ of rolled iron.321 

“ taps, dies, and stocks . . . 364 

“ of turbines.482 

“ of wrought-iron girders . 321 
Prime, vertical and parallel . . . 165 

Prism, refracting.641 

Progression, arithmetical . . . 30,31 
geometrical .... 34 

“ high order. 32 

Projectiles for guns.523 

Prony’s friction brake.546 

Propeller, centripetal.618 

Proportion, simple mean .... 22 

Puissance de cheval.58,60 

Pulleys.308 

“ cone.404 

Pumps, air, capacity of.540 

“ donkey, price of.557 

“ force, “ .540 

“ water-works.473 

Punching iron plates.585 

Puzzolano.630 

Pyrites, sulphurets, iron .... 638 
Quantity, definition of. ... 17 

“ different kinds . . 555 

Quartz-mills.638 

Ksidiat ion of heat, steam-pipes 596 

Kadi us of the earth.648 

Kails, spring of.151 

“ weight of.321 

Railway curves.148 

“ elevation of outer . . . 151 

“ gauges.158 

Rainfall. 158, 491 

Ram, hydraulic.475 

“ in pile-driving.422 

Range, horizontal, gunnery . . . 439 

Rebate or discount. 28 

Reciprocal of numbers . . . 112-142 
Recording formulas for ships . . 596 
Rectangular beam from a log . . 319 
Reduction for soundings .... 669 
“ of French and Eng¬ 
lish measures ... 46 

“ of inches to feet... 44 

Refining silver ..673 

Refraction of the atmosphere . . 663 

Refractive indexes.641 

Regulating time by stars .... 667 
Resistance of air to projectiles . 438 
“ of copper wire ... 646 

“ of electricity .... 644 

“ of water to a plan . . 473 

“ of wind.486 

Respiration.516 

Resultant forces.298 


page 

Retaining walls.300 

Retarded motion.437 

Right ascension, sun’s ..... 6ti0 

Kinging bells.622 

Rising and setting of stars . . . fi >7 

Rivers, obstruction in.472 

“ length.491 

“ velocity of water in . . . 472 

Riveted lap-joints. 567,568 

Rivets, iron and copper.365 

“ weight per 100 350 

Roads bad in Peru.396 

“ traction on.156 

Roasting sulphurets.638 

Roebling’s wire ropes .... 315-423 
Rolled iron, irregular forms . . . 321 

Rolling-mill.396 

Roman cement.630 

“ notation. 18 

Roof, wood and iron .... 331,332 

Roofing slate.349 

Roots.21, 112-143 

“ fourth and fifth.145 

Ropes,strength of . . . . 315,423-429 
Rotary cr circular motion . . . 394 
Rule for pattern-makers .... 365 

“ measure. 61 

Russian sheet iron.365 

Safety-valves.542 

Sailing distances. 652, 654 

Salt water in boilers.580 

Sasb-bars, iron, for windows . . 321 

Satellites of Jupiter.GG5 

Saturation in boilers .... 543,580 
Saw-mills, circular, reciprocating 396 

Saw-mill water-wheel.497 

Scales of music. 622-625 

Scientific and technical terms . . 554 

Screw, force by.311 

“ lielix.83, 84 

“ jack, force by.311 

“ propeller.618 

“ thread.362 

Seamless brass tubes.364 

Seasons.490 

Secant, natural .245 

Segments of circles.108-142 

Screw thread.363 

Setting and rising of stars . . . 667 
Shafts, diameter and revolutions 420 
“ sheering iron plates . . . 585 

“ strength of.319,370 

Sheet iron and copper ... . 459,359 
“ platinum, goid, silver . . . 635 
“ zinc and Russian iron . . 365 

Shells, piling of. 32 

Shield’s anti-friction curve . .75,180 

Shipbuilding,. 606-614 

Shop cranes.302 

Shrinkage of castings.365 

Sidereal time, clock, year . . 658,662 
Siding of railway tracks . . 153,167 

Signs, astronomical. 18 

Silver amalgam.638 




















































































Index. 



PAGE 


PAGE 

Silver and gold, to refine . . 

# . 

637 

Steam, expansion of. 


“ “ value of . . 

635 

636 

“ hammers. 


Silvering metals. 


631 

“ indicator-cards . . . 

. . 553 

Simple fellowship. 


28 

“ loss by radiation . . . 


“ interest . 


“ ports, area of. 


“ pendulum. 


452 

“ properties of. 

524-535 

“ substances . . . . . 


626 

“ ship performance . . 

597-611 

Simpson’s rule fur areas . . . 


146 

“ superheated. 


Sines and cosine. 


245 

“ tables. 

. . 530 

Size of engines. 


556 

Steel, tempering of. 

. . 464 

Slates for roofing ...... 


349 

Stock food. 


“ sizes of. 


350 

Stocks and dies, price of . . 

• • 3G4 

Slide valves. 


558 

Stone bridges. 


Slip of propellers. 


618 

“ perch of. 


Slope of embankments . . . 


154 

Strain on roofs. 


“ natural. 


301 

Strength of animals .... 

. . 396 

Sloping ground, measurement on 

161 

“ of belts. 


Smelting points, temperature of . 

507 

“ of boilers. 

560-571 

Snow-line, height of .... 


485 

“ of chains. 


Solar and sidereal time . . , 

662 

663 

of girders. 


Solders .. 


464 

“ of iron and copper 

. . 364 

“ for bracing. 


465 

“ of lateral. 


Soldering, acids for. 


631 

“ of materials . . . 

. . 312 

Soliditv of revolution .... 


146 

“ of Portland cement 

. . 630 

Solids, capacity of. 


90 

“ of ropes, hemp and 

wire 

Soundings. 


669 


315-423 

Sound, velocity of. 


621 

“ of teeth in wheels . 

. . 369 

South American ore measures 

• • 

634 

11 of timber. 


Space, linear, in mechanics . 

• • 

388 

“ of tubes forcollapse 

. . 565 

Specific gravitv. 

460 

-463 

“ of woods, S. American 320 

“ heat of gases .... 

513 

521 

String, musical.. . 


“ of substances .... 


520 

Structures, forces in .... 


Spectrum, decomposition ofli 

gbt 

484 

Stuttgart harmonic scale . . 

. . 623 

Speed in steamship performance 

610 

Subtraction in algebra . . . 

. . 19 

Spheres, capacity of. 



Sugar in food. 


Spherical distances. 


172 

Suiphurets, iron pyrites . . 

. . 638 

“ trigonometry . . . 


172 

Sun, cycle of . .. 


Spikes and nails. 

347 

348 

“ distance to. 


Spindle, circular. 


92 

“ parallax of. 


Spirals.. 

77, 83, 84 

“ properties of. 


Splices and bolts. 


348 

“ K. A. and declination . 

. . 660 

Spring of rails. 


151 

“ set and rise of. 


Square a circle, to. 


84 

Surface condensers. 


“ and hexagon nuts . . 

• • 

347 

“ heating, in boilers . . 

561,573 

“ inches and cm 2 . . . 


50 

“ of boiler-tubes . . . 

. . 579 

“ measures. 


38 

“ of revolution .... 


“ miles and knfi .... 


53 

“ of solids. 

. . 89 

“ yards and m3 ... . 


52 

“ plane figures .... 

. 86-88 

Squares and cubes. 

112- 

-142 

Surveying chain .. 

. . 38 

Stability, moment of . . . . 

299, 303 

Survev traverse table .... 

. . 161 

“ of floating vessels 

• • 

614 

Suspension bridges. 


Standard horse-power of boilers . 

564 

Tacks . 


“ pitch of gear-wheels 

, , 

372 

Talon, to construct a . . . . 

. . 72 

“ weights and measures . 

38 

Tangential angle for chord of 100 

Starch in vegetables .... 

• • 

633 

feet . 

. . 152 

Stars, It. A. and declination . 

• • 

666 

Tangents, logarithmic . . . 

. . 199 

Stars, setting and rising of . 

• • 

667 

“ natural. 


State law of interest .... 


26 

Taps and dies. 


Statics and stability. 

• • 

298 

Tarred hemp ropes. 

. . 425 

Staving of boilers. 


565 

Technical and scientific terms . 554 

Steam-boiler explosions . . . 



Teeth for wheels. 

307-385 

“ boilers. 


560 

Telescopes, astronomical . . 

641, 643 

“ condenser . 


543 

Temperature, boiling and smelt- 

“ engines . 


549 

ing .... 


































































































14 


Index. 


page 

Temperature, boiling water bar., 499 
“ color of steel . . . 464 

“ correction for. 498, 575 

“ force of.509 

“ fusion alloys . 464, 507 

“ mean, climate. 490, 502 

“ miscellaneous . . 511 

“ of distillation of 

oils.507 

“ • of the air. . . 490, 502 

“ on the ocean . . . 502 

“ table,con version of. 504 

Tempered intonation, music . . 628 

Tempering steel, colors.464 

Terms, scientific and technical . 554 
Tests for metals in solution . . . 637 
Tetrahedron, elements of .... 85 

Thermo-dynamics.522 

Thermometers,different kinds.503, 505 
Thickness of boiler-iron . . 569, 584 


Threads, screw, number per inch. 362 

Threshing-machine.396 

Timber, dimensions of, for roofs. 331 
“ green and seasoned . . 576 

“ strength of.325 

Time, apparent.665 

“ chronology.658 

“ counting seconds.620 

“ definition of.378 

“ equation of.660 

“ for cream to rise.632 

“ of high water.668 


“ passing the meridian, stars. 667 
“ sidereal and solar .... 658 
“ sunset and sunrise .... 670 
“ to regulate a watch .... 667 

Tin amalgam.638 

Tinder, temperature of ignition. 518 

Tinning, acids for.631 

Tin-plates, weight of, and marks. 434 
Ton, English and French .... 56 

Tonnage of vessels, English . . . 617 
“ U.S. measure. 615 


Towers, stability of.303 

Tracing paper ..434 

Traction ou roads.156,158 

Travelling distances .... 653-655 
Traverse sailing and survey¬ 
ing .159, 161 

Triangular formulas, plane . . . 170 
“ spherical . 174 

Triangulatiou for distances . 294-296 

Trigonometry, plane.168 

“ spherical .... 172 

Tropical year.658 

Troy weight. 39 

Truss bridges. 323, 324, 333 

Tubes, copper and brass .... 364 

“ for boilers, iron.579 

“ for plumbing, brass . . . 364 

Turbines, Jouval’s.480 

Turn-outs and sidings P. K,. P. 153, 167 

Tuyeres for furnaces.590 

Twisting of shafts.319 


PAGE 

Tying knots.431 

Type metal.464 

“ proportion of.351 

Unrtersliot wheel.478 

Units of electric force, volt . 555, 644 
“ “ intensity, am¬ 
pere. . . 554, 644 

“ of heat. 59 

“ of heat, different kinds . . 522 

“ of length.38,46 

“ of metric. 45 

“ of power ..387 

“ of work.389 

“ of workmandays.389 

U. S. boiler inspection.569 

“ tonnage law.615 


“ weights and measures ... 38 

Value ol silver and gold . . 635,636 
“ of steam-engines .... 557 

“ of various metals .... 620 


Valves, air-pump.540 

“ blast-engines.590 

“ safety.542 

“ slide.558 


Vegetable acids and salts . . 627, 628 
Vegetable growth per acre . . . 633 
Vein of water, contracted . . . 471 

Velocity, definition of.387 

“ of belts and pulleys.410-413 


U 

“ of falling bodies . . . 440 

“ of feet per second of 

revolution.410 

“ of feet per second = 

miles per hour . . . 4S6 

“ of light.6:33 

“ of projectiles.523 

“ of sound.621 

“ of steamships.610 

“ of the planets.666 

“ of water in circulating 

pump.543 

“ of water in pipes . . . 468 

“ “ rivers . . 472 

“ of wind.486 

Ventilation and warming .... 516 

Ventilator, fan for.592 

Vessels, construction of.606 

“ tonnage of.615 

Vibration, musical. 622, 625 

Vis-viva . 443, 555 

Volcanoes, height of.495 

Volt, I'lectric force. 555, 644 

Volume, air.517,519 

“ measure . 39 

“ steam.530 

“ water.526 

Vulgar fractions to decimals . . 44 

Wages per year, month, week, 

day. 41 

Working beam, Fig. 89. 83 

Walls, retaining.300 

Warming and ventilating . . . 516 




































































Index 


Id 


PAGE 


Warren’s girder.384 

Water, blasting under.629 

“ boiling point, barometer . 499 

“ colors.371 

“ Dead Sea.463 

“ ^evaporation of steam . . 524 

“ evaporation on lake . . . 492 

“ expansion of .... 508,524 

“ falls, effect of.480 

“ falls, heights.501 

“ legs for boilers.540 

“ fresh, condenser.453 

“ in food.663 

“ injector.598 

“ in the ocean.463 

“ motion in pipes.468 

“ motion in rivers.472 

“ properties of.628 

“ quantity of condensing . 540 

“ salt incrustation .... 580 

“ waves.597 

“ wheels.477 

“ works.473 

Watt, electric power.555 

Wave line. 583,597 

Weather, prediction of.487 

Wedge., . . 305, 311 

Weight and bulk of substances 460, 553 
“ and capacity, coefficients 

371, 460 

“ copper bolts.359 

“ cubic inch of substance. 462 


“ fiat bar iron. 356-358 

“ heavy ordnance .... 523 

“ of castings by patterns . 365 

“ of sphere and balls . . . 353 

“ of square and round iron 354 

“ of steam hammers . . . 333 


PAGE 

Weight per square foot of metals 

359, 365 


“ pipes and cylinders . . • 355 

“ ringing bells.622 

“ ropes and chains. 315, 423-429 

“ steam-boilers.584 

“ steam-engines.584 

“ tubes, copper and brass . 364 

- Weights and measures. 39 

Weir measurements of water . . 483 

Wharf cranes. 302 

Whitworth’s screw thread . . . 362 
Wind aerodynamics,force of . . 486 

“ velocity of.486 

Window glass.434 

“ sashes, iron.321 

Wire-gauge, American.360 

“ Birmingham . . . 361 

“ for silver and gold . 361 
“ New English stand¬ 
ard .646 

Wire ropes, steel, iron, copper 

315, 423-429 

Wood, charcoal from.465 

“ for combustion.576 

“ strength of S. Amer. . . . 320 
Work, ft.-pounds and kilogram- 

meter . 58 

“ ft.-tonsand tonnes-metres 59 
“ of transformation .... 555 

“ workmandays .389 

Yard and meter. 52 

“ feet and inches. 38 

Years, different kinds.658 


Yield of vegetables per acre . . 633 

Zinc, sheet.365 

Zone of a circle.89, 90 


LIST OF PLATES. 


Plate I.—Construction of teeth in gear-wheels 

Plate II.—Cone-pulleys and belting. 

Plate III.—Jouvat’s turbine. 

Plate IV.—Slide-valve motion. 

Plate V.—Eccentric motion. 

Plate VI.—Stub-ends and flanges. 

Plate X.—On the construction of ships . . . . 
Plate XI.—Centripetal propeller. 


.Facing page 368 

. “ “ 406 

. “ “482 

Bet ween pages 558 and 559 

“ “ 574 and 575 

.606 

.Facing page 618 




















































































. 










• - • 












































* 









l ■ 









































Mathematics. 


17 


INTRODUCTION. 

QuanfI ity is that which can be increased or diminished by addition or sub¬ 
traction of homogeneous parts. Quantities are of two essential kinds, Geometrical 
and Physical. 

1st, Geometrical quantities are those which occupy space; as lines, surfaces , 
solids, liquids, gases, &c. 

2nd, Physical quantities are those which exist, but occupy no space, they are 
known by their character and action upon geometrical quantities; as attraction 
light, heat, electricity and magnetism, colors, force, power, &c., &c. 

To obtain the magnitude of a quantity we compare it with a part of the same, 
this part is imprinted in our mind as a unit, by which the whole is measured 
and conceived. No quantity can be measured by a quantity of another kind, 
but any quantity can be compared with any other quantity, and by such com¬ 
parison arises what we call calculation or Mathematics. 


MATHEMATICS. 

Mathematics is a science by which the comparative value of quantities 
is investigated; it is divided into: 

1st, Arithmetic*—that branch of Mathematics, which treats of the nature 
and property of numbers; it is subdivided into Addition, Subtraction, Multiplicar 
turn, Division, Involution, Evolution and Logarithms. 

2nd, Algebra*—that branch of Mathematics which employs letters to repre¬ 
sent quantities, and by that means performs solutions without knowing or 
noticing the value of the quantities. The subdivisions of Algebra are the same 
as in Arithmetic. 

3rd, Geometry*—that branch of Mathematics which investigates the rela¬ 
tive property of quantities that occupies space; its subdivisions are Longimctry , 
Planimetry, Stereometry, Trigonometry, and Conic Sections. 

4th, Differehtial-calculus,—that branch of Mathematics which ascer¬ 
tains the ultimate ratio of two or more variables connected by an equation. 

5th, Integral-calculus,—the contrary of Differential, or that branch of 
Mathematics which effects tlio summation of infinitesimal quantities. 


ARITHMETIC. 

The art of computation with known quantities. 

Figures —1, 2, 3, 4, 5, 6, 7, 8, 9. Arabic digits, about nine hundred years old. 
Ciphers—0, 0, 0. Sometimes called noughts. 

Number is the expression of one or more figures and ciphers. 

Integer is a whole number or unit. 

Fraction is a part of a number or unit. 

When figures are joined together in a number, the relative dignity expressed 
ty each figure, depends upon its position to the others. Thus, 


OO 



674,385 ; 496,345 ; 695,216 ; 505,310 : 685, 36 7; 


2 















18 


Notation, 


ROMAN NOTATION. 

Tho Romans expressed their numbers by various repetitions and combinations 
of seven letters in the alphabet; as, 


1 = 1 . 

2 = 11 . 

3 = III. 

4 = IV. 

5 = V. 

6 = VI. 

7 = VII. 

8 = VIII. 

9 = IX. 

10 = X. 

20 = XX. 

30 = XXX. 

40 = XL. 

60 = L. 

60 = LX. 

70 = LXX. 

80 = LX XX. 

90 = XO. 

100 = c. 

Ex AMPLF.S.—1872.—M DCCCLXXII. 


500 = D, or LO. 

1,000= M, or CO. 

2,000 = MM, or IIOOO. , 
6,000 = V, or LOO. 

6,000 = VI, or MMM 
10,000 = X, or COO. 

60,000 = L, or LOGO. 

60,000 = EX, or MM MO. 

100,000 = IT, or COOO. 

1,000,000= M, or COOOO. 

2,000,000 = STM, or MMOOO. 

A bar, thus, — over any number, in¬ 
creases it 1000 times. 


524,365, DXXlVCCCLXV. 


Au imperfection in the Roman Notation consists in the fact that there is no sig¬ 
nification for the cipher, as in the Arabic Notation. 


Signification of Characters. 


= Equality, as 6 = 6, reads 6 is 
equal to 6. 

+ Plus, Addition, . 3 -j- 6 — 9 
— Minus, Subtraction, 6 — 2 = 4 
X Multiplication, . 3X4 = 12 
or: Division, . .15:5 = 3 
4 / Square root, . . . 4/9 = 3 

i = 2 

8>4 

6<9 
1 


Cube root, 
Greater, 
Less, . 

00 Infiuite, 


J Integral, . . 
dy Differential, 

ZJi Fraction, .... 

M. Ship sign, dead flat, 
a Furnace fire-grate. 

O Boiler heating-surface. 
# Sharp. High. 

[7 Flat. Low. 

7 r Periphery. 


fdy = y. 

dy — dx -f. 

- = *. 


Planets. 

O The Sun. 

<1 The Moon. 
$ Mercury, 

$ Venus, 

© The Earth, 
cf Mars. 

5 Ceres. 

$ Pallas. 

0 Juno. 

§ Vesta. 

1/ Jupiter. 

^2 Saturn, 
iji Uranus. 
Neptune. 


Astronomical Characters. 


(5 Conjunction in the 
same degree or sign, or 
having the same longi¬ 
tude or Right Ascension. 
Sextile, when two signs 
distant, or differing 60°' 
in longitude or Right 
Ascension. 

□ Quartile, when three 
signs distant, or differ¬ 
ing 90° in Longitude or 
Right Ascension. 

8 Opposition, when six 
signs distant, or differ¬ 
ing 180° in Longitude 
or Right Ascension. 

Q Ascending Node. 

£3 Descending Node. 

R. A. Right Ascension. 


Signs of the Zodiac. 

T Aries, . . 


8 Taurus, . . 


El Gemini, . . 

n 

0 Cancer, . . 


Leo, • • • 


IIP Virgo, . . . 


=cs Libra, . . 

A 

TTt Scorpio, . . 


f Sagittarius, 


Vy Capricornus, 


~ Aquarius, . 


X Pisces, . . . 

♦ 



< 















Algebra. 


19 


ALGEBRA. 

In Algebra we employ certain characters or letters to represent quantities. 
These characters are separated by signs, which describe the operations ; and by 
that means, simplify the solution. 

1. Whatever the value of any quantity may be, it can be represented by a 
character, as a. Another quantity of the same kind, but of different value, be¬ 
ing* represented by b. The sum of these two quantities is of the same kind but 
of different value. 

For Addition we have the algebraical sign +, (plus) which, when placed 
between quantities, denotes they shall be added; as a+6, reads in the 
algebraical language, “a plus b,” or a is to be added to b. 

Another algebraical sign =, (Equal) denotes that quantities which are placed 
on each side of this sign, are equal. Let the sum of a and b be denoted by the 
letter c ; then we have, 

a+6=c. 

This composition is called an algebraical equation. The quantity on each side 
of the equal sign is called a member, as a-\-b, is one member, andc, the other. When 
one of the members contains only one quantity, that member is generally 
placed on the first side of the equal sign, and its value commonly unknown; 
but the value of the quantities in the other member being given, as a— 4, and 
6=5, then the practical mode, to insert numerical values in algebraical equa¬ 
tions, will appear; as, 

Equation, c—a + b, 

4+5=9, the value of c. 

2. The sum of three quantities a, b, and c, is equal to d, then 

Equation, d=a+6+c, 

4+5-|-9=18, the value of d. 

3. For Subtraction we have the algebraical sign,—, (minus) which, when 
placed before a quantity, denotes it is to be subtracted as, a — b, reads in the 
algebraical language “a minus b,” or from a, subtract b. Let the difference be 
denoted by the letter c; and a=8. 6=3 

Equation, c—a —6, 

8—3=5, the value of c. 

4. From the sum of a and 6, subtract c, and the result will be d; then, 

Equation, d—a-\-b — -c, 

8+3—5=6, the value of d. 

5. When two equal quantities are to be added, as a+a. it is the same as to 
take one of them twice, and is marked thus 2a. The number 2 is called the 
coefficient of the quantity a. If there are more than two equal quantities to be 
added, the coefficient denotes how many there are of them; as, 

Equation , - - * * a+a=2a, 

“ a+a+a=3a, 

“ a+a+a+a=4a, 

etc., dx. 

When the quantities are separated by the signs, plus, or minus, they are 
called terms. 

6. Multiplication—When a quantity a, is to be multiplied by another 
quantity 6, then a and 6 are called factors; and separated by no sign as ab; 
which denotes that a is to be multiplied by 6; but when the values of a and 6 
are expressed by numbers, they are separated by the sign X (Multiplication); the 
result from Multiplication is called the product. Let o=8, and 6=6, and the pro¬ 
duct of a and 6, to be c, then, 

Equation, c=ab, 

8X6=48, the value of c. 

7. The product of a and 6, is to be multiplied by c, and the latter product will 
be equal to d ; then, 

Equation, d=abc, 

8X6X^8=2304, the value of d. 









20 


Algebra. 


8. The sum of a and 6, is to be multiplied by c, and the product will be d; 
then. 


Equation, d — c (a+6), 

48 (8-(-6) = 672 the value of d. 


When the sum of two or more quantities is to be multiplied by another quan¬ 
tity, the sum is to be enclosed in parentheses, and is called one factor. The other 
factor is to be placed on the outside of the parentheses, as seen in the preceding 
example. 


9. To the product of a and c, add b, and the result will be d; then, 
Equation, d— ac +6, 

8X48+6 = 390 the value of d. 


Be particular to distinguish the two Examples 8, and 9. 

10. The sum of a and b, to be multiplied by the sum of a and c; the product 
will be d; then, 

Equation, d = (a+6) (a + c), 

(8+6) (8+48) = 784. 

11. The sum of c and b, to be multiplied by the difference of c and a; the re¬ 
sult will be d ; then, 

Equation, d — (c+ h) (c— a), 

(48 + 6) (48—8) = 2160. 

12. Division.—When aquantity a, is to be separated into b equal parts, the 
numbers of parts or b, is called the divisor, and the value of each part, is called 
the quotient. The sum of the parts or the whole quantity a, is called the dividend ; 
a and 6, is separated by the sign : (Division); jus a: b, reads in the algebraical 
language, “a divided by b.” Let the quotient be denoted by the letter c; and 
a=18, 6=6, then, 

Equation, c= a : b, 

18 : 6 = 3 the quotient c. 

In Algebra it is found more convenient to set up Division as a fraction, then 
it will appear as, 

13. Divide a, by c, and the quotient will be b. Then, 

a. 

Equation, b = —» 

18 

j = G the quotient b. 


14. The product of a and b, to be divided by c; and the product will be d. 
Then, 

ab 

Equation, d =-» 

c 


18X6 

3 


= 36. 


15. The sum of d and b, to be multiplied by c, and the product divided by a; 
then the result will be e. 


Equation, 


c (d+b) 
a * 
3 (36+6) 
18 


16. From the product of a and c, subtract 36; divide the remainder by the 
difference of a, and c; the result will be h. 








Algebra. 


21 


Equation , h — 


ac — 36 
a — c ’ 


18X3 — 3X6 
18 — 3 


An old man said to a boy, “How old are you?” to which be replied,—“To 
seven times my father’s age add yours, divide the sum by double the differ¬ 
ence of yours and his, and the result will he my age.” 


Letters will denote, 
a = the old man’s age, 

6 — the lather’s age, 
c = the boy’s age. Then, 


Equation , 


c 


76 + rt 

■-— the bov’s age. 

2(« —6) ' b 


a = 73 years, the age of the old man, 7 X 57 + 73 

6 = 57 years, the father’s age. c = —==r- = 14f years. 

Required the boy’s age. 0 > 


Powers.—When a number or a quantity is to be multiplied by itself, the 
operation is called power and denoted by a small number at the right-hand 
corner of the quantity, like a 2 , which denotes that a. or the numerical value 
of a must he multiplied by itself. Suppose a = 4; then a- = a X « — 4 X 4 = 16. 
When a represents the length of the side of a square, then a- represents the 
area of that square. Suppose the side of t he square is a = 12 inches; then the 
area of that square will be a 2 = 12 2 = 12 X 12 = 144 sq. in. The small number 
is called the exponent of the power. When the exponent is 2, the power is 
called the square; when 3, it is called the cube; and when 4, the bisquare. 
When the side of a cube is a=12 inches, then the volume of that cube is 
a'i = a X a X « = 12 3 = 12 X 12 X 12 = 1728 cubic inches. The squares and 
cubes of numbers will be found in tables farther on. 

When quantities are separated bysigns, + or —, enclosed within parentheses 
and with an exponent outside of the parentheses—like this: (a -h 6) 2 —then a 
and 6 must first be added and the sum squared. If a = 4 and 6 = 9, then 
(4 + 9) 2 = 13 2 = 169. (a 6 — c d)3, a = 4, 6 = 9, d = 5, and c = 3. 

Then, (4 X 9 — 3 X 5)3 = (36 —15)3 = 213 = 9261. 

The operation within the parentheses must be accomplished before the power 
is used. 

Roots.—A root is a number from which a given power is raised, and is 
denoted by the sign y which means the square root. That is, yT=2, because 
22 = 4 . The number 4 is the power and 2 the root. •/16 = 4, because 4 2 = 16. 
Roots of higher order than the square root are indicated by a small number 

in the root-mark; as, jK _ , which means the cube root. Thus, the j^8 = 2, be¬ 
cause 23 = 2 X 2 X 2 = 8. 

The fourth root is marked or ^ 16— 2, because2 4 = 2 X 2 X 2 X 2 = 16. 
The small number in the root-mark is called the index of the root. 

A common formula for a right-angled triangle is « = j/6 2 + c 2 , in which a 
is the hypothenuse, 6 and c the two legs forming the right angles. Suppose 

6 = 4 and c=3; then a = >/4 2 + 3 2 = y' 16 f-9 = |/25 = 5. That is to say, the 
hypothenuse is 5. The operation under the root-mark must be accomplished 
before the root is extracted. 

The extraction of roo s by arithmetic is very complicated and not often 
resorted to in practice, but tables and logarithms are generally used for that 
purpose. 


. — 

















22 


Proportion. 


PROPORTION. 


Tne relative value of two quantities is obtained by dividing one into the 
other, and the quotient is called the ratio of their relationship. If the ratio 
of two quantities is equal to the ratio of two other quantities, they are said 
to be in the same proportion ; as, 

a . b c . rf, 

reads in the algebraical language “o is to b as c is to d." a, b, c, and d are 
called terms, of which a is the first, b the second, c the third, and d the lourth 
term. The first and fourth are called “the outer terms," and the second and 
third “ the inner terms." The whole is called an “analogy." 

A property in the nature of analogies is that the product of the outer terms, 
ad, is equal to the product of the inner, be. Suppose a = 4, 6 = 9, c = 12, 
d = 27. 

4:9 = 12: 27, 
ad = bc, 4X27 = 9X12. 

If any one of the four quantities is unknown, its value can be calculated 
by the other three; as, 


b c 

9X12 

d 

27 

ad 

4X27 

c 

12 

ad 

4X27 

b 

9 

b c 

9 X 12 

a 

4 


Proportion is generally used in commercial calculations, and in arithmetics 
it is called the rule of three, because in simple proportion thereare three given 
quantities by which the fourth one is calculated. The fourth or unknown 
quantity is generally denoted by x. 

Example. If 3 yards of cloth cost 11 dollars, how much will 7 yards of the 
same cloth cost? 

From these three given quantities we can find the fourth or unkuown price 
of 7 yards. 

Proportion, 3:7 = 11: x, 

the product of the inner terms being equal to that of the outer ones, or 
3x = 7 X 11. Move the 3 of the first member under the second member; thus, 

7X11 

x = — - — = 25.GG, or $25 and GG cents, the price of 7 yards. 

u 

Mean Proportion. 

The mean proportion between two quantities, a and b, is set up as follows: 
a:x — x:b, x~ = ab, or x = y'ab. 

The square root of the product of the two quantities is the mean propor¬ 
tion x. 

Suppose a = 2 and b = 8; then the mean proportion, x, between 2 and 8, is 

x = )/2X8= ]/l 6 = 4. 


Mean Difference. 

The mean difference or average between two quantities Is the sum of the 

Q uantities divided by 2. Let the quantities be a = 2 and 6 = 8; then the mean 
itference, x, is 

a -f- b 2 + 8 10 

* = 2 = 2 = Y = 5 ' 

The mean proportion between 2 and 8 is 4, but the mean difference Is 5. 

It is of great importance to clearly distinguish mean proportion from mean 
difference, for otherwise calculation may lead to erroneous results. 









Simple Interest. 


23 


SIMPLE INTEREST. 

Interest is money paid for use of money which is lent for a certain time. 

Notation, 
c = the amount lent, 
r = interest on the amount, c, 
p — per cent, in the certain time. 

Analogy, c : r = 100 : p. 

If p is the per cent, on 100, in one year, then t = time in years fbr the stand¬ 
ing capital c, and the interest r. 

Analogy, c:r = 100 : pt. 

From this analogy we obtain the equations, 


Interest, 


cpt 


T — 

100 ’ * 

• * • * lj 

Per cent., 


100 r 

i 

• • • • 

. p — 

tc ’ 

Capital, 

c = - 

100 r 

pt * 

- - * - 3, 

Time in years, 

t - 

100 r 
cp * 

• * • • t 


Now for any question in Simple Interest, there is one equation which gives the 
answer. If the time is given in months, wedcs, or days, multiply the 100 cor¬ 
respondingly by 12, 52,365. 

Example 1. What is the interest on $3789.35, for 3 years and five months, at 
6 per cent, per annum? 

t = 3X12+5 = 41 months, from the Equation 1, we have, 


Interest, 


r = 


37S9.35X6X41 

12X100 


=776.81 Dollars. 


Example 2. A capital c = $469.78, returned interest r = 150.72 dollars, in time 
t = 4 years and 7 months. Required the per centage per annum ? 

t = 4 X 12 + 7 = 55 months, from Equation 2, we have, 


Per cent.. 


p = 


12 X 100 X 150.72 
469.78 X 55 


: 7 per cent. 


Example 3. What amount is required to return interest r = 345 dollars in 6 years, 
at 5 per cent, per annum ? From the Equation 3, we have, 


Capital, 


100 X 345 
5X6 


= $1150. 


Example 4. An amount c = $2365 is to stand until the interest r = 550 dollars, 
at p — 6 per cent, per annum. How long must the amount stand? 

From the Equation 4, we have, 


Time, 


100X550 
1 ~ 23(35X6 


3.876 years. 




12X0.876 = 10.512 months, 4X0.512 = 2.048 weeks, the time t = 3 years, 10 
months, and 2 weeks. 



















24 


Simple Interest Table. 


Kive per cent, per Annum. 


Time. 

$100. 

$200. 

$300. 

$400. 

$.500. 

$600. 

$700. 

$800. 

$900. 

$1000. 

1 dav. 

0.01 

0.03 

0.04 

0.06 

0.07 

0.08 

0.10 

0.11 

0.13 

0.14 

2 

days. 

0.03 

0.06 

0.08 

0.11 

0.14 

0.17 

0.19 

0.22 

0.25 

0.28 

3 

u 

0.04 

0.08 

0.13 

0.17 

0.21 

0.25 

0.29 

0.33 

0.37 

0.42 

4 

u 

0.06 

0.11 

0.17 

0.22 

0.28 

0.33 

0.39 

0.45 

0.50 

0.56 

5 

u 

0.07 

0.14 

0.21 

0.28 

0.35 

0.42 

0.49 

0.56 

0.63 

0.70 

6 

it 

0.08 

0.17 

0.25 

0.33 

0.42 

0.50 

0.58 

0.67 

0.75 

0.83 

7 

u 

0.10 

0.19 

0.29 

0.39 

0.49 

0.58 

0.68 

0.78 

0.88 

0.97 

8 

a 

0.11 

0.22 

0.33 

0.44 

0.56 

0.67 

0.78 

0.89 

1.00 

1.11 

9 

u 

0.13 

0.25 

0.38 

0.50 

0.63 

0.75 

0.88 

1.00 

1.12 

1.25 

10 

a 

0.14 

0.28 

0.42 

0.55 

0.69 

0.83 

0.97 

1.11 

1.25 

1.39 

11 

« 

0.15 

0.31 

0.46 

0.61 

0.76 

0.92 

1.07 

1.22 

1.38 

1.53 

12 

IC 

0.17 

0.33 

0.50 

0.67 

0.83 

1.00 

1.17 

1.33 

1.50 

1.67 

13 

cc 

0.18 

0.36 

0.54 

0.72 

0.90 

1.08 

1.26 

1.44 

1.62 

1.S0 

14 

u 

0.20 

0.39 

0.59 

0.79 

0.98 

1.18 

1.38 

1.57 

1.77 

1.96 

15 

cc 

0.21 

0.43 

0.62 

0.S3 

1.04 

1.25 

1.45 

1.67 

1.87 

2.08 

16 

« 

0.22 

0.44 

0.67 

0.89 

1.11 

1.33 

1.56 

1.78 

2.00 

2.22 

17 

cc 

0.24 

0.47 

0.71 

0.94 

1.18 

1.41 

1.65 

1.89 

2.12 

2.36 

18 

cc 

0.25 

0.50 

0.75 

1.00 

1.25 

1.50 

1.75 

2.00 

2.25 

2.50 

19 

cc 

0.26 

0.53 

0.79 

1.06 

1.32 

1.59 

1.85 

2.11 

2.38 

2.64 

20 

cc 

0.2S 

0.56 

0.83 

1.11 

1.39 

1.67 

1.95 

2.22 

2.50 

2.78 

21 

cc 

0.29 

0.58 

0.88 

1.17 

1.46 

1.75 

2.04 

2.33 

2.63 

2.92 

22 

4C 

0.31 

0.61 

0.92 

1.22 

1.53 

1.83 

2.14 

2.45 

2.75 

3.06 

23 

u 

0.32 

0.64 

0.96 

1.28 

1.60 

1.92 

2.24 

2.56 

2.87 

3.19 

24 

CC 

0.33 

0.67 

1.00 

1.33 

1.67 

2.00 

2.33 

2.67 

3.00 

3.33 

25 

cc 

0.35 

0.69 

1.04 

1.39 

1.74 

2.09 

2.43 

2.78 

3.13 

3.47 

26 

cc 

0.36 

0.72 

1.0S 

1.44 

1.80 

2.17 

2.53 

2.89 

3.25 

3.61 

27 

cc 

0.38 

0.75 

1.13 

1.50 

1.88 

2.25 

2.63 

3.00 

3.37 

3.75 

28 

cc 

0. lu 

0.78 

1.17 

1.56 

1.94 

2.33 

2.72 

3.10 

3.50 

3.89 

29 

cc 

0.40 

0.81 

1.21 

1.61 

2.02 

2.42 

2.82 

3.23 

3.63 

4.03 

30 

cc 

0.12 

0.83 

1.25 

1.67 

2.08 

2.50 

2.92 

3.33 

3.75 

4.17 

1 month. 

0.42 

0.83 

1.25 

1.67 

2.08 

2.50 

2.92 

3.33 

3.75 

4.17 

2 months 

0.83 

1.67 

2.50 

3.33 

4.17 

5.00 

5.83 

6.67 

7.50 

8.33 

3 

cc 

1.25 

2.50 

3.75 

5.00 

6.25 

7.50 

8.75 

10.00 

11.25 

12.50 

4 

cc 

1.67 

3.33 

5.00 

6.67 

8.33 

10.00 

11.67 

13.33 

15.00 

16.67 

5 

cc 

2.08 

4.17 

6.25 

S.33 

10.41 

12.50 

14.58 

16.67 

18.75 

20.83 

6 

cc 

2.50 

5.00 

7.50 

10.00 

12.50 

15.00 

17.50 

20.00 

22.50 

25.00 

7 

cc 

2.92 

5.83 

8.75 

11.67 

14.58 

17.50 

20.42 

23.33 

26.25 

29.17 

8 

cc 

3.33 

6.67 

10.00 

13.33 

16.67 

20.00 

23.33 

26.67 

30.00 

33.33 

9 

cc 

3.75 

7.50 

11.25 

15.00 

18.75 

22.50 

26.25 

30.00 

33.75 

37.50 

10 

cc 

4.17 

8.33 

12.50 

16.67 

20.83 

25.00 

29.17 

33.33 

37.50 

41.67 

11 

cc 

4.58 

9.17 

13.75 

18.33 

22.92 

27.50 

32.08 

36.67 

41.25 

45.83 

1 

year. 

5.00 

10.00 

15.00 

20.00 

25.00 

30.00 

35.00 

40.00 

45.00 

50.00 



Example ..—Required the interest on $8978 in 27 days at five per cent, 
per annum ? 

Interest on $8000 = 30.00 


900 = 3.37 
70= 0.26 
8= 0.03 

$8978 = 33.66 the answer. 










































Simple Interest Table. 


20 


Six per cent, per Annum. 


Time. 

$100. 

$200; 

$300. 

$400. 

$500. 

$G00. 

$700. 

$800. 

$900. 

$1000. 

1 day. 

0.02 

0.03 

0.05 

0.07 

0.08 

0.10 

0.12 

0.13 

0.15 

0.17 

2 days. 

0.03 

0.07 

0.10 

0.13 

0.17 

0.20 

0.23 

0.27 

0.30 

0.33 

3 

a 

0.05 

0.10 

0.15 

0.20 

0.25 

0.30 

0.35 

0.40 

0.45 

0.50 

4 

a 

0.07 

0.13 

0.20 

0.27 

0.33 

0.40 

0.47 

0.53 

0.60 

0.67 

5 

U 

0.08 

0.17 

0.25 

0.33 

0.42 

0.50 

0.58 

0.67 

0.75 

0.83 

6 

a 

0.10 

0.20 

0.30 

0.40 

0.50 

0.60 

0.70 

0.80 

0.90 

1.00 

7 

a 

0.12 

0.23 

0.35 

0.47 

0.58 

0.70 

0.81 

0.93 

1.05 

1.17 

8 

a 

0.13 

0.27 

0.40 

0.53 

0.67 

0.80 

0.93 

1.07 

1.20 

1.33 

9 

a 

0.15 

0.30 

0.45 

0.60 

0.75 

0.90 

1.05 

1.20 

1.35 

1.50 

10 

a 

0.17 

0.33 

0.50 

0.67 

0.83 

1.00 

1.17 

1.33 

1.50 

1.67 

11 

a 

0.18 

0.37 

0.55 

0.73 

0.91 

1.10 

1.28 

1.47 

1.65 

1.83 

12 

a 

0.20 

0.40 

0.60 

0.80 

1.00 

1.20 

1.40 

1.60 

1.80 

2.00 

13 

a 

0.22 

0.43 

0.65 

0.87 

1.08 

1.30 

1.52 

1.73 

1.95 

2.17 

14 

a 

0.23 

0.47 

0.70 

0.93 

1.17 

1.40 

1.63 

1.87 

2.10 

2.33 

15 

(( 

0.25 

0.50 

0.75 

1.00 

1.25 

1.50 

1.75 

2.00 

2.25 

2.50 

16 

a 

0.27 

0.53 

0.80 

1.07 

1.33 

1.60 

1.86 

2.13 

2.40 

2.67 

17 

a 

0.28 

0.57 

0.85 

1.13 

1.41 

1.70 

1.98 

2.27 

2.55 

2.83 

18 

a 

0.30 

0.60 

0.90 

1.20 

1.50 

1.80 

2.10 

2.40 

2.70 

3.00 

19 

ii 

0.32 

0.63 

0.95 

1.27 

1.58 

1.90 

2.21 

2.53 

2.85 

3.17 

20 

a 

0.33 

0.67 

1.00 

1.33 

1.67 

2.00 

2.33 

2.67 

3.00 

3.33 

21 

a 

0.35 

0.70 

1.05 

1.40 

1.75 

2.10 

2.45 

2.80 

3.15 

3.50 

22 

a 

0.37 

0.73 

1.10 

1.47 

1.83 

2.20 

2.53 

2.93 

3.30 

3.67 

23 

a 

0.38 

0.77 

1.15 

1.53 

1.92 

2.30 

2.68 

3.07 

3.45 

3.83 

24 

a 

0.40 

0.80 

1.20 

1.60 

2.00 

2.40 

2.80 

3.20 

3.60 

4.00 

25 

a 

0.42 

0.83 

1.25 

1.67 

2.08 

2.50 

2.91 

3.33 

3.75 

4.17 

26 

ii 

0.43 

0.87 

1.30 

1.73 

2.17 

2.60 

3.03 

3.47 

3.90 

4.33 

27 

ii 

0.45 

0.90 

1.35 

1.80 

2.25 

2.70 

3.13 

3.60 

4.05 

4.50 

28 

ii 

0.47 

0.93 

1.40 

1.S7 

2.33 

2.80 

3.26 

3.73 

4.20 

4.67 

29 

a 

0.48 

0.97 

1.45 

1.93 

2.42 

2.90 

3.38 

3.87 

4.35 

4.83 

30 

a 

0.50 

1.00 

1.50 

2.00 

2.50 

3.00 

3.50 

4.00 

4.50 

5.00 

1 month. 

0.50 

1.00 

1.50 

2.00 

2.50 

3.00 

3.50 

4.00 

4.50 

5.00 

2 months 

1.00 

2.00 

3.00 

4.00 

5.00 

6.00 

7.00 

8.00 

9.00 

10.00 

3 

a 

1.50 

3.00 

4.50 

6.00 

7.50 

9.00 

10.50 

12.00 

13.40 

15.00 

4 

a 

2.00 

4.00 

6.00 

8.00 

10.00 

12.00 

14.00 

16.00 

18.00 

20.00 

5 

a 

2.50 

5.00 

7.50 

10.00 

12.50 

15.00 

17.50 

20.00 

22.50 

25.00 

6 

a 

3.00 

6.00 

9.00 

12.00 

15.00 

1S.00 

21.00 

24.00 

27.00 

30.00 

7 

a 

3.50 

7.00 

10.50 

14.00 

17.50 

21.00 

24.50 

27.00 

31.50 

35.00 

8 

a 

4.00 

8.00 

12.00 

16.00 

20.00 

24.00 

28.00 

32.00 

36.00 

40.00 

9 

u 

4.50 

9.00 

13.50 

18.00 

22.50 

27.00 

31.50 

36.00 

40.50 

45.00 

10 

a 

5.00 

10.00 

15.00 

20.00 

25.00 

30.00 

35.00 

40.00 

45.00 

50.00 

11 

u 

5.50 

11.00 

16.50 

22.00 

27.50 

33.00 

38.50 

44.00 

49.50 

55.00 

1 

year. 

6.00112.00 

18.00 

24.00 

30.00 

36.00 

42.00 

48.00 

54.00 

60.00 


Example .—The interest of $700, at 6 per cent, per annum, for five 
months, is $17.50. The interest on $70 in the same time is $1.75, and 
for $7, 17£ cents. For $7000 the interest is $175 in five months. Thus, 
the six per cent, interest for any sum and time can be found by this 
table. 










































26 


Interest Laws of all the States. 


INTEREST LAWS OF ALL THE STATES. 


Statics and 
Territories. 


Alabama. 

Arizona. 

Arkansas.. 

California. 

Colorado. 

Connecticut.. 

Dakota.. 

Delaware. 

Dist. of Columbia 

Florida. 

Georgia. 

Idaho . 

Illinois. 

Indiana. 

Iowa. 

Kansas. 

Kentucky. 

Louisiana. 

Maine. 

Maryland. 

Massachusetts. 

Michigan. 

Minnesota . 

Mississippi. 

Missouri. 

Montana. 

Nebraska. 

Nevada . 

New Hampshire... 

New Jersey. 

New Mexico. 

New York. 

North Carolina.... 

Ohio. 

Oregon. 

Pennsylvania. 

Rhode Island...... 

South Carolina.... 

Tennessee.^. 

I exas . 

Utah. 

Vermont. 


Virginia.. 

Washington ... 
West Virginia. 

Wisconsin. 

Wyoming . 


Penalty of Usury. 


Loss of interest. 
No penalty. 






Forfeiture of all interest. 
“ of contract..... 

tt tt <• 


tt 


of all interest. 

No penalty. 

Forfeiture of excess. 

$300 fine or imprisonment 6 ms. or both'® 

Forfeiture of all interest. 

of interest and costs. 

of excess. 

“ over 12 per ct. 

of all interest. 

of interest. 

No penalty. 

Forfeiture of excess. 

No penalty—6 per ct. on judgments_ 

Forfeiture of excess. 

“ “ u over 7 per ct. 

Forfeiture of all interest. 

U U ii 


u 

u 

u 

u 

u 


No penalty.*. 

Forfeiture of all interest and costs. 

No penalty. 

Forfeit of three times interest received 

Forfeit of all interest. 

No penalty.. 

Forfeiture of contract f. 

Forfeiture of interest. 

“ of excess. 

“ of principal, int., and costs... 

of excess ; Act of 1858. 

unless by contract j. 

No penalty. 

Forfeit of over G per ct. and $100 fine.. 
No penalty. 


tt 

tt 


Forfeit of excess on R. R. bonds. 

“ of contract. 

No penalty. 

Forfeit of excess. 

“ of all interest. 

No penalty. 


C5 

a. 

8 

10 

6 

10 

10 

7 

7 
6 
6 

8 
7 

10 

0 

6 

6 

7 

6 

5 

6 
6 
6 
7 
7 
6 
6 

10 

10 

10 

6 

7 

6 

6 

G 

6 


Special. 


No limit. 

u 


u 

u 

u 


a 

(£ 

a 


18 per ct. 
6 per ct. 
10 per ct. 
No limit. 
12 per ct. 
21 per ct. 
10 per ct. 
10 per ct. 
10 per ct. 
12 per ct. 
10 per ct. 
8 per ct. 
No limit. 
6 per ct. 
No limit. 
10 per ct. 
12 per ct. 
10 per ct. 
10 per ct. 

12 per ct. 
No limit. 

6 per ct. 

7 per ct. 
12 per ct. 

6 per ct. 

8 per ct. 


* Liable to arrest for misdemeanor. t Also punishabl 
interest only, or double the interest if charged in advance. 


10 

12 per ct. 

6 

6 per ct. 

6 

No limit. 

10 

tt tt 

6 

10 per ct. 

8 

No limit. 

10 

tt tt 

6 

7 per ct. 

6 

12 per ct. 

10 

No limit. 

G 

6 per ct. 

7 

10 per ct. 

10 

No limit. 

r. Hanks forfeit 


t Also 0 per ct. on judgments. 


































































































Rate of Income on Investment. 27 



Par Value being $100, Bearing Interest at 


Price 

paid. 

4 per ct. 

5 per ct. 

6 per ct. 

7 per ct. 

8 per ct. 

9 per ct. 

10 per ct. 

$50 

8.00 

10.00 

12.00 

14.00 

16.00 

18.00 

20.00 

55 

7.28 

9.09 

. 10.90 

12.72 

14.55 

16.36 

18.18 

60 

6.66 

8.33 

10.00 

11.66 

13.33 

14.99 

16.66 

65 

6.15 

7.69 

9.23 

10.76 

12.30 

13.85 

15.38 

70 

5.71 

7.14 

8.57 

10.00 

11.42 

12.85 

14.28 

75 

5.33 

6.66 

8.00 

9.33 

10.66 

12.00 

13.35 

80 

5.00 

6.25 

7.50 

8.75 

10.00 

11.25 

12.50 

82 J 

4.85 

6.06 

7.27 

8.48 

9.69 

10.91 

12.12 

85 

4.71 

5.88 

7.05 

8.23 

9.41 

10.58 

11.76 

87 ^ 

4.57 

5.71 

6.85 

8.00 

9.14 

10.28 

11.42 

90 

4.44 

5.55 

6.66 

7.77 

8.88 

10.00 

11.11 

92 ^ 

4.32 

5.40 

6.48 

7.56 

8.64 

9.72 

10.80 

95 

4.21 

5.26 

6.31 

7.36 

8.42 

9.47 

10.52 

96 

4.16 

5.20 

6.25 

7.29 

8.33 

9.37 

10.41 

97 

4.12 

5.15 

6.16 

7.21 

8.24 

9.27 

10.30 

97 J 

‘ 4.10 

5.12 

6.15 

7.17 

8.20 

9.22 

10.25 

98 

4.08 

5.10 

6.12 

7.14 

8.16 

9.18 

1 D .20 

99 

4.04 

5.05 

6.06 

7.07 

8.08 

9.09 

10.10 

100 

4.00 

5.00 

6.00 

7.00 

8.00 

9.00 

10.00 

101 

3.96 

4.95 

5.94 

6.93 

7.92 

8.91 

9.90 

102 

3.92 

4.90 

5.88 

6.86 

7.84 

8.82 

9.80 

103 

3.88 

4.85 

5.82 

6.79 

7.76 

8.73 

9.70 

104 

3.84 

4.80 

5.76 

6.73 

7.69 

8.65 

9.61 

105 

3.80 

4.76 

5.71 

6.66 

7.61 

8.57 

9.52 

110 

3.63 

4.54 

5.45 

6.36 

7.27 

8.18 

9.09 

115 

3.47 

4.34 

5.21 

6.08 

6.95 

7.82 

8.69 

120 

3.33 

4.16 

5.00 

5.83 

6.66 

7.50 

8.33 

125 

3.20 

4.00 

4.80 

5.60 

6.40 

7.20 

8.00 

130 

3.07 

3.84 

4.61 

5.38 

6.15 

6.92 

7.69 

135 

2.96 

3.70 

4.44 

5.18 

5.92 

6.66 

7.40 

140 

2.86 

3.57 

4.28 

5.00 

5.71 

6.43 

7.14 

. 145 

2.76 

3.44 

4.13 

4.82 

5.51 

6.20 

6.89 

150 

2.66 

3.33 

4.00 

4.66 

5.33 

6.00 

6.66 

. 155 

2.58 

3.23 

3.87 

4.52 

5.17 

5.80 

6.45 

' 160 

2.25 

3.13 

3.75 

4.38 

5.00 

5.62 

6.25 

165 

2.42 

3.03 

3.63 

4.24 

4.85 

5.45 

6.06 

170 

2.35 

2.95 

3.53 

4.12 

4.71 

5.30 

5.88 

175 

2.28 

2.86 

3.42 

4.00 

4.57 

5.14 

5.71 

180 

2.22 

2.78 

3.33 

3.89 

4.45 

5.00 

5.55 

185 

2.16 

2.70 

3.24 

3.79 

4.33 

4.86 

5.40 

190 

2.11 

2.64 

3.16 

3.69 

4.22 

4.73 

5.26 

195 

2.05 

2.57 

3.08 

3.60 

4.11 

4.62 

5.13 

200 

2.00 

2.50 

• 

3.00 

3.50 

4.00 

~i - 

4.50 

5.00 

























28 


Discount.—Partnership. 


DISCOUNT. 

Discount is interest on money which is paid before due. a == amount of 
money to be paid at the time t. Ry agreement the amount is paid with a capital 
e, at. the beginning of the time t, but discounted a Rebate r, at p per cent., so that 
the interest on the capital c, at p per cent., should be equal to the Rebate r, in the 

a = c r. 


time t. 
Discount, 
Capital, 
Per cent., 


apt 

r— ---. 

100 +/>* 

_ loQq 

~ 100 + pi 

p _ 100(q — c) - 

c t 


5. 

6 . 
7. 


Time , . t= 10 °( a — c ) ... . . 8. 

cp 

Amount, a= (100 +p t). . 9. 

100 

Amount, a — — (100 f-p t). . . 10. 

pt 


Now, for any question in Rebate or Discount, there is one equation that will give 
the answer. 

Example 5. A sum of money, a = 78460 dollars, is to be paid after 3 years and 
6 months, but by agreement payment is to bo made at the present time. 'What 
will be the Rebate, at 7 per cent.? 


Amount of discount, 


r = 


78460 X 7 X 3.5 


: SI5439.91. 


100 + 7 X 3.5 

. PARTNERSHIP. 

Partnership or Fellowship is a rule by which companies ascertain each 
partner’s profit or loss by their stock. Each partner’s part in the stock is called 
liis share. The sum of shares is called the stock. 

Partnerships are of two kinds, Simple and Double. 

Simple Partnership, when there is no regard to the time the shares or 
stock is employed. 

Letters denote, 

A = share of either one. , I stock or the sum of the shares. 

a = profit or loss on the share A. | s = gain or loss on the stock & 

A : a = S:s. 


Then, 


Stock, 

a 

. . 11. 

Gain or loss, 

A 

. . 12. 


Share, 

Profit or loss, 


A = 


a = 


a S 
s 

A s 


13. 

14. 


Example 1. A person had invested A =$11645 in a stock S — $64800, which 
gave a gain of s — $13864. What will be the profit of the person’s share ? 


Profit, 


a = 11645 X 13864 =$2491.45. 
64800 


When the different shares are employed at a different length of time, each share 
is multiplied by its time employed, and the product is the effect of the share. 


Letters denote, 


f = time for the employed share A. 

T= mean time for the employod stock S. 
e = effect of the share A. 


a = profit of the effect e. 
E= effect of the stock, 
s = gain of the effect E. 


Then, 


e : a — E: s. 














Permutation—Combination. 


29 


Effect of A, 

O — -• « « 

s 

15. 

Time, 

A s 

. . 19. 

Profit of e, 

a-* s 

- • • • • 

E 

16. 

Share , 

A — — 
t s 

. . . 20. 

Effect of S, 

E= e X . . 

a 

17. 

Meantime, 

T=±±. 
a S 

. . 21. 

Gain of E, 

a E 

® -• • • 

e 

M 

OO 

Stock, 

<8= —. 
a T 

. . . 22, 


Example 2. A canal is to bo dug, and requires an effect E = 76850 (men and 
days) to be accomplished; after that it will give a gain s =12390 dollars. An em¬ 
ployer has A = 108 laborers. How many days must those laborers be employed at 
tho canal, that the employer will obtain a profit a = 5000 dollars ? 

Time, t = ft 000 -* =184.6 days. 

168 X 1-390 


PERMUTATION. 

Permutation is to arrange a number of things in every possible position. 
It is commonly used in games. 

Example 1. How many different values can be written by the three figures 
1, 2, 3. 

1 X 2 X 3 = 6 different values, namely, 

123, 132, 213, 231, 312, 321. 

With any three different figures can be written six different values. Any three 
things can be placed in 6 different positions. 

Example 2. How many names can be written by the three syllables, mo, ta, la? 
The answer is, Motala, Molata, Tamola, Talamo, Lamota, Latamo. 

Example 3. How many words can be written by tho five syllables, mul, tip, li, 
ca, lion f 

1X2X3X4X& = 120 words, tho answer. 


COMBINATION. 

Combination is to arrange a less number of things out of a greater in every 
possible position. It is commonly used in games. 

Example 1. How many different numbers can be set up by the nine figures, 
1, 2, 3, 4, 5, 6 , 7, 8 , 9, and three figures in each number? 

- x — 84 different numbers. 

1X2X3 

Example 2. How many different variations can a player obtain his cards, when 
the set contains 52 cards, of which he receives 8 at a time ? 

S2X61 X 50 X49 X48X 47 X4r.X 45 = 75253S150 varia.io n3 . 

1 X 2 X 3 X 4 X 5 X o X 7 X 8 

If there are four players, and p r . 4 = 24, they can play 24X 752538150 = 
18,060,915,600 different plays. 

If it takes half an hour for each play, and they play 8 hours per day, it will |»ke 

_1 8060915600 _ j428807225 days = 3092622 vears. 

2X8 


















30 


Arithmetical Progression, 


ARITHMETICAL PROGRESSION. 

Arithmetical Progression is a scries of numbers, as 2, 4, 6, 8, 10,12, 
&c., or 18, 15, 12, 9, 6, 3, iu which every successive term is increased or dimin¬ 
ished by a constant number. 

Letters denote , 
a = the first term of the series. 

1) = any other term whose number from a is ». 
n = number of terms within a and b. 

$ — the difference between the terms. 

S = the sum of all the terms. 

In the series, 2, 5, 8,11, a = 2, b — 11, n = 4, $ = 3,and S = 26. 

4®"When the series is decreasing, take the first term = b and the last term 
== a. 

The accompanying Table contains all the formulas or questions in Arithmeti¬ 
cal Progressions, and the nature of the question will tell which formula is to be 
used. 


Formulas for Arithmetical Progressions. 


a, o= b—S (n —1), • 

• 1, 

„ b —a 

5 = -, • • • 

n—1’ 


2S R 

& = —C>J • 

n 

• * 2, 

„ (&-f-a)(5—a), 

2S — <i —6 , * 

10, 

S 4 

a = - ——(»—1), - 
n & 

• 3, 

f 2 (5—dfi) 

J - nl«-i) ’ * * 

11, 

b = a-f-J (n—1), 

• * *> 

s 2 (bn-S), 
n («—1) 

12, 

. 2 S 

b --a, - 

n 

. 5, 

s _n(a±b) t . . . 

2 

13, 

■ 6 = |+| (n - 1 )’ 

• • 6, 

(«+5)(&-fA—a) t 

21 

14, 

b—a. 

" = _+!, - . 

* 7, 

S = n £a+- (»—1) J 

15, 

2 S 

n =*= —. y - 

Cl-\-b 

• - 8, 

—-(n— 1)J - 

16, 





a = 

4+\/( 

6+i)-2!S, .... 

17, 

b — 

“ 2 ± \, 

/ («- |-) 2 +26S, • . - 

18, 

n = 

1 a 

2-i-±\ 


19, 

n 

§- + T±\ 

/(H) L £ • • • 

20. 






















Arithmetical Progression’. 


31 


Examples In Aritlimetical Progression. 

The preceding table furnishes a means for practicing the insertion of 
numerical values in algebraical formulas which illustrate the different 
arithmetical operations. 

Example, 1. An arithmetical progression of n — 20 terms has a difference 
of 6 = 2 between each term, and the last term is 6 = 166. Required the value 
of the first a ? 

Find in the table the formula containing the given quantities, and of 
which the first member is a. Formula 1 corresponds to this. 

The first term « = 6 — 6(n — 1) = 166 — 2(20 — 1) = 166 — 2 X 19 = 166 — 38 
= 128, which is the required first term. 

The calculation is given in detail merely for illustration, but in practice 
we write only 

a = 166 — 2(20 —1) = 128. 

Example 2. The sum of n = 9 terms in an arithmetical progression is 
S— 1787, and the last term is b — 360. Required the value of the first term a? 



n 


2 X 1787 
9 


360 = 37.1. 


Example 9. Required the difference 5 in the progression of example 2? 


6 — a 360 — 37.1 
n—l~ 9 — 1 


322.9 

8 


40.3625. 


Example 7. In an arithmetical progression the first term is a = 85, the last, 
term b = 163, and the difference between each term is 6 = 17. Required the 
number of terms n ? 

Find the proper formula in the table; insert the given quantities and per¬ 
forin the calculation, which will give n = 6. 


Example 8. The first term of a progression is a = 44, and the last terra 
6 = 256. How many terms must be inserted between a and 6, inclusive, in 
order that the sum of all the terms shall be 8= 1800? 

Find the proper formula in the table; insert the given quantities, and the 
calculation will give n = 12 terms. 


Example 15. A man was engaged to dig a well at one dollar ($1) for the first 
foot of the depth of the well, $1,84 for the second, and 84 cents more per every 
successive foot in depth until he reached the water, which was found at a 
depth of 25 feet. How much money is due to the man ? 

This will be answered by the formula 15, in which o = 1, d=0.84, and 
n = 25; then the sum, 

S— 25 £l + ^p-(25 — 1) J = $277 the answer. 


Example 20. A ship cost $116,500, of which the company agreed to pay on 
account $14,075 on her first trip; and for every successive trip they paid 
$650 less than the former. How many trips must the vessel make till fully 
paid for? 

This will be answered by the formula 20, in which 6 =$14,075, 5 = 650, and 
S = 116,500. 


1 , 14075 

YL — "T" - ■ 

2 ' 650 



14075 

650 


1 \ 2 2 X 116500 

■**2/ 65fT~ 


10.6 or 11 trips. 















32 


Arithmetical Progression. 


Arithmetical Progressions of a Higher Order. 

Arithmetical Progressions are of the first order, when the difference S is a 
coustant number, but when the difference $ progresses itself with a constant 
number, the Progression is of the second order. 

When the difference S' progresses in a second order, the Progression is of the 
third order, &c., &c., and is thus explained: 

1, 2, 3, 4, 5, 6, . . . n, - • Arith. Prog., first order. 

1, 3, 6, 10, 15, 21, . . . .... 2d. order. 

2 


1, 4, 10, 20, 35, 56, 


»(»+l)(n+2) 
2X3 * 


• 3d. order. 


1, 5, 15, 35, 70, 126, 


w (re+D(w+2)(rt+3 ) 

2X3X4 1 


4th. order. 


Here you will discover that the sum of n terms in one order, is equal to the 
same nth term in the next higher order. Arithmetical Progressions of the first, 
second, and third orders, are applied to 


PILES OF BALLS AND SHELLS. 

Triangular Piling. 

Example 1. A complete triangular pile of balls has n = 12 balls in each side. 
Require how mauy bafts in the base, and how many in the whole pile? 


In the base, 

= V>r 2 2 —= 7S balls, - . 

* 2d. order. 

Whole pUe, • 

^ 304 bolls, . 

• 3d. order. 


Square Piling. 


9, 16, 25, 36, . 

• • • • • • 

- 2d. order. 

114, 30, 55, 91, - 

n(n+lX2»+l) 

2X3 * 

• 3d. order. 


Example 2. 
rows? 


IIow many halls are contained in a complete square pile, n 


10(10 +1)(2X10+1) 10XHX21 
2X3 6 


= 385 balls. 


10 


Rectangular Piling. 

Let m he the number of balls on the top of the complete pile, and tt = num¬ 
ber of rows in the same, then the number of halls in the whole pile will 

be, 


n(n+l)(2n+3wt—2} 
2X3 > 


3d. order. 


The number of balls in the longest bottom side will be = m+n — L 

Example 3. The rectangular pile having 15 rows and 23 balls on the the top, 
how many in the whole pile? 


1505+1X2X15 +3X23—2) 15X10X67 

_ . « *5 


2X3 


2680 ball*. 




















Alligation. 


33 


ALLIGATION. 


Alligation is to mix together a number of different things of different price 
or value, and ascertain the mean value of the mixture; or from a given mean 
value of a mixture ascertain the proportion and value of each ingredient. 

Let the different things be a , b, c and d, etc., their respective price or value per 
unit, z, y, x and w, etc. 

A = a-\-b-\-c-\-d, etc., the sum of the things. 

P=meau value or price per unit of A. 

Then, A P= az + by + cx + dw+, etc.1. 


and 


p_ az + feft+c:c-|-dw-f 


, etc. 


Example 1. If 3 gallons of wine, at $1.37 per gallon, 2, at $2.18, and 5, at $1.75> 
be mixed together, what is a gallon of the mixture worth? 

A = 3 + 2-f5 = 10 gallons. 

P= 3 X 1-37 + 2 X 2.18 + 5 X 1.75 = $ 172 per ga n on . 

10 

Alligation of two ingredients, a and b, with their respective prices or value per 
unit, z and y. z>P>y. A = a-\-b. 

a:b = ( P — y):(z — P) .3. 

W-Pl and a = S P ~y) . . 


a ■■ 


4 & 5. 


{z = py ( z-y) 

Example 2. A silversmith mixes two sorts of silver, one at 54 and one at 64 cents 
per ounce. How much must be taken of each sort to make the mixture worth 60 
cents per ounce ? (Formula 3.) P = 60. x = 54. y = 64. 

a : b = (60 — 54) : (64 — 60) = 6 : 4, or 

4 ounces, at 54 cents, and 6 ounces, at 64 cents. 

Alligation of three ingredients, a, b and c, with their prices or value per unit, 
z , y and x. 

a':c' — (P — x) : (z —P). ....... 6 

a" :b =(P—y) :(z — P) when z>P>y>x. . . 7. 

b :c" = (P—x): (y — P) when £>y>P>:». . . 8. 

a = a' + a", c = c' + c". 

Example 3. A farmer mixes wheat, at 94 cents per bushel, with barley, at 72 
cents, rye, at 64 cents per bushel. How much of each sort mugt be taken to make 
the mixture worth 80 cents per bushel? 

(Formula 6.) z = 94, y = 72, x = 64, and P= 80. 

a': c'=(80—64): (94 — 80) = 16 : 14. 


a’ 


b = (80 — 72) : (94 — 80)= 8:14. 


The wheat a = 16 4- 8 = 24 bushels, at 94 cents per bushel. 

“ barley b = 14 “ “ 72 “ 

c= 14 “ “ 64 “ “ 


rye 


Alligation of four ingredients a, b, c and d, respective prices or value per unit, 
, y, x and w. 

ten*>v>r>x>w> {£ 

d 


a 

b 


a’ 
a" 


a 


/// 


= (P—w ): (z — P)) 

b = (P— y)’(z — P) >when s>P>y> x>w 
= (P— x) : (z —P)j 


a — a'-\-a"-\- a'", 
aid' =(P — w):(z — P)) 

b : d" =(P—w): ( y — P ) Vwhen z>y>x>P>w 
c: d"'=(P— w): (x — P)J 
d =d' + d" + d"'. 

In the same manuer, formulae can be set up for any number of ingredients. 


3 














u 


GEOMETRICAL PROORESSXOW. 


GEOMETRICAL PROGRESSION. 

G eometrical Progression ia a series of numbers, as 2 :4 : 8 :16: 32 : Ac., 
or 729; 243:81: 27 : 9 : &c., in which every successive term is multiplied ot divided 
by a constant factor. 


Notation. 

a = the first term of the series. 
b = any other term whose number from a is ». 
n = number of terms within a and b. 

r = ratio, or the factor by which the terms are multiplied or divided. 

S =* Sum of the terms. 

In the series 1: 3: 9 : 27 : a = 1, 6 = 27, n = 4, r = 3, S = 40. 

The accompanying Table contains all the formulas or questions in Geometrical 
Progressions. The nature of the question will tell which formula is to be 
used. 


Formulas for Geometrical Progressions. 


b 

a= *v^ * ■ * 

1, 

/S~ 

r= V r ’ * ■ 

• 7, 

« - 3—r(S—b},‘ • 

2, 

$-a 

r ~ X-b’ * * * 


# =s S - r, • • 

r *—V 

3, 

ar"-f 5— — a = 0, 

- 5 > 

4 =» or"— 1 , • • • 

4, 

„ hr— a 

* * * 

*>, 

g ^ ° T 

& 

a(r"_ 1) 

-11, 

r 


r — 3 ' 

9 

* - 

6, 

sr h(r"-l) 

* (^-“1 )rn-l» * * 

12, 


ToffJ>— log. g 
log.r ' 

log.b — log.a 


'log.(S-a) — log.(J3 — bf 

log.\a+S(r —1 )] — log.a 

log.r. * 


i , logh—logfbr — S(r— 1)] 


b --l/V- q~y'T 


• • - 18 , 

• • • 14, 

♦ 

• • • 15, 

• • • 16, 

• L - • IT, 



















Geometrical Progression. 


35 


Example 1. Required the 10th term in the Geometrical Progression 4:12: 
Given a = 4, n = 10, and r = 3. We have, 

Ibrmula 4. b = ar*-i = 4X 3 9 = 78732, the tenth term. 

Example. 2. Required the sum of the 10 terms in the preceding example? 


Formula 11, 


S = 


a(n 1) 4(3io—l) 

r — 1 ~ 2 


= 118096, the sum. 


Example 3. Insert 6 proportional terms between 3 and 384 ? 
Given a = 3, b — 384, and n = 6+2 = 8. 

n-i 


Formula 7, 


then 


r — 



384 



= 2 , 


\/ « \/ 3 

3 : 6 : 12 : 24 : 48 : 96 : 192 : 384, the answer. 


Example 4. A man had 16 twenty dollar gold pieces, which he agreed to ex¬ 
change for copper in such a way, that he gets one cent on the first $20, two on 
the second, four on the third, and eight on the fourth, &c., &c.; until the sixteen 
$20 pieces were covered. How many cents will come on the sixteenth gold 
piece, and what will be the whole amount of copper on the gold? 


In the progression 1 : 2 : 4 : 8 : &c., we have, 
Given n — 16, r =2, and a = 1, then, 


6 = 


Formula 4. 
sixteenth piece. 

The total sum of cents will he found by the 

32768X2- 


2563 


= 32768 cents, on the 


Formula 10. 


S= 


2—1 


65535 cents = $655 - 35. 


Table of Geometrical Progression. 

The ratio r — 2. 


1 

1 

16 

32768 

31 

1073741824 

46 

35184372088832 

2 

2 

17 

65536 

32 

2147483648 

47 

70368744177664 

3 

4 

18 

131072 

33 

4294967296 

48 

140737488355328 

4 

8 

19 

262144 

34 

8589934592 

49 

281474976710656 

5 

16 

20 

524288 

35 

17179869184 

50 

562949953421312 

6 

32 

21 

1048576 

36 

34359738368 

51 

1135899906842624 

7 

64 

22 

2097152 

37 

68719476736 

52 

2271799813685248 

8 

128 

23 

4194304 

38 

137438953472 

53 

4543599627370496 

9 

256 

24 

8388608 

39 

274877906944 

54 

9087199254740992 

10 

512 

25 

16777216 

40 

549755813888 

55 

18174398509481914 

11 

1024 

26 

33554432 

41 

1099511627776 

56 

36348797018963968 

12 

2048 

27 

67108864 

42 

2199023255552 

57 

72697594037927936 

13 

4096 

28 

134217728 

43 

4198046511104 

58 

145395188075855872 

14 

8192 

29 

268435456 

44 

8796093022208 

59 

290790376151711744 

15 

16384 

30 

536870912 | 

45 

17592186044116 

60 

581580752303423488 


Any power of 2 can be found by this table, up to the 60th power. 

Example. The 10th power of 2, or 2 10 , = 1024. 2 8 = 256. 

The 9th root of 512 = 2. The 20th root of 1048576 = 2. 






























36 


Compound Interest. 


COMPOUND INTEREST. 

Compound Interest is when the interest is added to the capita] for each 
year, aud the sum is the capital for the following year. 


Amount, a=c(l+j>) n . . 1. 

Capital, c *= ---. . . 2. 


Percentage, 


»- 


.3. 


Number of years, n = IfLf - 4 . 

%•( 1 +P) 


(i +P) n 

JQGT’ In these formulas p must be expressed in hundredths. 

Example 1. A capital c = 8650 standing with compound interest, at p = 5 per 
cent. What will it amount to in n = 9 years ? 

Amount a = 8650 (1.05)® = 13419 dollars. 

Example 2. A man commenced business with c = 300 dollars; after n = 5 years 
he had a = 6875 dollars. At what rate did his money increase, and how soon will 
he have a fortune of 50000 dollars? 

The first question, or the percentage, will be answered by the formula 3. 

p = \/ — 5 — 1 = V 22.9166 — 1 = 0.87, or 87 per cent. 

. v 300 v 

The time from the commencement of business until the fortune is completed 
will be answered from the formula 4. 


n _ log. 50000■ 


■log.300 4.69897 — 2.47712 


log. 1.87 
or 8 years and 2 months. 


0.2720048 


■ 8.169 years, 


Compound Interest Table, calculated from formula 1. 


n 

Compound Interest. 

Years. 

5 per ct. 

6 per ct. 

7 per ct. 

1 

1.0500 

1.0600 

1.0700 

2 

1.1025 

1.1236 

1.1449 

3 

1.1576 

1.1910 

1.2250 

4 

1.2155 

1.2625 

1.3108 

5 

1.2770 

1.3382 

1.4025 

6 

1.3400 

1.4185 

1.5007 

7 

1.4071 

1.5036 

1.6058 

8 

1.4774 

1.5938 

1.7182 

9 

1.5513 

1.6895 

1.8385 

10 

1.6289 

1.7908 

1.9671 

11 

1.7103 

1.8983 

2.1048 

12 

1.7958 

2.0122 

2.2522 

13 

1.8856 

2.1329 

2.4098 

14 

1.9799 

2.2609 

2.5785 

15 

2.07S9 

2.3965 

2.7599 

16 

2.1829 

2.5403 

2.9522 

17 

2.2920 

2.6928 

3.15S8 

18 

2.4066 

2.8543 

3.3799 

19 

2.5269 

3.0256 

3.6165 

20 

2.6533 

3.2071 

3.8697 

21 

2.7859 

3.3995 

4.1406 

22 

2.9252 

3.6035 

4.4304 

23 

3.0715 

3.8197 

4.7405 

24 

3.2251 

4.0487 

5.0724 

25 

3.3864 

4.2919 

5.4274 

30 

4.3219 

5.7435 

7.6123 

35 

5.5166 

7.6861 

10.6766 

40 

7.0400 

10.2858 

14.9745 

45 

8.9850 

13.7646 

21.0025 

50 

11.6792 

18.4190 

29 4570 

60 

18.6792 

32.9878 

57.9466 


This table shows the value of one unit of 
money at the rates of 5, 6 and 7 per cent, per 
annum, compound interest, up to 60 years. 

Example 1. What is the amount of 864 pounds 
sterling for 12 years, at 6 per cent, compound 
interest? 

Table, 2.01219 x 864 = 1738.53216, or £1738 
10 s. 7.7 d. 

Example 2. What is the amount of 3450 dollars 
for IS years, at 5 per cent, compound interest? 
Table, 2.40661 X 3150 = 8302.80 dollars. 

When the interest is compounded in more or 
less than one year, at the rate of interest per 
year, and m = the number of months in which 
the interest is compounded; 

Then, instead of p in the formulas, put PLP . 

19m 12 

and instead of n, put -—* 

• m 

Example 3. A capital of 500 dollars bears 
compound interest semi-annually, at 5 per cent, 
per annum; what will it amount to in 10 years? 

m = 6 months, p — 7 P-P = — 0.025 

12 12 

and n = — 0 = 20, 

6 

then, a = c(l + p) n = 500(1’+ 0.025)20 = 

8193.11 dollars, the answer. 

log.( l + 0.025) = 0.0107239 

20 


Amount, 


log. 500 
8193.11: 


0.2144780 

2.69S9700 

2.9134480 





















Annuities. 


37 


ANNUITIES. 


Annuity is a certain sum of money to be paid at regular intervals. 

A yearly payment or annuity b is standing for n years; to find the whole amount 
a, at p per cent, interest. 


Amount, 


Amount, 


a = b n + -^(n + l)"j Simple Int. 

« = +i>) n — lj Comp. Int.. 


. 1 . 


2 . 


A yearly payment or annuity b is to be paid for n years; to find the present 
worth, or the amount a, which would pay it in full at the beginning of the time 
n, deducting/; per cent, interest. 

Amount , a= bn 1 1 — P(^-t^ \ 1 Simple Int. ... 3. 

Amount, a = -Tl-1—1 Comp. Int. .... 4. 

Pi (1 + P) n J 

A debt D, standing for interest, is diminished yearly by a sum b; to find the 
debt d after n years, and the time n when it is fully paid 1 

The debt d after n years will be— 

d = iR P~ &X 1 +P) n + & Comp. Int. ... 5. 

P 

The time n until fully paid will be— 


n = loff-b — loff-ib — Dp) . 
log.{I + p) 


. 6 . 


If b — D p, then n= oo, or the debt D will never bo paid. If b<^Dp, the debt 
D will be increased. 

To find the yearly annuity b which will pay a debt D in n years, at j; per cent, 
compouud interest ? 

.Dp(l +pf 

w — ' -- • • • • « • 4 . 

(l + p) n — l 


Annuity Table, 

Showing the present worth of an annuity or rent of one unit of money, at 5, 6 
and 7 per cent, compound interest for years up to CO, calculated from formula 4. 


Years. 

5 per ct. 

6 per ct. 

7 per ct. 

Years. 

5 per ct. 

6 per ct. 

7 per ct. 

1 

0.9524 

0.9434 

0.9345 

17 

11.2741 

10.4772 

9.7632 

2 

1.8594 

1.8333 

1.8080 

18 

11.6896 

10.8276 

10.0591 

3 

2.7232 

2.6730 

2.6243 

19 

12.0853 

11.1581 

10.3356 

4 

3.5459 

3.4651 

3.3872 

20 

12.4622 

11.4699 

10.5940 

5 

4.3295 

4.2123 

4.1001 

21 

12.8211 

11.7641 

10.8355 

6 

5.0757 

4.9173 

4.7 665 

22 

13.1630 

12.0416 

11.0612 

7 

5.7S64 

5.5824 

5.3892 

23 

13.4881 

12.3034 

11.2722 

8 

6.4632 

6.2098 

5.9712 

24 

13.7986 

12.5503 

11.4693 

9 

7.1078 

6.8017 

0.5152 

25 

14.0939 

12.7833 

11.6536 

10 

7.7217 

7.3601 

7.0235 

30 

15.3724 

13.7648 

12,4090 

11 

8.3064 

7.8868 

7.49S6 

35 

16.3742 

14.4982 

12.9476 

12 

8.8632 

8.3838 

7.9426 

40 

17.1591 

15.0463 

13,3317 

13 

9.3936 

8.8527 

8.3576 

45 

17.7741 

15.4558 

13.6055 

14 

9.8986 

9.2950 

8.7454 

50 

18.2559 

15.7618 

13.8007 

15 

10.3796 

9.7122 

9.1079 

55 

18.6334 

15.9905 

13.9399 

16 

10.8378 

10.1059 

9.4466 

60 

18,9292 

16.1014 

14.0389 





















38 


United States’ Standard Measures and Weights 


UNITED STATES’ STANDARD MEASURES AND WEIGHTS. 

MEASURE OF LENGTH. 

The Standard Measure, of Length is a brass rod = 1 yard at the temperature of 
32° Fahrenheit. The length of a pendulum vibrating seconds in vacuo, at 
Philadelphia is 1-08014 yards, at + 32° Fahrenheit. 

The Surveying Chain is —22 yards = G 6 feet. It consists of 100 links, 
and each link = 7-02 inches. 

ROPES AND CABLES. 

1 Cable length = 120 fathoms = 720 feet. 

1 fathom = 6 feet. 

GEOGRAPHICAL AND NAUTICAL MEASURES. 

1 Degree of the great circle of the Earth round the Equator = 69*032 statute 
miles = 60 Nautical miles. • 

1 Statute mile = 5280 feet = 0-S6 Q 75 Nautical miles. 

1 Nautical mile = 6037-424 = 1-150 Statute miles. 

LOG LINE. 

The Log Line should be about 150 fathoms long, and 10 fathoms from 
the Log to the first knot on the line. If half a minute glass is used, it will be 
• 51 feet between each succeeding knot. For 28 seconds glass it will be 47 -6 feet 
= 7-93 fathoms per knot. This is the length of knot by calculation, but prac¬ 
tically it is shortened to 7-5 fathoms per knot for 28 seconds glass. 

MEASURE OF CAPACITY. 

Gallon* The standard Gallon measures 231 cubic inches, and contains 
8-3388822 pounds Avoirdupois = 58372T757 grains Troy, of distilled water, at its 
maximum density 39-83° Fahrenheit, and 30 inches barometer height. 

Ruslicl* The standard Bushel measures 2150-42 cubic inches = 77"627413 
pounds Avoirdupois of distilled water at 39-83° Fahrenheit, barometer 30 inches. 
Its dimensions are I 85 inches inside diameter, 19£ inches outside, and 8 inches 
deep; and when heaped, the cone must not be less than 6 inches high,equal 
2747-70 cubic inches for a true cone. 

Pound. The standard Pound Avoirdupois is the weight of 27*7015 cubic 
inches of distilled water, at 39-83° Fahrenheit, barometer 30 inches, and 
weighed in the air. 


MEASURE OF LENGTH. 


Miles. 

Furlongs. 

Chains. 

Rods. 

Yai-ds. 

Feet. 

Inches. 

1 

8 

80 

320 

1760 

5280 

63360 

0125 

1 

10 

40 

220 

660 

7920 

0-0125 

01 

1 

4 

22 

66 

792 

0-003125 

0025 

025 

1 

5.5 

16-5 

198 

0 00056818 

0 0045454 

0-045454 

0-181818 

1 

3 

36 

0 00018939 

0 00151515 

001515151 

0-0606060 

0-33333 

1 

12 

0 000015783.0 000126262,0 001262626 

000505050 

,0-0277777 

0083333 

1 


MEASURE OF SURFACE. 


Sq. Miles. 

Acres 

S.Chains. 

Sq. Rods. 

Sq. Yards. 

Sq. Feet. 

Sq. Inches. 

1 

640 

G400 

102400 

3097600 

27878400 

4014489600 

0 001562 

1 

10 

160 

4840 

43560 

6272640 

0 0001562 

01 

1 

16 

484 

4356 

627264 

0000009764 

0-00625 

00625 

1 

30-25 

272 25 

39204 

0000000323 

00002066 

0002066 

00330 

1 

9 

1296 

0 0000000358 

0 00002296 

0-0002290 

0-00367 

0 Till 11 J 

1 

144 

000000000025 

0 000009159,0-00000159 jo-00002552 200007710 

000G944 

1 


























Measure of Capacity and Weights. 


39 


MEASURE OF CAPACITY. 


I 

> 


Cub. Yard. 

Bushel. 

Cub. Feet. 

Pecks. 

G a! Ions. 

Cub. inch. j 

1 

21-0962 

27 

100*987 

201*974 

46656 

0*03961 

1 

1-24445 

4 

9*30918 

2150*42 1 

0-037037 

0-803564 

1 

3*21425 

7*4805 

1728 

0*000259 

025 

0-31114 

I 

2-32729 ! 

537 605 : 


0-107421 , 

0 133681 

0*000547 

0*429684 

0-001860 

1 

0*004329 

231 



JL 


i MEASURE OF LIQUIDS. 

Gaily u. 

Quarts. 

Pints. 

Gills. 

Cub. inch. ! 

I 

4 

8 

32 

231 

025 

1 

2 

8 

67-75 

0*125 

0*5 

l 

4 

28-875 

0.03125 

0125 

0*25 

1 

7-21875 

0-004329 

0-017315 

003463 

0-1385S 

I 


MEASURES OF WEIGHTS. 

>* 



AVOIRDUPOIS. 


Ton. 

Cwt. 

Pounds. 

Ounces. 

Drams. 

1 

20 

2240 

35840 

573440 

0*05 

1 

112 

1792 

28672 

0*00044642 

0-0089285 

1 

16 

256 

0*00002790 

0-000558 

00625 

I 

16 

0*00000174 

0 0000348 ' 

0-0016 

0-0625 

1 

TIIOY. 

Pounds. 

Ounces. 

Dwt. 

Grains. 

• 

Pound Avoir. 

1 

12 

240 

5760 

0*822861 

0-0S3333 

1 

20 

480 

0*068571 

0*004166 

0*05000 

I 

24 

0-0034285 

0*0001736 

0-002083333 

0-0416666 

1 

0-00014285 

1*215275 

14-58333 

291-6666 

7000 

1 

APOTHECARIES. 

Pounds. 

Ounces. 

Drams. 

Scruples. 

Grains. 

1 

12 

96 

288 

6760 

0G8333 

1 

8 

24 

480 

0*01041606 

0*125 

I 

3 

60 

0*0034722 

0-0416666 

0-3333 

1 

20 

0*00017301 

0-0020833 

0-016666 

0-05 

1 


















































40 


Weight per Bushel op Grain, etc. 


WEIGHT PER BUSHEL OF GRAIN, ETC. 

The following table shows the number of pounds per bushel 
required by law or custom, in the sale of articles specified, in the 
several States of the Union. (Official.) 


States. 

Barley. 

Buckwheat.. 

Coal. 

Corn, shelled. 

Corn-meal. 

Onions. 

Oats. 

Potatoes. 

a; 

hH 

Wheat. 

c5 

m 

Turnips. 

Beans, whole. 

Clover-seed. 

Timothy. 

Maine. 

48 

48 


56 

50 

52 

30 

60 


60 


50 

64 



New Hampshire. 




56 

50 


30 

60 56 

60 



60 



Vermont. 

48 

48 




32 

60 56 

60 

70 


64 

60 

42 

Massachusetts. 

48 

48 


56 

50 

52 

32 

60 56 

GO 






Connecticut. 


45 


56 



28 

00 50 

56 






New York. 

48 

48 

• • • 

58 



32 

60 

56 

60 

... 

... 

62 

60 

44 

New Jersey. 

48 

50 


56 



30 

60 

fjf) 

60 




64 


Pennsylvania. 

47 

48 


56 



30 

56 

56 

60 

85 



62 


Delaware. 




56 





60 






Maryland. 

48 

48 

• • • 

56 


57 

32 

60 56 

60 

56 

... 

62 

64 

45 

Dist. Columbia. 

47 

48 

« • • 

56 

48 

57 

32 

56 56160 

50 

55 

62 

60 

45 

■Virginia. 

48 

48 

... 

56 

50 

... 

32 

60 

56 

60 


56 60 

64 

45 

West Virginia. 

48 

52 

80 

56 

48 

... 

32 

60 

50 

60 


60 

60 

60 

45 

North Carolina. 

48 

50 


54 

46 


30 


56 

60 




64 


South Carolina. 

48 

56 

80 

56 

50 

57 

33 

60 

56 

60 

50 

• •• 

60 

60 


Georgia. 

40 

... 

80 

56 

48 

75 

35 

56 

... 

60 

56 



60 

45 

Louisiana. 

32 



56 



32 


32 

60 






Arkansas. 

48 

52 

80 

56 

50 

57- 

32 

60 

56 

60 

50 


60 

60 

45 

Tennessee. 

48 

50 

... 

56 

50 

56 

32 

60 

56 

60 

... 

... 

60 

.. 

45 

Kentucky. 

48 

52 

... 

56 

50 

57 

33 

56 

56 

60 

50 

... 

60 

60 

45 

Ohio. 

48 

50 

... 

56 

... 

... 

32 

60 

56 

60 

... 

... 

60 

60 

45 

Michigan. 

48 

48 

80 

56 

... 

54 

32 

60 

56 

60 

56 

58 

60 

60 

45 

Indiana. 

48 

50 

70 

56 

50 

48 

32 

60 

56 

60 

50 

... 

60 

60 


Illinois. 

48 

52 

... 

56 

48 

57 

32 

60 

56 

60 

50 


60 

60 

• • • 

Wisconsin. 

48 

50 

... 

56 



32 

60 

56 60 




60 


Minnesota. 

48 

42 


56 



32 

60 

56 60 




60 


Iowa. 

48 

52 

... 

56 

... 

57 

33 

60 

56 60 

50 

... 

60 

60 

45 

Missouri. 

48 

52 


56 

... 

57 

32 

60 

5660 

50 


60 

60 

45 

Kansas. 

50 

50 

... 

56 

50 

57 

,32 

60 

56 60 

50 

155 

60 


45 

Nebraska. . 

48 

52 

... 

56 

50 

57 

34 

60 

56 [60 

50 

55 

60 

60 

45 

California. 

50 

40 


52 



'32 


154 60 






Oregon. 

46 

42 

• • • 

56 

... 

... 

36 

60 

5660 




60 































































































































Wages per Year, Month, Week, and Day. 


41 


Year. 

Month. 

Week. 

Day. 

Year. 

Month. 

Week. 

Day. 

$100 

$8.33 

$1.92 

$0.33 

$560 

$46.67 

$10.56 

$1.87 

110 

9.17 

2.12 

0.37 

570 

47.50 

10.96 

1.90 

120 

10.00 

2.31 

0.40 

580 

48.30 

11.15 

1.93 

130 

10.83 

2.50 

0.43 

590 

49.17 

11.34 

1.97 

140 

11.67 

2.70 

0.47 

600 

50.00 

11.53 

2.00 

150 

12.50 

2.89 

0.50 

610 

50.S3 

11.73 

2.03 

160 

13.33 

3.08 

0.53 

620 

5 J .67 

11.93 

2.07 

170 

14.17 

3.27 

0.57 

630 

52.50 

12.12 

2.10 

180 

15.00 

3.46 

0.60 

640 

53.33 

12.31 

2.13 

190 

15.83 

3.65 

0.63 

650 

54.17 

12.50 

2.17 

200 

16.67 

3.84 

0.67 

660 

55.00 

12.70 

2.20 

210 

17.50 

4.04 

0.70 

670 

55.83 

12.88 

2.23 

220 

18.33 

4.23 

0.73 

680 

56.67 

13.07 

2.27 

230 

19.17 

4.42 

0.77 

690 

57.50 

13.26 

2.30 

240 

20.00 

4.61 

0.80 

700 

58.33 

13.45 

2.33 

250 

20.83 

4.80 

0.83 

710 

59.17 

13.64 

2.37 

260 

21.57 

5.00 

0.87 

720 

60.00 

13.84 

2.40 

270 

22.50 

5.19 

0.90 

730 

60.83 

14.03 

2.43 

2S0 

23.33 

5.38 

0.93 

740 

61.67 

14.23 

2.47 

290 

24.17 

5.57 

0.97 

750 

62.50 

14.42 

2.50 

300 

25.00 

5.76 

1.00 

760 

63.33 

14.60 

2.53 

310 

25.83 

5.95 

1.03 

770 

64.17 

14.80 

2.57 

320 

26.67 

6.14 

1.07 

780 

65.00 

15.00 

2.60 

330 

27.50 

6.34 

1.10 

790 

65.83 

15.19 

2.63 

340 

28.33 

6.53 

1.13 

800 

66.67 

15.38 

2.67 

350 

29.17 

6.72 

1.17 

810 

67.50 

15.57 

2.70 

360 

30.00 

6.91 

1.20 

820 

68.33 

15.77 

2.73 

370 

30.83 

7.10 

1.23 

830 

69.17 

15.96 

2.77 

380 

31.67 

7.30 

1.27 

840 

70.00 

16.15 

2.30 

390 

32.50 

7.49 

1.30 

850 

70.83 

16.34 

2.83 

400 

33.33 

7.68 

1.33 

860 

71.67 

16.54 

2.87 

410 

34.17 

7.87 

1.37 

870 

72.50 

16.73 

2.90 

420 

35.00 

8.06 

1.40 

S80 

73.33 

16.92 

2.93 

430 

35.83 

8.25 

1.43 

890 

74.17 

17.11 

2.97 

440 

36.67 

8.44 

1.47 

900 

75.00 

17.30 

3.00 

450 

37.50 

8.64 

1.50 

910 

75.83 

17.49 

3.03 

460 

38.33 

8.83 

1.53 

920 

76.67 

17.68 

3.07 

470 

39.17 

9.02 

1.57 

930 

77.50 

17.87 

3.10 

480 

40.00 

9.23 

1.60 

940 

78.33 

18.06 

3.13 

490 

40.83 

9.42 

1.63 

950 

79.17 

18.25 

3.17 

500 

41.67 

9.61 

1.67 

960 

80.00 

18.44 

3.20 

510 

42.50 

9.80 

1.70 

970 

80.83 

18.64 

3.23 

520 

43.33 

10.00 

1.73 

9S0 

81.67 

18.83 

3.27 

530 

44.17 

10.19 

1.77 

990 

82.50 

19.03 

3.30 

540 

45.00 

10.38 

1.80 

1000 

83.33 

19.23 

3.33 

550 

45.83 

10.57 

1.83 

1010 

84.17 

19.42 

3.37 


When the salary per yoar exceeds hundreds and is less than $10,000, 
then move the decimal-point one space to the right in the columns 
Months, Weeks, and Days,- for instance, if the salary is $4200 per year, 
then it will be $350 per month, $80.00 per week, and $14 per day. 


































42 


Money. 


MONEY AND COINS OF THE UNITED STATES. 

10 mills = 1 cent. I 10 dimes = 1 dollar. 

10 cents = 1 dime. | 10 dollars = 1 eagle. 

The standard gold and silver coins contain 900 parts of pure metal and 100 parts 
of base metal in 1000 parts of the alloy. 

The remedy of the Mint is the allowance for deviation from the exact standard 
fineness and weight of coins. 

The nickel cent contains 88 parts of copper and 12 of nickel. 

The new bronze cent contains 95 parts of copper and 5 of tin and zinc. 

Pure gold, 23.22 grains = $1, or $20.67.183 = 1 ounce. 

Pure silver, 357.03 grains = $1, or $1.36.166 = 1 ounce. 

Silver coins of less value than one dollar are issued at the rate of 384 grains to 
the dollar. 

Standard alloyed gold = $18.60.465, and silver = $1.22.5 per ounce. 


Gold coins. Grains. Silver coins. . Grains. 


Double eagle,. 
Eagle, . 

Dollar, . 


516. 

258. 

25.8 


One dollar, . 
Fifty cents,. 
Twenty-five cents, 


For silver and gold tables see pages 000. 


412.5 

192. 

96. 


Copper coins. 

Cent (old), . 
Cent (new), 
Cent(bronze), 


Grains. 

. 168. 
72. 

. 48. 


WEIGHT AND FINENESS OF DIFFERENT COINS, AND 
THEIR VALUE IN AMERICAN MONEY. 


Country. 

Austria, . 

Baden,. . 
Belgium, 
Brazil,. . 
Canada, . 
China, . . 

Chili,. . 

Denmark, 
England, 
East Indies 
France, . 
Greece, . 
Hamburg, 

Holland, . 

Italy, . 

Mexico, . 

Norway,. 
Peru, . . 

Portugal, 

Prussia, . 
Rome, . 

Russia, . 

Spain, . 

Sweden, . 
Turkey, . 


{ 

{ 

{ 

{ 

{ 

{ 

i 


Piece and Divisions. 

Crown,. 

Florins, .... 

Ducat,. 

25 Francs, .... 
2000 Reis, . 

20 Cents, 1851, . . . 

Tael. 

10 Pesos, 1855, . . . 

1 Peso, 1854-6, 

2 Rix dollars, 

Pound sterling = 20 shillings, 
Company’s Rupee,. 

Napoleon, 20 Francs,. . 

20 Drachms, . . . . 

Rix Dollar, . . 

Ducat,. 

Florin, .... 

20 Lire. 

Doubloon = 8 Escudos, . 
Peso = 8 Reals, 

2 Rigsdaler, . . . 

1 Sol = 100 Centavos, . . 

Corona (Crown), 1838, . 

1000 Keis, . . . . 

Thaler, .... 

2.5 Scudi = 250 Bajochi, 
Imperial = 5 Roubles, 
Rouble silver = 100 Copecks, 
100 Beals, . 

80 Reals = 4 Dollars, . 

Ducat,. 

Rix Dollar =100 Ore, 
Piastres, 1845, . . . 


Weight. 

Grains. 

171.36 

190.56 

47.5 

121.92 

393.6 

96.0 


236.16 
384.48 
444.96 
123.21 
180. 

99.5 

88.8 

450. 

53.75 

50. 

99.36 

416.4 

415.68 

444.96 

385.82 

147.84 

45.6 
268.46 

67.2 

100.8 

320.16 
128.64 

103.2 
53. 

112.3 

110.88 


Fineness 
in 1000 . 

900. 

900. 

987. 

899. 

918.5 

925. 


900. 

900.5 
877. 

916.5 
892. 
898. 

900. 
860. 
9S2. 
787. 
898. 

870.5 

901. 
877. 
900. 
912. 
912. 
900. 
900. 
916. 
875. 
896. 

869.5 
979. 
873. 
900. 


Unitkd 

States. 

$ Cts. 

6.64.19 

0.48.63 

2.00.70 

4.72.03 

1.02.53 

0.18.87 

1.43.00 

9.15.35 

0.98.17 

1.10.65 
4.86.34 

5.10.49 
3.85.00 

3.44.29 

1.17.66 
2.29.7 

1.69.30 
3.84.26 

15.61.05 

1.06.20 

1.10.65 
0.95.41 

5.80.66 
1.18.00 
0.72.89 
2.60.47 
3.97.64 
0.79.44 
4.96.39 
3.86.44 

2.23.50 
0.26.10 
4.36.93 


Fine 

metal. 

Grains. 

154.22 

171.5 
46.9 

109.6 
361.5 

88.8 


212.5 

346.2 

390.2 
112.9 

16.5 

87.4 

80.9 

397.5 
52.77 
39.32 
89.22 

362.5 

374.5 

390.2 
347.24 

134.8 

41.6 

243.6 

60.5 
92.3 

2 5 6.8 

115.2 
89.73 

51.9 
97.15 
99.79 






— 























Foreign Money, 


43 


THE CURRENCY OF DIFFERENT COUNTRIES COMPARED 
WITH ENGLISH AND AMERICAN MONEY. 


Engl’nd. 

France. 

Belgi’m. 

Sw’land. 

Prussia. 

Austria. 

(in notes.) 

Sweden. 

Ger¬ 

many. 

Russia, 
(in paper) 

Ham¬ 

burg. 

U. S. 

£ 

e. 

d. 

Frs. 

Cts. 

Th. 

Sgr. 

PC. 

FI. 

Kr. 

Rix. Ore. 

Fl. 

Kr. 

Rhl. 

Kop. 

Mrk 

Sell. 

$ Cts. 

0 

0 

1 

0 

l(|i 

0 

0 

10 

0 

5 

0.07 

0 

3 

0 

3 

0 

0 

0.02 

0 

0 

2 

0 

21 

0 

1 

8 

0 

10 

0.14 

0 

6 

0 

5 

0 

2 

0.04 

0 

0 

3 

0 

32 

0 

2 

6 

0 

16 

0.21 

0 

9 

0 

8 

0 

2 f 

0.06 

0 

0 

4 

0 

42 

0 

3 

4 

0 

21 i 

0.28 

0 

12 

0 

12 

0 

2 £ 

0.08 

0 

0 

5 

0 

53 

0 

4 

2 

0 

27 

0.36 

0 

15 

0 

16 

0 

H 

0.10 

0 

0 

6 

0 

64 

0 

5 

1 

0 

31 

0.44 

0 

18 

0 

19 

0 

51 

0.12 

0 

0 

7 

0 

74 

0 

5 

11 

0 

364 

0.51 

0 

21 

0 

22 

0 

6 * 

0.14 

0 

0 

8 

0 

85 

0 

6 

10 

0 

42 j 

0.59 

0 

24 

0 

26 

0 

7 

0.16 

0 

0 

9 

0 

96 

0 

7 

7 

0 

47 i 

0.66 

0 

27 

0 

27 

0 

8 

0.18 

0 

0 10 

1 

6 

0 

8 

6 

0 

53 

0.73 

0 

30 

0 

33 

0 

8 # 

0.20 

0 

0 

11 

1 

16 

0 

9 

5 

0 

57£ 

0.80 

0 

34 

0 

36 

0 

n 

0.22 

0 

1 

0 

1 

27 

0 

10 

3 

0 

62 

0.89 

0 

36 

0 

39 

0 

ii 

0.24 

0 

o 

w 

0 

2 

55 

0 

20 

6 

1 

25 

1.78 

1 

13 

0 

79 

1 

6 

0.48 

0 

3 

0 

3 

82 

1 

0 

9 

1 

87 

2.67 

1 

49 

1 

18 

2 

1 

0.72 

0 

4 

0 

5 

10 

1 

10 

11 

2 

50 

3.56 

2 

24 

1 

58 

2 

12 

0.96 

0 

5 

0 

6 

36 

1 

21 

3 

3 

12 

4.45 

2 

59 

1 

97 

3 

7 

1.21 

0 

6 

0 

7 

64 

2 

1 

6 

3 

74 

5.34 

3 

38 

2 

37 

4 

2 

1.45 

0 

7 

0 

8 

92 

2 

11 

9 

4 

36 

6.23 

4 

12 

2 

77 

4 

12 

1.69 

0 

8 

6 

10 

20 

2 

22 

0 

4 

95 

7.12 

4 

47 

3 

18 

5 

7 

1.93 

0 

9 

0 

11 

46 

3 

2 

0 

5 

58 

8.09 

5 

22 

3 

58 

6 

2h 

2.18 

0 10 

0 

12 

72 

3 

12 

4 

6 

25 

8.90 

5 

58 

3 

94 

6 

13 i 

2.42 

0 11 

0 

13 

99 

3 

22 

6 

6 

87 

9.79 

6 

34 

4 

38 

7 

8 ? 

2.66 

0 

12 

0 

15 

27 

4 

2 

9 

7 

49 

10.68 

7 

11 

4 

75 

8 

3.1 

2.90 

0 

13 

0 

16 

55 

4 

13 

0 

8 

12 

11.57 

7 

46 

5 

15 

8 

14 j 

3.14 

0 14 

0 

17 

84 

4 

23 

3 

8 

75 

12.66 

8 

24 

5 

55 

9 

9a 

3.39 

0 

15 

0 

19 

8 

5 

3 

5 

9 

37 

13.45 

8 

57 

5 

96 

10 

4i 

3.63 

0 

16 

0 

20 

40 

5 

13 

8 

10 

0 

14.24 

9 

33 

6 

35 

10 

15£ 

3.87 

0 17 

0 

21 

66 

5 

23 

11 

10 

65 

15.13 

10 

9 

6 

74 

11 

10 * 

3.12 

0 18 

0 

22 

92 

6 

4 

2 

11 

28 

16.02 

10 

46 

7 

14 

12 

5 a 1 

4.36 

0 

19 

0 

24 

18 

6 

14 

4 

11 

88 

17.01 

11 

21 

7 

44 

13 

Oh 

4.60 

1 

0 

0 

25 

45 

6 

24 

6 

12 

50 

17.80 

11 

57 

7 

88 

13 

9 

4.84 

o 

0 

0 

50 

90 

13 

19 

0 

25 

0 

35.60 

23 

54 

15 

77 

27 

2 

9.68 

3 

0 

0 

76 

35 

20 

13 

6 

37 

50 

53.40 

35 

51 

23 

95 

40 

11 

14.52 

4 

0 

0 

101 

80 

27 

8 

0 

50 

0 

71.20 

47 

48 

31 

54 

54 

4 

17.36 

5 

0 

0 

127 

25 

34 

3 

0 

62 

50 

89.00 

59 

46 

39 

42 

67 

11 

24.20 

6 

0 

0 

152 

70 

40 

27 

6 

75 

0 

106.80 

71 

42 

47 

31 

81 

4 

29.04 

7 

0 

0 

178 

15 

47 

22 

6 

87 

50 

124.60 

83 

39 

55 

20 

94 

13 

33.88 

8 

0 

0 

202 

60 

54 

16 

6 

100 

0 

142.40 

95 

36 

63 

9 

108 

6 

38.72 

9 

0 

0 

229 

5 

61 

11 

6 

112 

50 

160.20 

107 

34 

70 

96 

121 

15 

43.56 

10 

0 

0 

254 

50 

68 

6 

0 

125 

0 

178.00 

119 

30 

78 

84 

135 

8 

48.40 


The mark of Finland is equal to the French franc. 


Carat. 

DIAMOND. 

Grain. 

Parts. 

Grains (Troy). 

1 . 

4. 

64 

3.2 

0.25 

1 . 

16 

0.8 

0.015625 

0.0625 

1 

0 05 

0.3125 

12.5 

20 

1 . 





















































































44 Rule Measure. 


Conversion of Indies anil 

Eigliltos into 

Decimals of a Foot. 




Fractions 

of an Inch. 



Inches. 

0 

1 

i 

¥ 

3 

J 

i 

5 

8 

3 

¥ 

\ 

0 

.0000 

.01041 

.02083 

.03125 

.04166 

.05208 

.0625 

.07291 

1 

.08333 

.09375 

.10416 

.11458 

.125 

.13541 

.14588 

.15639 

2 

.16666 

.17707 

.1875 

.19792 

.20832 

.21873 

.22914 

.23965 

3 

.25 

.26041 

.270 

.28125 

.29166 

.30208 

.3125 

.32291 

4 

.33333 

.34375 

.35416 

.364 

.375 

.38541 

.39588 

.40639 

5 

.41666 

.42707 

.437 

.44792 

.45832 

.46873 

.47914 

.48965 

6 

.5 

.51041 

.520 

.53125 

.54166 

.55208 

.5625 

.57291 

7 

.58333 

.59375 

.60416 

.614 

.625 

.63541 

.64588 

.65639 

8 

.66666 

.67707 

.685 

.69792 

.70832 

.71773 

.72914 

.73965 

9 

.75 

..76041 

.770 

.78125 

.79169 

.80208 

.8425 

.82291 

10 

.83333 

.84375 

.85416 

.864 

.875 

.8S541 

.89588 

.90039 

11 

.91666 

.92707 

.937 

.94792 

.95832 

.96873 

.97914 

.98965 

12 

1 foot. 

foot. 

foot. 

foot. 

foot. 

foot. 

foot. 

foot. 

T 6 ' n 

= 0.0U5208 ft 

.; sV in. = 0.00265 ft. 

i' 1 * 

= 0.001375 ft. 

Angle Measurement toy tlie Opening of a Tivo-foot Rule. 

Opening 



Fractions of an Inch. 



Rule. 

0 

1 

8 

i 

¥ 

3 

ST 

l 

? 

5 

% 

3 

¥ 

7 

8 

Inch’s. 

o / 

O ' 

O f 

o / 

o / 

O / 

O f 

o / 

0 

00 00 

0 36 

1 12 

1 47 

2 23 

2 59 

3 35 

4 11 

1 

4 46 

5 22 

5 59 

6 34 

7 10 

7 46 

8 22 

8 58 

2 

9 34 

10 10 

10 46 

11 22 

11 58 

12 34 

13 10 

13 4G 

3 

14 22 

14 58 

15 34 

16 10 

16 46 

17 22 

17 59 

18 35 

4 

19 12 

19 48 

20 24 

21 0 

21 37 

22 13 

22 50 

23 27 

5 

24 3 

24 39 

25 16 

25 53 

26 30 

27 7 

27 44 

28 21 

6 

28 58 

29 35 

30 12 

30 49 

31 26 

32 3 

32 40 

33 17 

7 

33 54 

34 33 

35 8 

35 46 

36 25 

37 3 

37 40 

38 18 

8 

38 56 

39 34 

40 12 

40 50 

41 29 

42 7 

42 46 

43 24 

9 

44 4 

44 42 

45 21 

45 59 

46 38 

47 17 

47 56 

48 35 

10 

49 15 

49 54 

50 34 

51 13 

51 53 

52 33 

53 13 

53 53 

11 

54 34 

55 14 

55 55 

56 35 

57 16 

57 57 

58 38 

59 19 

12 

60 0 

60 41 

61 23 

62 5 

62 47 

63 28 

04 10 

64 52 

13 

65 35 

66 18 

67 1 

67 44 

68 28 

69 12 

69 55 

70 38 

14 

71 20 

72 6 

72 51 

73 35 

74 21 

75 6 

75 51 

76 36 

15 

77 20 

78 8 

78 54 

79 40 

80 27 

81 14 

82 1 

82 49 

16 

83 38 

84 26 

85 14 

86 3 

86 52 

87 41 

88 31 

89 21 

17 

90 12 

91 3 

91 55 

92 41 

93 39 

94 31 

95 34 

96 17 

18 

97 11 

98 5 

99 0 

99 55 

100 51 

101 47 

102 44 

103 42 

19 

104 40 

105 39 

105 39 

107 40 

108 41 

109 43 

110 46 

111 49 

20 

112 53 

113 58 

115 4 

116 11 I 

117 20 

118 30 

119 41 

120 53 

Conversion of Vulgar Fractions into Decimals. 


Fract ns. 

Decimals. 

Fract ns. 

Decimals. 

Fract’ns. 

Decimals. 

Fract’ns. 

Decimals. 

1:2 

.5 

1 :16 

.0625 

1:32 

.03125 

1:64 

.015625 

1:3 

.33333 

3: 1 G 

.1875 

3:32 

.09375 

3:64 

.046875 

2:3 

.66666 

5:16 

.3125 

5:32 

.15625 

5:64 

.078125 

1:4 

.25 

7:16 

.4375 

7:32 

.21875 

7:64 

.109375 

3:4 

.75 

9:16 

.5625 

9:32 

.28125 

9:64 

.140625 

1 :5 

.2 

11 :16 

.6875 

11:32 

.34375 

11:64 

.171875 

3:5 

.6 

13:16 

.8125 

13:32 

.40625 

15:64 

.234375 

1:6 

.16666 

15:16 

.9375 

15:32 

.4687 5 

19:64 

.296875 

5 :6 

.83333 

1:24 

.04166 

17 :32 

.53125 

23:64 

.359375 

1:8 

.125 

5:24 

.20833 

19:32 

.59375 

27:64 

.421875 

3:8 

.375 

7:24 

.20166 

21:32 

.65625 

31:64 

.484375 

5:8 

.625 

11:24 

.45833 

23:32 

.71875 

35:64 

.546875 

7:8 

.875 

13:24 

.54166 

25:32 

.78125 

39: 64 

.609375 

5:12 

.41666 

17:24 

.70833 

27:32 

.84375 

43:64 

.671875 

7:12 

.58333 

19:24 

.79166 

39:32 

.90625 

57:64 

.891625 

11:12 

.925 

23:24 

.95833 

31:32 

.96875 

61:64 

.953125 
































































f 


Metrical System, 


45 


To Determine an Angle by tlie Aid of a Two-foot Iinle. 

b — opening of the rule in inches; 
v = angle formed by the rule; 

Sin. ^ ; and b = ‘24 sin. \v. 

Example 1. IIow much (6 = ?) must a two-foot rule be opened to form an angle 
of 48° 40' ? 

b = 24 X sin. 24° 20' = 24 X 0.412 = 9.8S8 inches. 

Example 2. A two-foot rule is opened to b = 8 inches. Required the angle 
formed by the rule. 

Sin. iv = — = 0.3333 = sin. 19° 30'. and v = 39°. 

24 

THE FRENCH METRICAL SYSTEM. 

The French units of weight, measure and coin are arranged into a perfect deci¬ 
mal system, except those of time and the circle. The division and multiplication 
of the units are expressed by Latin and Greek names, as follow: 


Latin , Division. 

Milli = 1000th of the unit. 

Ceuti = 100th of the unit. 

Deci = 10th of the unit. 

Metre, Litre, Stere, Are, Franc, Gramme. 


Greek , Multiplica lion. 
Deca = 10 times the unit. 
Hecato = 100 times the unit. 
Kilio = 1000 times the unit. 
Myrio = 10000 times the unit. 


French Measure of Length. 


1 Millimetre = 0.03937079 inches. 

1 Centimetre = 0.3937079 inches. 

1 Decimetre = 3.937079 inches. 

1 Metre (unit) = 39.37079 inches. 

1 Sea mile or ] 
knot 

1 Kilometre = 0.541343 sea miles. 


•}= 


1.8472 kilometre. 


1 Metre (unit) = 3.280899 feet. 

1 Decametre = 32.80899 feet. 

1 Hectometre = 328.0899 feet 
1 Kilometre = 32S0.899 ft. •- * 0.62138 

mile. 

1 Statute mile = 1.609315 kilometres. 

1 Kilometre = 49.7106 chains. 


1 Sq. metre = 10.7643 square feet. 
1 Are = 100 square metres. 

1 Decare =10 ares. 

1 Hectare = 100 ares. 


French Measure of Surface. 

1 Are = 1076.43 square feet. 

1 Decare = 107.643 square feet. 

1 Hectare = 2.47114 Eng. acres. 

1 Sq. mile = 258.989 hectares. 


French Measure of Volume. 


1 Stere (cubic) 10decastere8> 


metre) 

1 Stere 
1 Litre 
1 Decistere 


1 Ton 


; 


= 1000 litres. 

= 1 cubic decimetre. 
= 3.53166 cubic feet. 


1 Stere = 35.3166 Eng. cubic feet. 

1 Litre = 61.0271 Eng. cubic inches. 

1 Gallon = 3.7852 litres. 

1 Decistere = 2.84 bushels. 


French Measure of Weight. 


dis- 


1 cubic metre 
tilled water. 

1000 kilogrammes. 
1000 grammes. 

100 grammes. 

10 grammes. 

1 cubic centimetre 
distilled water. 

1 French ton = 0.984274 Eng. tons. 


Ton 

Kilogramme 
Hectogramme 
Decagramme 


1 Gramme 


1 Gramme = 10 decigrammes. 

1 Decigramme = 10 centigrammes. 

1 Centigramme = 10 milligrammes. 

1 Kilogramme = ‘2.2047 pounds avoir¬ 
dupois. 

1 Eng. pound = 0.45358 kilogrammes 


Gramme 
English ton 


15.43315 grains troy. 
1.01598 French tons, 


French Coin. 

1 Franc 100 centimes = 19.06 cents of an American dollar. 









A TABLE OF UNITS FOR ENGLISH LONG MEASURE. 




46 


English Long Measures. 


Barley¬ 

corns. 

570240 

220176 

© 

oo 

© 

o 

03 

r—t 

23760 

2376 

594 

to 

rH 

Cl 

108 

36 

23.76 

12 

CO 

rH 

nches. 

o 

CO 

o 

o 

CS 

CO 

03 

CO 

co 

63360 

7920 

792 

198 

Cl 

l- 

to 

CO 

12 

7.92 



h|co 

>—< 

t—H 

r- 











a 

a 

o 

Cl 

no 

I — 

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co 

00 

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r 

03 

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0861 

198 

© 

18 

© 

CO 

oo 

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rH 

—H* 

co 

co 

CO 

© 

M 

Tfl 

rH 

— 







rH 



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no 


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25 

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r-H 



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rH 


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oo 

Feet. 

1584( 

6116 

5280 

099 

66 

Hn 

to 

1—t 

© 

CO 

rH 

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co 

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ci 

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Metric System. 


47 


A TABLE OF METRIC UNITS OF 

LENGTH. 

Kilo¬ 

metre. 

Hecto¬ 

metre. 

Deca¬ 

metre. 

Metre. 

Deci¬ 

metre. 

Centi¬ 

metre. 

Milli¬ 

metre. 

1 

0.1 

0.01 

0.001 

0.0001 

0.00001 

0.000001 

10 

1 

0.1 

0.01 

0.001 

0.0001 

0.00001 

100 

10 

1 

0.1 

0.01 

0.001 

0.0001 

1,000 

100 

10 

1 

0.1 

0.01 

0.001 

10,000 

1,000 

100 

10 

1 

0.1 

0.01 

100,000 

10,000 

1,000 

100 

10 

1 

0.1 

1,000,000 

100,000 

10,000 

1,000 

100 

10 

1 


These two tables represent a fair comparison of the English and French 
systems of measurement. 

In a lecture on “Wave Theory of Light,” delivered at the Academy of 
Music, Philadelphia, Sept. 29, 1884, Sir William Thomson said, “I look upon 
our English system as a wickedly brain-destroying piece of bondage under 
which we suffer. J say this seriously. I do not think any one knows how 
seriously I speak of it.” 

The English table is a heap of rubbish unworthy the title of system, 
whilst the French table exhibits a regular and simple system. 

The metric system is now generally used in European scientific and tech¬ 
nical publications, for which the following tables have been calculated, to 
euable the reader to convert quantities from one system to the other. 


















48 


Feet and Metres. 


Conversion of" English Indies into Centimetres. 


Inch’s 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


Ct.mt. 

Ct.mt-. 

Ct.mt. 

Ct.mt. 

Ct.mt. 

Ct.mt. 

Ct.mt. 

Ct.mt. 

Ct.mt. 

Ct.mt . 

0 

0.000 

2.540 

5.080 

7.620 

10.16 

12.70 

15.24 

17.78 

20.32 

22.86 

10 

25.40 

27.94 

30.48 

33.02 

35.56 

38.10 

40.64 

43.18 

45.72 

4S.26 

20 

50.S0 

53.34 

55.88 

58.42 

60.96 

63.50 

66.04 

68.58 

71.12 

73.66 

30 

76.20 

78.74 

81.28 

83.82 

86.36 

88.90 

91.44 

93.98 

96.52 

99.06 

40 

101.60 

101.14 

106.68 

109.22 

111.76 

114.30 

116.84 

119.38 

121.92 

124.46 

50 

127.00 

129.54 

132.08 

134.62 

137.16 

139.70 

142.24 

144.78 

147.32 

149.86 

60 

152.40 

154.94 

157.48 

160.02 

162.56 

165.10 

167.64 

170.18 

172 72 

175.26 

70 

177.80 

180.34 

182.88 

185.42 

187.96 

190.50 

193.04 

195.58 

198.12 

200.96 

80 

203.20 

205.74 

208.28 

210.82 

213.36 

215.90 

218.44 

220.98 

223.52 

226.06 

90 

228.60 

231.14 

233.68 

236.22 

238.76 

241.30 

243.84 

246.38 

248.92 

251.46 

10) 

254.00 

256.54 

259.08 

261.62 

264.16 

266.70 

269.24 

271.78 

274.32 

276.85 


Conversion of Centimetres 

into English Inches. 


flt. mt. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


Inches. 

Inches. 

Inches. 

Inches.- 

Inohes. 

Inches. 

Inches. 

Inches. 

Inches. 

Inches. 

0 

0.000 

0.394 

0.787 

1.181 

1.575 

1.969 

2.362 

2.756 

3.150 

3.543 

10 

3.937 

4.33 L 

4.742 

5.118 

5.512 

5.906 

6.299 

6.693 

7.087 

7.480 

20 

7.S74 

8.268 

8.662 

9.055 

9.449 

9.843 

10.236 

10.630 

11.024 

11.418 

30 

11.811 

12.205 

12.599 

12.992 

13.386 

13.780 

14.173 

14.567 

14.961 

15.355 

40 

15.748 

16.142 

16.536 

16.S29 

17.323 

17.717 

18.111 

18.504 

18.898 

19.292 

50 

19.685 

20 079 

20.473 

20.867 

21.260 

21.654 

22.048 

22.441 

22.835 

23.229 

60 

23.622 

24.016 

24.410 

24.804 

25.197 

25.591 

25.9S5 

26.378 

26.772 

27.166 

70 

27.560 

27.953 

28.347 

28.741 

29.134 

29.528 

29.922 

30.316 

30.709 

31.103 

80 

31.497 

31.890 

32.284 

32.678 

33.071 

33.465 

33.859 

34.253 

34.646 

35.040 

90 

35.434 

35.827 

36.221 

36.615 

37.009 

37.402 

37.796 

38.190 

38.583 

38.977 

100 

39.370 

39.764 

40.158 

40.552 

40.945 

41.339 

41.733 

42.126 

42.520 

42.914 


Conversion of English Feet into Metres. 


Feet. 

0 

1 

2 

3 

4 

5 

G 

7 

8 

9 

0 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

Met. 

0.000 
3.0479 
6.0359 
9.1438 
12.192 
15.239 
18 287 
21.335 
24.383 
27.431 
30.479 

Met. 

0.3048 

3.3527 

6.4006 

9.44.86 

12.496 

15.544 

18.592 

21.640 

24.688 

27.736 

30.784 

Met. 

0.6096 

3.6575 

6.7055 

9.7534 

12.801 

15.849 

18.897 

21.945 

24.993 

28.041 

31.089 

Met. 

0.9144 

3.9623 

7.0102 

10.058 

13.106 

16.154 

19.202 

22.250 

25.298 

28.346 

31.394 

Met. 

1.2192 

4.2671 

7.3150 

10.363 

13.411 

16.459 

19.507 

22.555 

25.602 

28.651 

31.698 

Met. 

1.5239 

4.5719 

7.6198 

10.668 

13.716 

16.763 

19.811 

22.859 

25.907 

28.955 

32.003 
—■- 

Met. 

1.8287 

4.8767 

7.9246 

10.972 

14.020 

17.068 

20.116 

23.164 

26.212 

29.260 

32.308 

Met. 

2.1335 

5.1815 

8.2294 

11.277 

14.325 

17.373 

20.421 

23.469 

26.517 

29.565 

32.613 

Met. 

2.4383 

5.4863 

8.5342 

11.582 

14.630 

17.678 

20.726 

23.774 

26.822 

29.870 

32.918 

Met. 

2.7431 

5.7911 

8.8390 

11.887 

14.935 

17.983 

21.031 

24.079 

27.126 

30.174 

33.222 


Conversion of Metres into English Feet. 


Metres. 

0 

1 

2 

3 

4 

5 

G 

7 

8 

9 

0 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 


0.000 

3.2809 

6.5618 

9.8427 

13.123 

16.404 

19.685 

22.966 

26.247 

29.528 

10 

32.809 

36.090 

39.371 

42.651 

45.932 

49.213 

52.494 

55.775 

59.056 

62.337 

20 

65.618 

68 899 

72.179 

75 461 

78.741 

82.022 

85.303 

88.584 

91.865 

95.146 

30 

98.427 

101.71 

104.99 

108 27 

111.55 

114.83 

118.11 

121.39 

124.67 

127.96 

40 

131.24 

134.52 

137.80 

141.08 

144.36 

147.64 

150.92 

154.20 

157.48 

160.76 

50 

164 04 

167.33 

170.61 

173.S9 

177.17 

180.45 

183.73 

187.01 

190.29 

193.57 

60 

196.85 

200.13 

203.42 

206.70 

209.98 

213 26 

216.54 

219.82 

223.10 

226.38 

70 

229.66 

232.94 

236.22 

239.51 

242.79 

246.07 

249.35 

252.63 

255.91 

259.19 

80 

262.47 

265.75 

269.03 

272.31 

275.60 

278.88 

282.16 

285.44 

288.72 

292.00 

90 

295.28 

298.56 

391.84 

305.12 

308.40 

311.69 

314.97 

318.25 

321.53 

324.81 

100 

328.09 

331.37 

334.65 

337.93 

341.21 

344.49 

347.78 

351.06 

354.34 

357.62 














































































































Miles and Kilometres. 


49 


Conversion of English Statute-miles into Kilometres. 


Miles. 

0 

1 

2 

3 

4r 

5 

G 

7 

8 

9 


ICilom. 

Kilont. 

Kilom. 

Kilom. 

Kilom. 

Kilom. 

Kilom. 

Kilom. 

Kilom. 

Kilom. 

0 

0.0000 

1.6093 

3.2180 

4.8279 

6.4372 

8.0465 

9.6558 

11.2652 

12.8745 

14.4848 

10 

10.093 

17.702 

19.312 

20.921 

22.530 

24.139 

25.749 

27.358 

28.967 

30.577 

20 

32.186 

33.795 

35.405 

37.014 

38.623 

40.232 

41.842 

43.451 

45.060 

46.670 

:',o 

48.279 

49.888 

51.498 

53.107 

54.716 

56.325 

57.935 

59.544 

61.153 

62.763 

40 

64.372 

65.9S1 

67.591 

09.200 

70.809 

72.41S 

74.028 

75.637 

77.246 

78.856 

50 

80.465 

82.074 

S3.084 

85.293 

86.902 

88.511 

90.121 

91.730 

93.339 

94.949 

00 

90.558 

98.107 

99.777 

101.39 

102.99 

104.60 

100.21 

107.82 

109.43 

111.01 

70 

112.05 

114.26 

115.87 

117.48 

119.08 

120.69 

122.30 

123.91 

125.52 

127.13 

80 

128.74 

130.35 

131.96 

133.57 

135.17 

136.78 

138.39 

140.00 

141.61 

143.22 

90 

144.85 

146.44 

148.05 

149.06 

151.26 

152.87 

151.48 

156.09 

157.70 

159.31 

100 

100.93 

162.53 

164.14 

165.75 

167.35 

168.96 

170.57 

172.18 

173.79 

175.40 


Conversion of Kilometres into English Statute-miles. 


Kilom. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


Miles. 

Miles. 

M lies. 

Miles. 

Miles. 

Miles. 

Miles. 

Miles. 

Miles. 

Miles. 

0 

0.0000 

0.6214 

1.2427 

1.8641 

2.4855 

3.1009 

3.7282 

4.3497 

4.9711 

5.5924 

10 

G.2138 

6.8352 

7.4505 

8.0780 

8.C994 

9.3208 

9.9421 

10.562 

11.185 

11.805 

20 

12.427 

13.049 

13.670 

14.292 

14.913 

15.534 

16.156 

16.776 

17.399 

18.019 

30 

18.641 

19.263 

19.884 

20.506 

21.127 

21.748 

22.370 

22.990 

23.613 

24.233 

40 

24.855 

25.477 

2G.09S 

26.720 

27.341 

27.962 

28.584 

29.204 

29.827 

30.447 

50 

31.069 

31.690 

32.311 

32.933 

33.554 

34.175 

34.797 

35.417 

36.040 

36.G60 

60 

37.282 

37.904 

38.525 

39.147 

39.768 

40.389 

41.011 

41.031 

42.254 

42.874 

70 

43.497 

44,118 

44.739 

45.361 

45.982 

46.603 

47.225 

47.845 

48.468 

49.088 

80 

49.711 

50.332 

50.953 

51.575 

52.196 

52.817 

53.439 

54.059 

54.082 

55.302 

90 

55.024 

50.545 

57.166 

57.788 

58.409 

59.030 

59.652 

00.272 

60.895 

01.515 

100 

62.138 

02.759 

63.380 

64.002 

04.023 

65.244 

65.866 

66.480 

07.109 

07.729 


Conversion of Sea-miles, Knots or Minutes into Kilometres. 


Knots. 

0 

1 

2 

3 

4 

5 

G 

7 

8 

9 


Kilom. 

Kilom. 

Kilom. 

Kilom. 

Kilom. 

Kilom. 

Kilom. 

Kilom. 

Kilom. 

Kilom. 

0 

0.0000 

1.8472 

3.6944 

5.541G 

7.3888 

9.23G1 

11.083 

12.930 

14.777 

16.625 

10 

18.472 

20.319 

22.166 

24.013 

25.801 

27.708 

29.555 

31.402 

33.249 

35.097 

20 

36.944 

38.791 

40.638 

42.485 

44.333 

46.180 

48.027 

49.874 

51,721 

53.569 

30 

55.416 

57.203 

59.110 

60.957 

62.805 

64.652 

60.499 

68.346 

70.193 

72.041 

40 

73.888 

75.735 

77.582 

79.429 

81.277 

83 124 

84.971 

86.818 

88.665 

90.513 

50 

92.361 

94 207 

96.054 

97.901 

99.749 

101.59 

103.44 

105.29 

107.14 

108.98 

60 

110.83 

112.08 

114.53 

116.37 

118.22 

120.06 

121.91 

123.76 

125.01 

127.45 

70 

129.30 

131.15 

133.00 

134.84 

136.70 

138.54 

140.39 

142.24 

144.09 

145.94 

80 

147.77 

149.62 

151.47 

153.31 

155.18 

157.02 

158.87 

160.72 

162.57 

164.43 

90 

106.25 

168.09 

109.94 

171.78 

173.05 

175.49 

177.34 

179.19 

181.04 

182.90 

100 

184.72 

186.56 

188.41 

190.25 

192.12 

193.96 

195.81 

198.60 

199.51 

201.37 


Conversion of Kilometres into Sea-miles, Knots or Minutes. 


Kilom. 

O 

1 

2 

3 

4 

5 

G 

7 

8 

9 


Knots. 

Knots. 

Knots. 

Knots. 

Knots. 

Knots. 

Knots. 

Knots. 

Knots. 

Knots. 

0 

0.0000 

0.5413 

1.0S27 

1.6240 

2.1653 

2.7066 

3.2480 

3.7894 

4.3307 

4.8721 

10 

5.4134 

5.9547 

6.4961 

7.0374 

7.5787 

8.1200 

8.6614 

9.2028 

9.7441 

10.285 

20 

10.827 

11.368 

11.909 

12.451 

12.992 

13.533 

14.075 

14.616 

15.157 

15.702 

30 

16.24 

16.781 

17.322 

17.864 

18.406 

18.946 

19.488 

20.029 

20.570 

21.115 

40 

21.653 

22.194 

22.735 

23.277 

23.819 

24.359 

24 901 

25.442 

25.983 

26.528 

50 

27.066 

27.607 

28.148 

28.690 

29.232 

29.772 

30.314 

30.855 

31.396 

31.941 

60 

32.480 

33.020 

33.561 

34.103 

34.645 

35.185 

35.727 

36.268 

36.809 

37.364 

70 

37.894 

38.433 

38.974 

39.510 

40.058 

40.598 

41.140 

41.681 

42.222 

42.777 

80 

43.307 

43.846 

44.387 

44.929 

45.471 

46.011 

46.553 

47.094 

47.635 

48.190 

90 

48.721 

49.259 

49.800 

50.342 

50.884 

51.424 

51.966 

52.507 

63.048 

54.603 

100 

64.134 

54.672 

55.213 

55.755 

56.297 

56.837 

57.379 

57.920 

58.401 

60.016 


4 















































































































50 Inches and Centimetres. 


Conversion of Square Inches into Square Centimetres. 

In 2 . 

0 

1 

2 

3 

4 

5 

G 

7 

8 

9 


Cm 2 . 

Cm*. 

Cm 2 . 

Cm 2 . 

Cm 2 . 

Cm 2 . 

Cm* 

Cm 2 . 

Cm 2 . 

Cm* 

0 

0.0000 

6.4515 

12.903 

19.354 

25.805 

32.257 

38.709 

45.160 

51.612 

58.063 

10 

64.515 

70.967 

77.418 

83.869 

95.321 

96.772 

103.22 

109.67 

116.12 

122.57 , 

20 

129.03 

135.48 

141.93 

148.38 

154.83 

161.29 

167.74 

174.19 

180.64 

187.09 

30 

193.54 

199.99 

206.44 

212.89 

219.34 

225.80 

231.25 

238.70 

245.15 

251.60 

40 

258.06 

264.51 

270.96 

277.41 

283.86 

290.32 

296.77 

303.22 

309.67 

316.12 

50 

322.57 

329.02 

335.47 

341.92 

348.37 

354.83 

361.28 

36 1 .73 

374.18 

380.63 1 

60 

387.09 

393.54 

399.99 

406.44 

412.89 

419.35 

425.80 

432.25 

438.70 

445.15 

70 

451.60 

458.05 

464.50 

470.95 

477.40 

483.86 

490.31 

496.76 

503.21 

509.66 

80 

516.12 

522.57 

529.02 

535.47 

541.92 

548.38 

554.83 

561.28 

567.73 

574.18 

90 

580.63 

587.08 

593.53 

599.98 

606.43 

612.89 

619.34 

625.79 

632.24 

638.69 1 

100 

645.15 

651.60 

658.05 

664.50 

670.95 

677.41 

683.86 

690 31 

696.76 

703.21 ( 

Conversion of Square Centimetres into Square Indies. 

Cm 2 . 

0 

1 

2 

3 

4: 

5 

0 

7 

8 

» 


In 2 . 

In 2 . 

In 2 . 

In 2 . 

In 2 . 

In 2 . 

In 2 . 

In 2 . 

In 2 . 

In 2 

0 

0.0000 

0.1550 

0.3100 

0.4650 

0.6200 

0.7750 

0.9300 

1.0850 

1.2400 

1.3950 

10 

1.5500 

1.7050 

1.86001 2.0150 

2.1700 

2.3250 

2.4800 

2.6350 

2.7900 

2.9450 ' 

20 

3.1000 

3.2550 

3.4100 

3.5650 

3.7200 

3.8750 

4.0300 

4.1850 

4.3400 

4.4950 

30 

4.6501 

4.8051 

4.9601 

5.1151 

5.2701 

5.4251 

5.5801 

5.7351 

5.8901 

6.0451 

40 

6.2001 

6.3551 

6.5101 

6.6651 

6.8201 

6.9751 

7.1301 

7.2851 

7.4401 

7.5951 

50 

7.7501 

7.9051 

8.0601 

8.2151 

8.3701 

8.5251 

8.6801 

8.8351 

8.9901 

9.1451 

60 

9.3002 

9.4552 

9.6102 

9.7652 

9.9202 

10.075 

10.230 

10.385 

10.540 

10.695 , 

70 

10.850 

11.040 

11.160 

11.315 

11.470 

11.625 

11.780 

11.935 

12.090 

12.245 i 

80 

12.400 

12.555 

12.710 

12.865 

13.020 

13.175 

13.330 

13.485 

13.6-10 

13.795 j 

90 

13.950 

14.105 

14.260 

14.415 

14.570 

14.725 

14.880 

15.035 

15.190 

15.345 j 

100 

15.500 

15.655 

15.810 

15.965 

16.120 

16.275 

16.430 

16.585 

16.740 

16.895 1 

Conversion of Cubic Indies into Cubic 

Cen ti metres. 

Cub. In. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

.1 


Cm 3 . 

Cm 3 . 

Cm 3 . 

Cm 3 . 

Cm 3 . 

Cm 3 

Cm 3 . 

Cm 3 . 

Cm 3 . 

Cm*. 

0 

0.0000 

16.383 

32.773 

49.160 

65.546 81.933 

98.320 

114.71 

131.01 

147.48 

10 

163.87 

1S0.26 

196.64 

213.03 

229 41 

245.80 

262.19 

278.58 

294.88 

311.35 

20 

327.73 

314.12 

360.50 

376.89 

393.27 

409.66 

426.05 

442.44 

458.74 

475.21 

30 

491.60 

507.99 

524.37 

549.76 

557.14 

573.53 

569.92 

606.31 

622.61 

639.08 

40 

655.46 

671.85 

688.23 

701.52 

721.00 

737.39 

753.78 

770.17 

786.47 

802.94 

50 

819.33 

835.72 

851.10 

868.49 

884.87 

901.26 

917.65 

934.04 

950.34 

966.81 

60 

983.20 

999.59 

1016.0 

1032.4 

1018.7 

1065.1 

1081.5 

1097.9 

1114.2 

1130.7 

70 

1147.1 

1153.5 

1179.9 

1196.3 

1212.6 

1229.0 

1245.4 

1261.8 

1278.1 

1294.6 

80 

1310.9 

1327.3 

1343.7 

1360.1 

1376.4 

1392.8 

1409.2 

1425.6 

1441.9 

1458.4' 

90 

1474.8 

1491.2 

1507.6 

1524.0 

1540.3 

1556.7 

1573.1 

1589.5 

1605.8 

1622.3 

100 

1638.7 

1655.1 

1671.5 

1687.9 

1704.2 

1720.6 

1737.0 

1753.4 

1769.7 

1786.2 

Conversion of Cubic Centimetres into Cubic Indies. 

Cm 2 

0 

1 

2 

3 

4 

5 

G 

7 

8 

9 


In 3 . 

In 3 . 

In 3 . 

In 3 . 

I 11 3 . 

In 3 . 

In 3 . 

In 3 . 

In 3 . 

In 3 . 

0 

0.0000 

0.0610 

0.1221 

0.1831 

0.2441 

0.3051 

0.3661 

0.4272 

0.4882 

0.5492 

10 

0.6102 

0.6712 

0.7323 

0.7933 

0.8543 

0.9153 

0.9763 

1.0374 

1.0984 

1.1594 

20 

1.2205 

1.2215 

1.3426 

1.4036 

1.4646 

1.5256 

1.5S66 

1.6477 

1.7087 

1.7697 

30 

1.8308 

1.8918 

1.9529 

2.0139 

2.0749 

2.1359 

2.1969 

2.2580 

2.3190 

2.3800 

40 

2.4410 

2.5020 

2.5631 

2.6241 

2.6851 

2.7461 

2.8071 

2.8682 

2.9292 

2.9902 

50 

3.0513 3.1123 

3.1734 

3.2344 

3.2954 

3.3564 

3.4174 

3.4785 

3.5395 

3.6005 

60 

3.6615 

3.7225 

3.7836 

3.8446 

3.9056 

3.9666 

4.0276 

4.0887 

4.1497 

4,2107 

70 

4.2718 

4.3328 

4.3939 

4.4549 

4.5159 

4.5769 

4.6379! 4.6990 

4.7600 

4.8210 

80 

4.8820 

4 9430 

5.0041 

5.0651 

5.1261 

5.1871 

5.2481 

5.3092 

5.3702 

4.4312 

90 

5.4923 

5.5533 

5.6144 

5.6754 

5.7364 

5.7974 

5.8584 

5.9195 

5.9805 

6.0415 

100 

6.1025 

6.108o 

6.2246 

6.28561 6.3166 

6.4076 

6.4686 

6.5297 

6.5907 

6.6517 


















































































































Yards and Metres. sj 


Conversion of Cubic Yards into Cubic Metres. 

Cub. yd. 

0 

1 

2 

3 

4 

5 

G 

7 

8 

9 


M3. 

M 3 , 

M3. 

M3. 

M». 

M». 

M 3 . 

M 3 . 

M 3 . 

M* 

0 

0.0000 

0.7645 

1 .5291 

2.2936 

3.0581 

3.8226 

4.5872 

5.3517 

6.1163 

6.8808 

10 

7.6453 

8.4098 

9.1744 

9.9389 

10.703 

11.468 

12.232 

12.997 

13.761 

114.526 

20 

15.291 

16.055 

16.820 

17.585 

18.341 

19.114 

19.878 

20.643 

21.407 

22.172 

30 

22.93G 

23.700 

24.455 

25.231 

25.994 

26.759 

27.523 

28.288 

29.052 

29.817 

40 

30.581 

31.345 

32.11( 

32.875 

33.639 

34.404 

35.168 

35.933 

36.797 

37.462 

50 

38.22G 

38.99( 

39.755 

40.520 

41.284 

42.049 

42.813 

43.578 

44,342 

45.107 

GO 

45.872 

46.631 

47.401 

48.160 

48.930 

49.695 

50.459 

51.224 

51.988 

52.753 

70 

53.517 

54.281 

55.040 

55.811 

56.575 

57.340 

58.104 

58.869 

59.633160.398 

80 

61.163 

61.927 

62.692 

63.457 

64.221 

64.986 

65.750 

66.515 

67.279 

68.044 

90 

68.808 

69.572 

70.337 

71.102 

71.860 

72.631 

73.395 

74.160 

74.924 

75.6S9 

100 

76.453 

77.217 

77.982 

78.747 

79.511 

80.276 

81.040 

81.805 

82,569 

183.334 


Conversion of 

Cubic Metres into Cubic Yards. 


M 3 . 

0 

] 

2 

3 

4 

5 

G 

7 

8 

9 


Yd 3 . 

Yd 3 . 

Yd 3 . 

Yd 3. 

Yd 3 . 

Yd 3 . 

Yd 3 . 

Yd 3 . 

Yd 3 . 

Yd 3 . 

0 

0.0000 

1.3080 

2.6160 

3.9240 

5.2329 

6.5399 

7.8479 

9.1559 

10.464 

11.772 

10 

13.080 

14.3S8 

15.696 

17.004 

18.313 

19.620 

20.928 

22.236 

23.544 

24.852 

20 

26.1 GO 

27.468 

28.776 

30.084 

31.393 

32.700 

34.008 

35.316 

36.624 

137.932 

30 

39.240 

40.548 

41.856 

43.164 

44.473 

45.780 

47.088 

48.396 

49.704 

51.012 

40 

52.319 

53.627 

54.935 

56.243 

57.552 

58.859 

60.167 

61.475 

62.783 

63.091 

50 

65.399 

66.707 

68.015 

69.323 

70.632 

71.939 

73.247 

74.545 

75.863 

77.171 

GO 

78.479 

79.787 

81.095 

82.403 

83.712 

85.019 

86.327 

87.535 

88.943 

90 251 

70 

91.559 

92.867 

94.175 

95.483 

96.792 

98.099 

99.407 

100.71 

102.02 

103.33 

80 

104.63 

105.94 

107.25 

108.56 

109.87 

111.17 

112.48 

113.79 

115.10 

116.41 

90 

117.72 

119.03 

120.34 

121.64 

122.95 

124.26 

125.57 

126.88 

128.18 

129.49 

100 

130.80 

132.11 

133.42 

134.72 

136.03 

137.34 

138.65 

139.96 

141.26 

142.57 

Conversion of Gallons into Litres. 

Gall. 

O 

1 

2 

3 

4 

5 

G 

7 

8 

9 


Lit. 

Lit, 

Lit. 

Lit, 

Lit. 

Lit, 

Lit. 

Lit. 

Lit. 

Lit. 

0 

0.0000 

3.7853 

7.5706 

11.356 

15.141 

18.946 

22.712 

26.497 

30.282 

34.068 

10 

37.853 

41.638 

45.423 

49.209 

52.994 

56.799 

60.565 

64.350 

68.135 

71.921 

20 

75.706 

79.491 

83.276 

87.062 

90.847 

94.652 

98.418 

102.20 

105.99 

109.77 

30 

113.56 

117.34 

121.13 

124.92 

128.66 

132.50 

136.27 

140.06 

143.84 

147.63 

40 

151.42 

155.22 

158.99 

162.78 

166.56 

170.36 

174.13 

177.92 

181.70 

185.49 

50 

189.46 

193.24 

197.03 

200.82 

204.60 

208.40 

212.17 

215.96 

219.74 

223.53 

GO 

227.12 

230.90 

234.69 

238.48 

242.26 

246.06 

249 83 

253.62 

257.40 

261.19 

70 

264.97 

26.8.75 

272.54 

276.33 

280.11 

283.91 

286.68 

291.47 

295.25 

299.04 

80 

302.82 

306.60 

310.39 

314.18 

317.96 

321.76 

324,53 

329.32 

333.10 

336.89 

90 

440.68 

444.46 

448.25 

452.04 

455.82 

459.62 

463.39 

467.18 

470.96 

474.75 

100 

478.53 

482.31 

486.101 

789.89 

493.67 

497.47 

501.24 

505.03 

508.81 

512.60 



Conversion of Litres 

into 

Gallons. 



Lit. 

O 

1 

2 

3 

4 

5 

G 

7 

8 

9 


G (\ 1 . 

Gal. 

Gal. 

Gal. 

Gal. 

Gal. 

Gal. 

Gal. 

Gal. 

Gal. 

0 

0.0000 

0.2642 

0.5284 

0.7925 

1.0567 

1 .3209 

1.5851 

1.8492 

2.1134 2.3776 

10 

2.6418 

2.9060 

3.1702 

3.4343 

3.6985 

3.9627 

4.2269 

4.4910 

4. / o52 

5.0194 

20 

5.2836 

5.5478 

5.8120 

6.0761 

6.3403 

6.6045 

6.8687 

7.1328 

7.3970 

7.6612 

30 

7.9254 

8.1896 

8.4538 

8.7179 

8.9821 

9.2463 

9.5105 

9.8746 

10.030 

10.303 

40 

10.567 

10.831 

11.095 

11.360 

11.624 

11.888 

12.152 

12.416 

12.680 

12.945 

50 

13.209 

13.473 

13.737 

14.002 

14.266 

14.530 

14.794 

15.058 

15.322 

15.587 

GO 

15.851 

16.115 

16.379 

16.644 

16.908 

17.172 

17.436 

17.700 

17.9641 

18.229 

70 

18.492 

18.756 

19.020 

19.284 

19.549 

19.813 

20.077 

20.341 

20.605 

20 870 

80 

21.134 

21.398 

21.662 

21.926 

22.191 

22.455 

22 710 

22.983 

23.247 

23.512 

90 

23.776 

24.040 

24.304 

24.568 

24.832 

25.097 

25.361 

25.625 

25.889 

26.154 

100 

26.418 

26.682 

26.946 

27.2101 

27.475 

27.739 

28.003 

28.267 28.5311 

28.796 







































































































52 


Yards and Metres. 




Conversion of Yards into 

Metres. 



Yards. 

0 

1 

a 

3 

4 

5 

G 

7 

8 

9 


Met. 

Met. 

Met. 

Met. 

Met, 

Met. 

Met. 

Met, 

Met. 

Met, 

0 

0.0000 

0.9144 

1.S288 

2.7432 

3.6576 

4.5719 

5.4863 

6.4007 

7.3iol 

8.2295 

10 

9.1439 

10.058 

10.973 

11.887 

12.801 

13.716 

14.630 

15.544 

16.458 

17.373 

20 

18.288 

19.202 

20.117 

21.031! 21.945 

22.860 

23.774 

24.689 

25.603 

26.518 

30 

27.432 

28.346 

29.260 

30.174 

31.088 

32.0031 

32.917 

33.832 

34.746 

35.661 

40 

3G.57G 

37.490 

38.404 

39.318! 

40.232 

41.147 

42.061 

42.976 

43.890 

44.805 

50 

45.719 

46.634 

47.548 

48.462 

49.376 

50.291 

51.205 

52.120 

53.034, 

53.949 

00 

54.8G3 

55.778 

56.692 

57.6061 

58.520 

59.435 

60.349 

61.264 

02.178 

63.093 

70 

64.007 

64.922 

65.836 

66.750 1 

67.664 

68.578 

69.493 

70.408 

71.322 

72.237 

80 

73.151 j 

74.066 

74.980 

75.894 

76.808 

77.723 

78.637 

79.552 

80.466 

81.381 

90 

82.295 

83.210 

84.124 

85.038 

85.952 

86.867 

87.781: 

88.696 

89.610 

90.525 

100 

91.439] 

92.353 

93.267 

94.181 

95.095 

96.010 

96.924] 

97.839 

98.753| 

99.668 



Conversion of 

Metres into Yards. 



Metres. 

O 

1 

a 

3 

4 

5 

G 

7 

8 

9 


Yds. 

Yds. 

Yds. 

Yds. 

Yds. 

Yds. 

Yds. 

Yds. 

Yds. 

Y r ds. 

0 

0.0000 

1.0936 

2.1872 

3.2809 

4.3745 

5.4681 

6.5617 

7.6553 

8.7490 

9.8426 

10 

10.936 

12.029 

13.122 

14.217 

15.310 

16.404 

17.498 

18.591 

19.685 

20.778 

20 

21.872 

22.966 

24.059 

25.153 

26.247 

27.340 

28.434 

29.527 

30.621 

31.715 

30 

32.809 

33.900 

34.993 

36.090 

37.184 

38.277 

39.371 

40.464 

41.558 

42.652 

40 

43.745 

44.839 

45.932 

47.026 

48.120 

49.213 

50.307 

51.400 

52.544 

53.588 

50 

54.681 

55.775 

56.868 

57.962 

59.056 

60.149 

61.243 

62.336 

63.430 

64.524 

60 

G5.G17 

66.711 

67.804 

68.898 

69.992 

71.085 

72.179 

73.272 

74.366 

75.460 

70 

76.553 

77.647 

78.740 

79.834 

80.928 

82.021 

83.115 

84.208 

85.302 

86.396 

80 

87.490 

88 584 

89.677 

90.771 

91.865 

92.958 

94.052 

95.145 

96.239 

97.333 

90 

98.426 

99.520 

100.61 

101.71 

102.80 

103.89 

104.99 

106.08 

107.17 

108.27 

100 

109.36 

110.45 

111.55 

112.64 

113.73 

114.83 

115.92 

117.02 

118.11 

119.20 

Conversion 

of Square Yards into Square Metres. 


Sq. Yds. 

O 

1 

a 

3 

4 

5 

6 

7 

8 

9 


M 2 . 

M 2 . 

M 2 . 

M 2 . 

M 2 . 

M 2 . 

M 2 . 

M 2 . 

M 2 . 

M 2 . 

0 

0.0000 

0.8361 

1.6722 

2.5803 

3.3444 

4.1805 

5.0167 

5.8528 

6.6889 

7.5250 

10 

8.3611 

9.1972 

9.0333 

10.941 

11.706 

12.542 

13.378 

14.214 

15.050 

15.886 

20 

16.722 

17.558 

19.230 

19.302 

20.066 

20.903 

21.730 

22.575 

23.411 

24.247 

30 

25.083 

25.919 

26.755 

27 663 

28.431 

29.264 

30.100 

30.936 

31.772 

32.608 

40 

33.444 

34.280 

35.116 

36.024 

36.788 

37.625 

38.461 

39.297 

40.133 

40.969 

50 

41.805 

42.641 

43.477 

44.385 

45.149 

45.986 

46.822 

47.658 

48.494 

49.330 

GO 

50.167 

51.003 

51.839 

52.747 

53.511 

54.348 

55.184 

56.020 

56.856 

57.692 

70 

58.528 

59.364 

60.190 

61.108 

61.872 

1 62.709 

63.545 

64.381 

65.217 

66.053 

80 

66.889 

67.725 

68.561 69.469 

70.233 

71.070 

71.906 

72.742 

73.578 

74.414 

90 

75.250 

76.086 

76.922 

77.830 

78.594 

79.431 

80.267 

81.103 

81.939 

82.775 

100 

83.611 

84.447 

85.283 

1 86.191 

86.955 

I 87.792 

| 88.628 

89.464 

90.3001 91.136 

Conversion of Square 

Metres into Square Yards. 


Sq* M. 

0 

1 

a 

3 

4 

5 

G 

7 

8 

9 


Yd 2 . 

Y d 2 . 

Y r d 2 . 

Yd 2 . 

Y'd 2 . 

Yd 2 . 

Y r d 2 . 

Yd 2 . 

Yd 2 . 

Yd 2 . 

0 

0.0000 

1.1960 

2.3920 

3.5880 

4.7840 

5.9800 

7.1760 

8.3720 

9.5681 

10.764 

10 

11.060 

13.156 

14.352 

15.548 

16.744 

17.940 

19.136 

20.332 

21.528 

22.724 

20 

23.920 

25.116 

26.312 

27.508 

28.704 

29.900 

31.096 

32.292 

33.488 

34.684 

30 

35.880 

37.076 

38.272 

39.468 

40.664 

41.860 

43.056 

44.252 

45.448 

46.644 

40 

47.840 

49.036 

1 50.232 

51.428 

52.624 

53.820 

55.016 

56.212 

57.408 

58.604 

50 

50.800 

| 60.996 

! 62,192 

63,388 

63.584 

65.780 

66.976 

68.172 

69.368 

70.564 

GO 

71,760 

72.056 

74452 

75,348 

76,544 

77.740 

78.936 

80.132 

81.328 

82.524 

70 

83,721 

.84.917 

86,113 

87,309 

87.505 

89.701 

90.897 

92.093 

93.289 

94.485 

60 

95,681 

! 96.877 

98,073 

99,269 

100.46 

101.66 

102.86 

104.06 

105.25 

106.44 

00 

107.64 

] 108.84 

110,03 

111.24 

112.42 

113,62 

114.81 

116.01 

117.21 

118.40 

100 

119.00 

I 120.80 

121.99 

123.19 

124.38 

125.58 

126.77 

127.97 

129.17 

130.36 


























































































































Acres and Hectares. 


53 




Conversion of Acres 

into Hectares. 



Acres. 

0 

1 

2 

3 

A 

5 

6 

7 

8 

9 


Hect. 

Hect. 

Hect. 

Hect. 

Hect. 

Hect. 

Hect. 

Hect. 

Hect, 

Hect. 

0 

0.0000 

2.4711 

4.9422 

7.4133 

9.8844 

12.355 

14.836 

17.298 

19.769 

22.240 

10 

24.711 

27.182 

29.653 

32.124 

34.695 

37.046 

39.547 

42.009 

44.480 

46.951 

20 

49.422 

51.893 

54.364 

56.835 

59.306 

61.757 

64.258 

66.720 

68.191 

71.662 

30 

74.133 

76.604 

79.075 

81.546 

84.017 

86.468 

88.969 

91.431 

93.902 

96.373 

40 

98.844 

101.31 

103.79 

106.26 

108.73 

111.18 

113.68 

116.14 

118.61 

121.08 

50 

123.55 

126.02 

128.49 

130.96 

133.43 

135.88 

138.38 

140.85 

143.32 

145.79 

60 

148.36 

150.83 

153.30 

155.77 

15S.24 

160.69 

163.19 

155.66 

168.13 

170.60 

70 

172.95 

175.45 

177.92 

180.39 

182.86 

185.31 

187.81 

190.28 

191.75 

195.22 

80 

197.69 

200.16 

202.63 

205.10 

207.57 

210.02 

212.52 

214.99 

217.46 

219.93 

90 

222.40 

224.87 

227.34 

229.81 

232.28 

234.73 

237.23 

239.70 

242.17 

244.64 

100 

247.11 

249.58 

252.05 

254.52 

256.99 

259.44 

261.94 

264.41 

266.88 

269.35 

Conversion of Hectares into Acres. 

Hect. 

0 

1 

2 

3 

A 

5 

6 

7 

8 

9 


Acres. 

Acres. 

Acres. 

Acres. 

Acres. 

Acres. 

Acres. 

Acres. 

Acres. 

Acres. 

0 

0.0000 

0.4047 

0.8093 

1.2140 

1.6187 

2.0234 

2.4280 

2.8327 

3.2374 

3.6420 

10 

4.0468 

4.4515 

4.8561 

5.2608 

5.6655 

6.0702 

6.4748 

6.8795 

7.2782 

7.6888 

20 

8.0936 

8.4983 

8.9029 

9.3076 

9.7123 

10.117 

10.521 

10.926 

11.331 

11.735 

30 

12.140 

12.545 

12.949 

13.354 

13.759 

14.163 

14.568 

14.973 

15.377 

15.782 

40 

16.187 

16.592 

16.996 

17.401 

17.806 

18.210 

18.615 

19.020 

19.414 

19.829 

50 

20.234 

20.639 

21.043 

21.448 

21.853 

22.257 

22.662 

23.067 

23.471 

23.876 

00 

24.280 

24.685 

25.089 

25.494 

25.899 

26.303 

26.708 

27.113 

27.517 

27.922 

70 

28.327 

28.732 

29.136 

29.541 

29.946 

30.350 

30.755 

31.160 

31.564 

31.969 

80 

32.374 

32.779 

33.183 

33.588 

33.993 

34.397 

34.802 

35.207 

35.611 

36.016 

90 

36.420 

36.825 

37.229 

37.634 

38.039 

38.443 

38.848 

39.253 

39.657 

40.062 

100 

40.468 

40.873 

41.277 

41.682 

42.087 

42.491 

42.896 

43.301 

43.695 

44.110 

Conversion of Square Miles into Square 

Kilometres. 

Sq. Mil. 

0 

1 

2 

3 

A 

5 

6 

7 

8 

9 


KiR 

KIR 

KiR 

KiR 

KiR 

KiR 

KiR 

KiR 

KiR 

KiR 

0 

0.0000 

2.5899 

5.1-798 

7.7697 

10.359 

12.929 

15.539 

18.129 

20.718 

23.309 

10 

25.899 

28.490 

31.079 

33.669 

37.259 

38.829 

41.439 

44.029 

46.619 

49.209 

20 

51.798 

54.388 

56.978' 

59.568 

63.158 

64.728 

67.338 

69.928 

72.518 

75.108 

30 

77.697 

80.287 

82.877 

85.467 

89.057 

90.627 

93.238 

96.828 

98.417 

101.01 

40 

103.59 

106.18 

108.77 

111.36 

114.95 

116.52 

119.13 

121.72 

124.31 

126.90 

50 

129.29 

131.88 

134.47 

137.06 

140.-65 

142.22 

144.83 

147.42 

150.01 

152.50 

00 

155.39 

157.98 

160.57 

163.16 

166.75 

168.32 

170.93 

173.52 

176.11 

178.70 

70 

181.29 

183.88 

186.47 

188.06 

192.65 

194.22 

196.83 

199.42 

202.01 

204.60 

80 

207.19 

209.77 

212.36 

214.95 

218.55 

220.11 

222.73 

225.31 

227.91 

230.50 

90 

233.09 

235.68 

238.27 

240.86 

244.45 

246.02 

248.63 

251.22 

253.81 

256.40 

100 

258.99 

261.58 

264.17 

266.76 

270.35 

271.92 

274.53 

277.12 

279.71 

282.20 

Conversion of Square Kilometres into Square 

Miles. 

KiR 

0 

1 

2 

3 

A 

5 

G 

7 

8 

9 


Sq. M. 

Sq. M. 

Sq. M. 

Sq. M. 

Sq. M. 

Sq. M. 

Sq. M. 

Sq. M. 

Sq. M. 

Sq. M. 

0 

0.0000 

0.3861 

0.7722 

1.1583 

1.5445 

1.9304 

2.3166 

2.702S 

3.0890 

3.4749 

10 

3.8612 

4.1873 

4.6334 

5.0195 

5.4057 

5.7916 

6.1778 

6.5640 

6.9502 

7.3362 

20 

7.7224 

8.0485 

8.4946 

8.SS07 

9.2669 

9.6528 

10.039 

10.425 

10.811 

11.197 

30 

11.583 

11.909 

12.355 

12.741 

13.127 

13.513 

13.899 

14.286 

14.672 

15.058 

40 

15.445 

15.771 

16.217 

16.603 

16.989 

17.375 

17.761 

18.146 

18.534 

18.920 

50 

19.304 

19.630 

20.076 

20.462 

20.848 

21.234 

21.620 

22.007 

22.393 

22.779 

60 

23.166 

23.492 

23.938 

24.324 

24.710 

25.096 

25.482 

25.869 

26.245 

26.641 

70 

27.028 

27.354 

27.800 

23.186 

28.572 

28.958 

29.344 

29.731 

30.117 

30.503 

80 

30.890 

31.216 

31.662 

32.048 

32.434 

32.820 

33.206 

33.593 

33.979 

34.365 

90 

34.749 

35.075 

35.521 

35.907 

36.293 

36.679 

37.065 

37.452 

37.838 

38.224 

100 

38.612 

38.938 

39.384 

39.770 

40.156 

40.542 

40.928 

41.315 

41.701 

42.087 















































































































54 


Cubic Feet and Cubic Decimetres 


Conversion 

of Cubie Feet into Cubic Decimetres. 


Cub. ft. 

0 

1 

2 

3 

4 

5 

(i 

7 

8 

9 


dm 3 . 

dm 3 . 

dm 3 . 

dm 3 . 

dm 3 . 

dm 3 . 

dm 3 . 

dm 3 . 

dm 3 . 

dm 3 

0 

0.0000 

28.316 

56.632 

84.948 

113.26 

141.58 

169.90 

198.21 

226.53 

254.84 

10 

283.16 

305.48 

339.79 

268.11 

396.42 

424.74 

453.06 

481.37 

509.69 

538.00 

20 

566.32 

6SS.64 

622.95 

551.27 

679.58 

707.90 

736.22 

764.53 

792.85 

821.16 

30 

849.48 

871.80 

906.11 

934.43 

962.74 

991.06 

1019.4 

1047.7 

1076.0 

1104.3 

40 

1132.6 

1154.9 

1189.2 

1217.5 

1245.9 

1274.2 

1302.5 

1 380,8 

1359.1 

1387.4 

50 

1415.8 

1438.1 

1472.4 

1500.7 

1529.1 

1557.4 

1585.7 

1614.0 

1642.3 

1670.6 

GO 

1698.9 

1721.2 

1755.5 

1783.8 

1712.2 

1840.5 

1868.8 

1897.1 

1925.4 

1953.7 

70 

1982.1 

2004.4 

2038.7 

2067.0 

2095.4 

2123.7 

2152.0 

2180.3 

2208.6 

2236.9 

80 

2265.3 

2287.6 

2321.9 

2350.2 

2378.6 

2406.9 

2435.2 

2468.5 

2491.8 

2520.1 

90 

2548.4 

2570.7 

2605.0 

2633.3 

2661.6 

2690.0 

2718.3 

2746.6 

2774.9 

2803.2 

100 

2831.6 

2853.9 

2888.2 

2916.5 

2944.9 

2973.2 

3001.5 

3029.8 

3058.1 

3086.4 

Conversion 

of Cubic Decimetres inio Cubic Feet. 


Dm3. 

0 

1 

2 

3 

4 

5 

G 

7 

8 

9 


ft 3 . 

ft 3 . 

ft 3 . 

ft 3 . 

ft 3 . 

ft 3 . 

ft 3 . 

ft 3 . 

ft. 3 . 

ft 3 . 

0 

0.0000 

0.0353 

0.0706 

0.1059 

0.1413 

0.1766 

0.2119 

0.2472 

0.2825 

0.3178 

10 

0.3531 

0.3884 

0.4237 

0.4590 

0.4944’ 

0.5297 

0.5540 

0.6003 

0.6356 

0.6709 

20 

0.7063 

0.7416 

1.4069 

0.8122 

0.8476 

0.8829. 

0.9182 

0.9585 

0.9888 

1.1241 

30 

1.0594 

1.0947 

1.1300 

1.1653 

1.2007 

1.2360 

1.2713 

1.3066 

1.3419 

1.3772 

40 

1.4126 

1.4479 

1.4832 

1.5185 

1.5539 

1.5892 

1.6245 

1.6608 

1.6951 

1.7304 

50 

1.7658 

1.8011 

1.8364 

1.8717 

1.9071 

1.9424 

1.9777 

2.0130 

2.0483 

2.0836 

60 

2.1189 

2.1542 

2.1895 

2.2248 

2.2602 

2.2955 

2.3308 

2.3661 

2.4014 

2.4367 

70 

2.4721 

2.5074 

2.5427 

2.5780 

2.6134 

2.6487 

2.6810 

2.7193 

2.7746 

2.7899 

80 

2.8252 

2.8605 

2.895S 

2.9311 

2.9665 

3.0018 

3.0371 

3.0724 

3.1077 

3.1430 

90 

3.1784 

3.2137 

3.2490 

3.2843 

3.3197 

3.3550 

3.3903 

3.4256 

3.4609 

3.4962 

100 

3.5315 

3.5668 

3.6021 

3.6374 

3.6728 

3.7081 

3.7434 

3.77871 3.8140 

3.8493 

Pounds per Sq. Foot, into Kilogrammes per §q. 

Metre. 

Lbs. ft*. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 


kg. m 2 

kg. m 2 

kg. in 2 

kg. m 2 

kg. m 2 

kg. in 2 

kg. m 2 

kg. m 2 

kg. m 2 

kg. m 2 

0 

0.0000 

4.8825 

9.7650 

14.647 

19.530 

24.413 

29.295 

34.177 

39.006 43.943 

10 

48.825 

53.707 

58.590 

63.472 

68.355 

73.238 

78.120 

83.002 

87.831 

92.768 

20 

97.650 

102.53 

107.41 

112.30 

117.18 

122.06 

126.94 

131.83 

136.66 

141.59 

30 

146.47 

151.35 

156.23 

161.12 

165.90 

170.88 

175.76 

ISO. 65 

185.47 

190.41 

40 

195.30 

200.13 

205.06 

209.95 

214.83 

219.71 

224.59 

229.48 

234.30 

239.24 

50 

244.13 

249.01 

253.89 

258.78 

263.66 

268.54 

273.42 

278.31 

283.13 

288.08 

60 

292.95 

297.83 

302.71 

307.60 

312.48 

317.36 

322.24 

327.13 

831.95 

336.89 

70 

341.77 

346.65 

351.53 

356.42 

361.20 

366.18 

371.06 

375.95 

380.77 

385.71 

80 

390.06 

394.94 

399.82 

404.71 

409.59 

414.47 

419.35 

424.24 

429.06 

434.00 

90 

439.43 

444.31 

449.19 

454.08 

458.96 

464.34 

468.72 

473.61 

478.43 

483.37 

100 

488.25 

493.13 

498.01 

502.90 

507.78 

512.66 

517.54 

522.43 

527.25 

532.19 

Kilogrammes per Sq. Metre into Pounds per Sq. Foot. 

Kg. m 2 . 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


lbs.ft 2 . 

lbs.ft 2 . 

lbs.ft 2 . 

lbs.ft 2 . 

lbs.ft 2 . 

lbs.ft 2 . 

lbs.ft 2 . 

lbs.ft 2 .1 

lbs.ft 2 . 

lbs.ft 2 

0 

0.0000 

0.2048 

0.4096 

0.6144 

0.8192 

1.0240 

1.2289 

1.4387 

1.6385 

1.8433 

10 

2.0481 

2.2529 

2.4577 

2.6625 

2.8673 

3.0721 

3.2770 

3.4818 

3.6866 

3.8914 

20 

4.0962 

4.3010 

4.5058 

4.7106 

4.9154 

5.1202 

5.3251 

5.5299 

5.7347 

5.9395 

30 

6.1444 

6.3492 

6.5540 

6.7588 

6.9636 

7.1684 

7.3733 

7.5781 

7.7829 

7.9877 

40 

8.1925 

8.3973 

8.6021 

8.8069- 

9.0117 

9.2165 

9.4214 

9.6262 

9.8310 

10.036 

50 

10.240 

10.445 

10.649 

10.854 

11.059 

11.264 

11.469 

11.674 

11.878 

12,083 

60 

12.2S9 

12.494 

12.698 

12.903 

13.108 

13.313 

13.518 

13.723 

13.927 

14.132 

70 

14.337 

14.542 

14.746 

14.951 

15.156 

15.361 

15.566 

15.771 

15.975 

16.180 

80 

16.385 

16.590 

16.794 

16.999 

17.204 

17.409 

17.614 

17.819 

18.023 

1S.228 

90 

18.433 

18.638 

18.842 

19.047 

19.252 

19.457 

19.662 

19.867 

20.071 

20.276 

100 

20.481 

20.686 

20.890 

21.095 

21.300 

21.505 

21.710 

21.915 

22.119 

22.324 




























































































































Inches and Kilogrammes. 


55 


Pressure per Sq. 

Inch 

into . 

atmospheric Pressure 

• 

Lbs. p. in. 

0 

1 

3 

3 

4 

5 

6 

7 

8 

9 


at. 

ai. 

at. 

at. 

at. 

at. 

at. 

at. 

at. 

at. 

0 

0.0000 

0.0680 

0.1361 

0.2041 

0.2722 

0.3402 

0.4082 

0.4763 

0.5443 

0.6124 

10 

0.6804 

0.7484 

0.8165 

0.8845 

0.9526 

1.0266 

1.0886 

1.1567 

1.2247 

1.2928 

20 

1.3608 

1.4288 

1.4969 

1.5649 

1.6330 

1.7070 

1.7690 

1.8371 

1.9051 

1.9732 

30 

2.0413 

2.1093 

2.1774 

2.2454 

2.3135 

2.3875 

2.4495 

2.5176 

2.5856 

2.6537 

40 

2.7217 

2.7897 

2.8578 

2.9258 

2.9939 

3.0679 

3.1299 

3.1980 

3.2660 

3.3341 

50 

3.4021 

3.4701 

3.5382 

3.6062 

3.6743 

3.7483 

3.8103 

3.8784 

3.9464 

4.0145 

(50 

4.0825 

4.1505 

4.2186 

4.2866 

4.3547 

4.4287 

4.4907 

4.5588 

4.6268 

4.6949 

70 

4.7630 

4.8310 

4.8991 

4.9671 

5.0352 

5.1092 

5.1712 

5.2393 

5.3073 

5.3754 

80 

5.4434 

5.5114 

5.5795 

5.6475 

5.7156 

5.7896 

5.8516 

5.9197 

5.9877 

6.0558 

90 

6.1238 

6.1918 

6.2599 

6.3279 

6.3960 

6.4690 

6.5320 

6.6001 

6.6681 

6.7362 

100 

6.8042 

6.8722 

6.9403 

7.0083 

7.0764 

7.1504 

7.2124 

7.2805 

7.3485 

7.4166 

Atmospheric Pressure into Pressure per Sq. 

Inch 

• 

At. p. 

O 

1 

3 

3 

4 

5 

6 

7 

8 

9 


lbs. iu. 

lbs. in. 

lbs. in. 

lbs.in. 

lbs. in. 

lbs. in. 

lbs. in. 

lbs. in. 

lbs. iu. 

lbs. in. 

0 

0.0000 

14.697 

29.393 

44.090 

58.787 

73.483 

88.180 

102.87 

117.57 

132.27 

10 

146.97 

161.67 

176.36 

191.06 

205.76 

220.45 

235.15 

249.84 

264.54 

279.24 

20 

293.93 

308.63 

323.32 

338.02 

352.72 

367.41 

382.11 

396.80 

411.50 

426.20 

30 

440.90 

455.60 

470.29 

484.99 

499.69 

514.38 

529.08 

543.77 

558.47 

573.17 

40 

587.87 

602.57 

617.26 

631.96 

' 646.66 

661.35 

676.05 

690.74 

705.44 

720.14 

50 

734.83 

749.53 

764.22 

778.92 

793.62 

808.31 

823.01 

837.70 

852.40 

867.10 

60 

881.80 

896.50 

911.19 

925.89 

940.59 

955.28 

969.98 

984.67 

999.37 

1014.1 

70 

1028.7 

1043.4 

1058.1 

1072.8 

1087.5 

1102.2 

1116.9 

1131.6 

1146.3 

1161.0 

80 

117n.7 

1190.4 

1205.1 

1219.8 

1234.5 

1249.2 

1263.9 

1278.6 

1293.3 

1308.0 

90 

1322.7 

1337.4 

1352.1 

1366.8 1381.5 

1396.2 

1410.9 

1425.6 

1439.3 

1455.0 

100 

1469.7 

1484.4 

1499.1 

1513.8 

1528.5 

1543.2 

1557.9 

1572.6 

1586.3 

1602.0 

Pounds per Sq. In. into Kilogrammes per 

Sq. Centimetre. 

Lbs. in 2 . 

O 

1 

3 

3 

4 

5 

6 

7 

8 

9 


k.cm 2 . 

k.cin 2 . 

k.cm 2 . 

k.cm 2 . 

k.cm 2 . 

k.cm 2 . 

k.cm 2 . 

k.cm 2 Ik.cm 2 . 

k. cm 2 . 

0 

0.0000 

0.0703 

0.1406 

0.2109 

0.2812 

0.3515 

0.4218 

0.4921 

0.5625 

0.6328 

10 

0.7031 

0.7734 

0.8437 

0.9140 

0.9843 

1.0546 

1.1249 

1.1952 

1.2655 

1.3358 

20 

1.4062 

1.4765 

1.5468 

1.6171 

1.6874 1.7577 

1 .8280 

1.8983 

1.9686 

2.0389 

30 

2.1092 

2.1795 

2.2498 

2.3202 

2.3905 

2.4608 

2.5311 

2.6014 

2.6717 

2.7420 

40 

2.8123 

2.8826 

2.9529 

3.0232 

3.0935 

3.1639 

3.2342 

3.3045 

3.3748 

3.4451 

50 

3.5154 

3.5857 

3.6560 

3.7263 

3.7966 

3.8669 

3.9372 

4.0075 

4.0779 

4.1482 

60 

4.2185 

4.2888 

4.3591 

4.4294 

4.4997i 4.5700 

4.6403 

4.7106 

4.7809 

4.8512 

70 

4.9216 

4.9919 

5.0622 

5.1325 

5.2028 

5.2731 

5.3434 

5.41371 

5.4840 

5.5543 

80 

5.6246 

5.6949 

5.7652 

5.8356 

5.9059, 

5.9762 

6.0465 

6.1168 

6.1871 

6.2574 

90 

6.3277 

6.3980 

6.4683 

6.5386 

6.6089 

6.6793 

6.7496 

6.8199 

6.8902 

6.9605 

100 

7.0308 

7.1011 

7.1714 

7.2417 

7.31201 7.3823 

7.4526 

7.5229 

7.5933 

7.6636 

Kilogrammes per Sq. 

Centimetre into Pounds per Sq 

. In. 

K. cm 2 . 

O 

I 

3 

3 

4 

5 

6 

7 

8 

9 


lbs.in 2 

lbs.in 2 

lbs.in 2 

lbs.in 2 

lbs.in 2 

lbs.in 2 

■ bs.in 2 

lbs.in 2 

lbs.in 2 

lbs. in. 2 

0 

0.0000 

14.223 

28.446 

42.670 

56.893 

71.116 

85.339 i 

99.562 

113.78 

128.01 

10 

142.23 

156.45 

170.68 

184.90 

199.12 

213.35 

227.57 

241.79 

256.02 

270.24 

20 

284.46 

298.69 

312.91 

327.13 

341.36 

355.58 

369.80 

384.03 

398.25 

412.47 

30 

426.70 

440.92 

455.14 

469.36 

483.59 

497.81 

512.03 

526.26 

540.48 

554.70 

40 

568.93 

583.15 

597.37 

611.60 

625.82 

640.04 

654.27 j 

668.49 

682.71 

696.94 

50 

711.16 

725.38 

739.61 

753.83 

768.05 

782.28 

796.50; 

810.72 

824.94 

839.17 

60 

853.39 

867.61 

881.84 

896.06 

910.28 

924.51 

938.73; 

952.95 

967.18 

981.40 

70 

995.62 

1009.8 

1024.1 

1038.3 

1052.5 

1066 7 

1081.0‘ 

1095.2 

1109.4 

1123.6 

80 

1137.8 

1152.1 

1166.3 

1180.5 

1194.7 

1209.0 

1223.2! 

1237.4 

1251.6 

1265.9 

90 

1280.1 

1294.3 

1308.5 

1322.7 

1337.0 

1351.2 

1365.4 

1379.6 

1393.9 

1408.1 

100 1 

1422.3 

1436.5 

1450.8 

1465.0 

1479.2 

1493.4 

1507.71 

1521.9 

1536.1 

1550.3 































































































56 


Pounds and Kilogrammes. 


Conversion of En, 

gltsli 

Pounds into Kilogrammes. 

n 

Eng. Lbs. 

0 

1 

3 

3 

4 

5 

G 

7 

8 

9 


Kilos. 

Kilos. 

Kilos. 

Kilos. 

Kilos. 

Kilos. 

Kilos. 

Kilos. 

Kilos. 

Kilos. 

0 

0.000 

0.453 

0.907 

1.361 

1.814 

2.268 

2.722 

3.175 

3.629 

4.082 

10 

4.536 

4.989 

5.443 

5.897 

6.350 

6.804 

7.258 

7.711 

8.165 

8.618 

20 

9.072 

9.525 

9.979 

10.43 

10.89 

11.34 

11.79 

12.25 

12.70 

13.15 

30 

13.61 

14.06 

14.52 

14.97 

15.42 

15.88 

16.33 

16.78 

17.24 

17.69 

40 

18.14 

18.59 

19.05 

19.50 

19.95 

20.41 

20.86 

21.31 

21.77 

22.22 

50 

22.08 

23.13 

23.59 

24.04 

24.49 

24.95 

25.40 

25.85 

26.31 

26.76 

60 

27.22 

27.67 

28.13 

28.58 

29.03 

29.49 

29.94 

30.39 

30.85 

31.30 

70 

81.75 

32.20 

32.66 

33.11 

33.56 

34.02 

34.47 

34.92 

35.38 

3q.S3 

80 

36.29 

36.74 

37.20 

37.65 

38.10 

38.56 

39.01 

39.46 

39.92 

40.37 

00 

40.82 

41.27 

41.73 

42.18 

42.63 

43.09 

43.54 

43.99 

41.45 

44.90 

100 

45.36 

45.81 

46.27 

46.72 

47.17 

47.63 

48.08 

48.53 

48.99 

49.44 

Conversion of Kilogrammes 

into English Pounds. 


Fr. Kil. 

0 

1 

a 

3 

4 

5 

G 

7 

8 

9 


Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

0 

0.000 

2.205 

4.410 

6.615 

8.820 

11 02 

13.23 

15.43 

17.64 

19.84 

10 

22.05 

24.25 

26.46 

28.67 

30.87 

33.07 

35.28 

37.48 

39.69 

41.89 

20 

44.10 

40.30 

48.51 

50.72 

52.92 

55.12 

o / .SS 

59.53 

61.74 

63.94 

30 

66.15 

68.35 

70.56 

72.77 

74.97 

77.17 

79.38 

81.58 

83.79 

85.99 

40 

88.20 

90.40 

92.61 

94.82 

97.02 

99.22 

101.4 

93.63 

105.8 

90.04 

50 

110.2 

112.5 

114.6 

116.8 

119.0 

121.2 

123.4 

125.6 

. 127.8 

130.0 

60 

132.3 

134.5 

136.7 

138.9 

141.1 

143.3 

145.5 

147.7 

149.9 

152.1 

70 

154.3 

156.5 

158.7 - 

160.9 

163.1 

165.3 

167.5 

169.7 

171.9 

174.1 

80 

176.4 

178.6 

180.8 

183.0 

185.2 

187.4 

189.6 

191.8 

194.0 

196.2 

90 

198.4 

200.6 

202.8 

205.0 

207.2 

209.4 

211.6 

213.8 

216.0 

21S.2 

100 

220.5 

222.7 

224.9 

227.1 

229.3 

231.5 

233.7 

235.9 

238.1 

240.3 

Conversion of English Tons into Metric Tons. 

Eng. Tons. 

0 

1 

3 

3 

4 

5 

G 

7 

8 

9 


M. toils. 

M.tons. 

M.tons. 

M.tons. 

M.tons. 

M.tons. 

M.tons. 

II .tons. 

M.tons. 

M.tons. 

0 

0.000 

1.016 

2.032 

3.048 

4.064 

5.080 

6.096 

7.112 

8.128 

9.144 

10 

10.16 

11.1S 

12.19 

13.21 

14.12 

15.24 

16.26 

17.27 

18.29 

19.30 

20 

20.32 

21.34 

22.35 

23.37 

24.38 

25.40 

26.42 

27.43 

28.45 

29.46 

30 

30.48 

31.50 

32.51 

33.53 

34.54 

35.56 

36.58 

37.59 

38.61 

39.62 

40 

40.64 

41.66 

42.67 

43.69 

44.70 

45.74 

46.74 

47.7$ 

4S.77 

49.78 

50 

50.80 

51.82 

52.83 

53.85 

54.86 

55.88 

56.90 

57.90 

58.93 

59.94 

60 

60.96 

61.97 

62.99 

64.01 

65.02 

66.04 

67.06 

68.07 

69.09 

70.10 

70 

71.12 

72.14 

73.15 

74.17 

75.18 

76.20 

77.22 

78.23 

79.25 

80.26 

80 

81.28 

82.29 

83.31 

84.33 

85.34 

806 

87.38 

88.39 

89.41 

90.42 

90 

91.44 

92.46 

93.47 

94.49 

95.50 

96.52 

97.54 

98.55 

99.57 

100.6 

100 

101.6 

102.6 

103.6 

104.6 

105.7 

106.7 

107.7 

108.7 

109.7 

110.7 


Conversion of Metric Tons into 

English Tons. 


Fr.M.Tons. 

0 

1 

3 

3 

4 

5 

G 

7 

8 

9 


E. tons. 

E. tons. 

E. tons. 

E. tons. 

E. tons. 

K. tons 

E. tons 

E. tons. 

E. tons. 

E. tons. 

0 

0.000 

0.984 

1.969 

2.953 

3.937 

4.921 

5.906 

6.890 

7.874 

8.858 

10 

9.843 

10.83 

11.81 

12.79 

13.78 

14.76 

15.75 

16.73 

17.72 

18.70 

20 

19.69 

20.67 

21.66 

22.64 

23.63 

24.61 

25.60 

26.58 

27.56 

28.55 

30 

29.53 

30.51 

31.50 

32.48 

83.47 

64.45 

35.44 

36.42 

37.40 

38.39 

40 

39.37 

40.35 

41.34 

42.32 

43.31 

44.29 

45.28 

46.26 

47.24 

48.23 

50 

49.21 

50.19 

51.18 

52.16 

53.15 

54.13 

55.12 

56.10 

57.08 

58.07 

60 

59.06 

60.04 

61.03 

62.01 

63.00 

63.98 

64.97 

65.95 

66.93 

67.92 

70 

68.90 

69.88 

70.87 

71.85 

72.8! 

73.82 

74.81 

75.79 

76.77 

77.76 

80 

78.74 

79.72 

80.71 

81.69 

82.68 

83.66 

84.65 

85.63 

86.61 

87.60 

90 

88.58 

89.56 

90.55 

91.53 

92.52 

93.50 

94.49 

95.47 

96.45 

97.44 

100 

98.43 

99.41 

100.4 

101.4 

102.4 

103.3 

104.3 

105.3 

106.3 

107.3 















































































































Ounces, Grains, and Grammes. 57 


Conversion 

of Eng. Ounces Avoirdupois into Fr. 

Grammes. 

English 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Ounces. 

Grams 

Grams 

Grams 

Grams 

Grams 

Grams 

Grams 

Grams 

Grams 

Grams 

0 

0.0000 

28.348 

56.697 

85.046 

113.39 

141.74 

170.09 

198.44 

226.79 

255.14 

10 

283.48 

311.83 

340.18 

368.52 

396.87 

425.22 

453.57 

481.92 

510.27 

538.62 

20 

566.97 

595.32 

623.67 

652.01 

680.36 

708.71 

737.06 

765.41 

793.76 

822.11 

30 

850.46 

878.81 

907.16 

93.550 

963.85 

992.20 

1020.5 

1048.9 

1077.2 

1105.6 

40 

1133.9 

1162.2 

1190.6 

1218.9 

1247.3 

1275.6 

1304.0 

1332.3 

1360.7 

13S9.0 

50 

1417.4 

1445.7 

1474.1 

1502.4 

1530.8 

1559.1 

1587.5 

1615.8 

1644.2 

1672.5 

GO 

1700.9 

1729.2 

1756.6 

1785.9 

1814.3 

1842.9 

1871.0 

1899.3 

1927.7 

1956.0 

70 

1984.4 

2012.7 

2041.1 

2079.4 

2097.8 

2126.1 

2154.5 

2182.8 

2211.2 

2239.5 

80 

2267.9 

2296.2 

2324.6 

2352.9 

2381.3 

2409.6 

2438.0 

2466.3 2494.7 

2523.0 

90 

2551.4 

2579.7 

2608.1 

2636.4 

2664.8 

2693.1 

2721.5 

2739.8 

2778.2 

2806.5 

100 

2834.8 

2863.1 

2891.5 

2919.8 

2948.2 

2976.5 

3004.9 

3033.2 

3061.6 

3089.9 

Conversion 

of Fr. Grammes into 

Eng. 

Ounces Avoirdupois. 

French 

0 

1 

2 

3 

4 

5 

G 

7 

8 

9 

Grammes. 

Oz. 

Oz. 

Oz. 

Oz. 

Oz. 

Oz. 

Oz. 

Oz. 

Oz. 

Oz. 

0 

0.0000 

0.0353 

0.0705 

0.1058 

0.1411 

0.1768 

0.2116 

0.2469 

0.2822 

0.3175 

10 

0.3527 

0.3880 

0.4232 

0.4585 

0.4938 

0.5295 

0.5643 

0.5996 

0.6349 

0.6702 

20 

0.7055 

0.7408 

0.7760 

0.8113 

0.8466 

0.8823 

0.9171 

0.9524 

0.9877 

1.0230 

30 

1.0582 

1.0935 

1.1287 

1.1640 

1.1993 

1.2350 

1.2698 

1.3051 

1.3404 

1.3757 

40 

1.4110 

1.4463 

1.4815 

1.5168 

1.5521 

1.5878 

1.6226 

1.6579 

1.6932 

1.7285 

50 

1.7687 

1.8040 

1.8392 

1.8745 

1.9098 

1.9455 

1.9803 

2.0156 

2.0509 

2.0862 

GO 

2.1165 

2.1518 

2.1870 

2.2223 

2.2576 

2.2933 

2.3281 

2.3634 

2.3987 

2.4340 

70 

2.4692 

2.5045 

2.5397 

2.5750 

2.6103 

2.6460 

2.6808 

2.7161 

2.7514 

2.7867 

80 

2.8220 

2.8573 

2.8925 

2.9278 

2.9631 

2.9988 

3.0336 

3.0689 

3.1042 

3.1395 

90 

3.1747 

3.2100 

3.2452 

3.2805 

3.3158 

3.3515 

3.3863 

3.4216 

3.4569 

3.4922 

100 

3.5275 

3.5628 

3.5980 

3.6333 

3.6686 

3.7043 

3.7391 

3.7744 

3.8097 

3.8450 

Conversion of Eng. Grains Troy into Fi 

*. Grammes. 

English 

O 

1 

2 

3 

4 

5 

G 

7 

8 

9 

Grains. 

Grams 

Grams 

Grams 

Grams 

Grams Grams 

Grams 

Grams 

Grams 

Grams 

0 

0.0000 

0.0648 

0.1296 

0.1944 

0.2592 

0.3240 

0.3888 

0.4535 

0.5183 

0.5831 

10 

0.6479 

0.7127 

0.7775 

0.8423 

0.9071 

0.9719 

1.0367 

1.1014 

1.1662 

1.2310 

20 

1.2959 

1.3607 

1.4255 

1.4903 

1.5551 

1.6199 

1.6847 

1.7494 

1.8142 

1.8890 

30 

1.9438 

2.0086 

2.0734 

2.1382 

2.2030 

2.2678 

2.3326 

2.3973 

2.4621 

2.5269 

40 

2.5918 

2.6566 

2.7214 

2.7862 

2.8510 

2.9158 

2.9806 

3.0453 

3.1101 

3.1749 

50 

3.2398 

3.3046 

3.3694 

3.4342 

3.4990 

3.5638 

3.6286 

3.6933 

3.7581 

3.8229 

60 

3.8877 

3.9525 

4.0173 

4.0821 

4.1469 

4.2117 

4.2765 

4.3412 

4.4060 

4.4708 

70 

4.5357 

4.6005 

4.6653 

4.7301 

4.7949 

4.8597 

4.9245 

4.9892 

5.0540 

5.1188 

80 

5.1830 

5.2484 

5.3132 

5.3780 

5.4428 

5.5076 

5.5724 

5.6371 

5.7019 

5.7667 

90 

5.8316 

5.8964 

5.9612 

6.0260. 

6.0908 

6.1556 

6.2204 

6.2851 

6.3499 

6.4147 

100 

6.4795 

6.5443 

6.6091 

6.6739 

6.7387 

6.8035 

6.8683 

6.9330 

6.9978 

7.0626 

Conversion of Fr. 

Grammes into Eng. Grains 

Troy 

« 

French 

O 

1 

2 

3 

4 

5 

0 

7 

8 

9 

Grammes. 

Grs. 

Grs. 

Grs. 

Grs. 

Grs. 

Grs. 

Grs. 

Grs. 

Grs. 

Grs. 

0 

0.0000 

15.433 

30.866 

46.299 

61.732 

77.165 

92.599 

105.03 

123.46 

138.90 

10 

154.33 

169.76 

185.19 

200.63 

216.06 

231.49 

246.93 

259.36 

277.79 

293.23 

20 

308.66 

324.09 

339.52 

354.96 

370.39 

385.82 

401.26, 

413.69 

432.12 

447.56 

30 

462.99 

478.42 

493.86 

509.29 

524.72 

540.15 

555.59 

568.02 

586.45 

601.89 

40 

617.65 

633.08 

648.51 

663.95 

679.38 

694.83 

710.25 

722.68 

741.11 

756.55 

50 

771.65 

787.08 

802.52 

817.95 

833.38 

848.82 

864.25 

876.68 

895.11 

910.55 

GO 

925.99 

941.42 

956.85 

972.29 

987.72 

1003.1 

1018.6 

1031.0 

1049.4 

1064.9 

70 

1050.3 

1065.7 

1081.1 

1096.6 

1112.0 

1127.5 

1142.9 

1155.3 

1173.7 

1189.2 

80 

1234.6 

1258.7 

1274.2 

1289.6 

1305.0 

1320.4 

1335.9 

1348.3 

1366.7 

1382.2 

90 

1389.0 

1404.4 

1419.8 

1435.3 

1450.7 

1466.1 

1481.6 

1494.0 

1512.4 

1527.9 

100 

1543.3 

1558.7 

1574.1 

1589.6 

1605.0 

1620.4 

1635.9 

1643.3 

1666.7 

1681.2 
















































































































58 Foot-Pounds and Kiloorammktres. 



Horse- 

power into 

Puissance de Clieval. 



IP 

0 

I 

2 

3 

4 

5 

O 

7 

8 

9 


P. c. 

P. C. 

P. C. 

P. C. 

P. C. 

P. c. 

P. c. 

P. 0. 

P. C. 

P. C. 

0 

0.0000 

1.0136 

2.0272 

3.0408 

4.0544 

5.0680 

6.0816 

7.0952 

8.1088 

9.1224 

10 

10.136 

11.150 

12.163 

13.176 

14.190 

15.204 

16.218 

17.231 

18.245 

19.258 

20 

20.272 

21.308 

22.299 

23.313 

24.326 

25.240 

26.354 

27.367 

28.381 

29.394 

30 

30.408 

31.422 

32.435 

33.449 

34.462 

35.476 

36.490 

37.503 

38.517 

39.530 

40 

40.544 

41.557 

42.571 

43.585 

44.598 

45.612 

46.626 

47.639 

48.653 

49.666 

50 

50.680 

50.781 

52.707 

53.721 

54.734 

55.748 

56.762 

57.775 

58.789 

59.802 

60 

60.816 

61.829 

62.843 

63.857 

64.870 

65.884 

66.898 

67.911 

68.925 

69.938 

70 

70.952 

71.905 

72.979 

73.993 

75.006 

76.020 

77.034 

78.047 

79.061 

80.074 

80 

8 LOSS 

82.102 

83.115 

84.129 

85.142 

86.156 

87.170 

88.183 

89.197 

90.210 

90 

91.224 

92.338 

93.251 

94.265 

95.278 

96.292 

97.396 

98.319 

99.333 

100.34 

100 

101.36 

102.37 

103.30 

104.40 

105.41 

106.43 

107.44 

108.45 

109.47 

110.48 


Puissance de Cliei 

al into Horse-power. 



P. c. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


IP 

IP 

IP 

IP 

IP 

IP 

IP 

IP 

IP 

IP 

0 

0.0000 

0.9863 

1.9726 

2.9589 

3.9452 

4.9315 

5.9178 

6.9041 

7.8904 

8.8767 

10 

9.8030 

10.849 

11.835 

12.822 

13.808 

14.794 

15.781 

16.767 

17.753 

18.739 

20 

19.726 

20.712 

21.698 

22.685 

23.671 

24.657 

25.644 

26.630 

27.616 

28.602 

30 

29.589 

30.575 

31.561 

32.548 

33.534 

34.520 

35.507 

36.493 

37.479 

38.465 

40 

39.452 

40.438 

41.424 

42.411 

43.397 

44.383 

45.370 

46.356 

47.342 

48.328 

50 

49.315 

50.301 

51.287 

52.274 

53.260 

54.246 

55.233 

56.219 

57.205 

58.191 

60 

59.178 

60.164 

61.150 

62.137 

63.123 

64.109 

65.096 

66.082 

67.068 

68.654 

70 

69.041 

70.027 

71.013 

71.990 

72.986 

73.972 

74.959 

75.945 

76.941 

77.917 

80 

78.904 

79.890 

80.876 

81.863 

82.849 

83.835 

84.822 

85.808 

86.794 

87.780 

90 

88.767 

89.753 

90.739 

91.726 

92.712 

93.698 

94.785 

95.671 

96.657 

97.643 

100 

98.630 

99.616 

100.60 

101.59 

102.57 

103.56 

104.55 

105.53 

106.52 

107.50 

Power or Work. Foot-pounds into 

Kilogram met res. 

Ft. lbs. 

0 

1 

2 

3 

4 

5 

G 

7 

8 

9 


k. in. 

k. ra. 

k. m. 

k. m. 

k. ra. 

k. in. 

k. in. 

k. rn. 

k. in. 

k. m. 

0 

0.0000 

0.1382 

0.2764 

0.4146 

0.5528 

0.6910 

0.8292 

0.9674 

1.1056 

1.2438 

10 

1.3820 

1.5202 

1.6584 

1.7966 

1.9348 

2.0731 

2.2112 

2.3494 

2.4876 

2.6259 

20 

2.7640 

2.9022 

3.0404 

3.1786 

3.3168 

3.4552 

3.5933 

3.7315 

3.8696 

4.0078 

30 

4.1460 

4.2842 

4.4224 

4.5606 

4.6988 

4.8370 

4.9751 

5.1134 

5.2517 

5.3897 

40 

5.5280 

5.6666 

5.8044 

5.9426 

6.0808 

6.2191 

6.3572 

6.4954 

6.6336 

6.7718 

50 

6.9100 

7.0482 

7.1864 

7.3246 

7.4628 

7.6010 

7.7393 

7.8775 

8.0155 

8.1538 

60 

8.2920 

8.4303 

8.5684 

8.7066 

8.8418 

8.9830 

9.1212 

9.2594 

2.8976 

9.5359 

70 

9.6740 

9.8122 

9.9504 

10.088 

10.227 

10.365 

10.503 

10,641 

10.779 

10.918 

80 

11.056 

11.194 

11.322 

11.570 

11.609 

11.717 

11.885 

12.023 

12.161 

12.300 

90 

12.438 

12.576 

12.714 

12.855 

12.991 

13.129 

13.267 

13.405 

13.544 

13.682 

100 

13.820 

13.958 

14.096 

14.235 

14.373 

14.511 

14.649 

14.787 

14.925 

14.064 

Power or Work. Kilogrammetreg into Foot-pounds. 

K. m. 

0 

1 

2 

3 

4 

5 

G 

7 

8 

9 


ft. lbs. 

ft. lbs. 

ft. lbs. 

ft. lbs. 

ft. lbs. 

ft. lbs. 

ft. lbs. 

ft. lbs. 

ft, 11)8. 

ft. lbs. 

0 

0.0000 

7.2334 

14.467 

21.700 

28.934 

36.166 

43.400 

50.734 

57.868 

65.100 

10 

72.334 

79.567 

87.101 

94.034 

101.27 

108.50 

115.74 

123.07 

130.20 

137.43 

20 

144.67 

151.90 

158.43 

166.37 

173.60 

180.84 

188.08 

195.40 

202.54 

209.77 

30 

217.00 

224.23 

231.77 

238.70 

245.93 

253.17 

260.41 

267.73 

274.87 

282.10 

40 

289.34 

296.57 

304.11 

311.04 

318.27 

325.50 

332.75 

340.07 

347.21 

354.44 

50 

361.66 

368.89 

376.43 

383.36 

390.59 

397.82 

405.07 

412.39 

419.53 

426.76 

60 

434.00 

441.23 

448.77 

455.70 

462.93 

470.17 

477.41 

484.73 

491.87 

499.10 

70 

507.34 

514.57 

522.11 

529.04 

536.27 

548.50 

550.75 

558.07 

565.21 

572.44 

80 

578.68 

585.91 

593.45 

599.38 

607.61 

614.85 

622.09 

629.41 

636.55 

643 78 

90 

651.00 

658.23 

665.77 

672.70 

679.93 687.17 

694.41 

701.73 

708.87 

716 10 

100 

723.34 

730.57 

738.11 

745.04 

752.27 

759.51 

766.75 

774.07 

781.21 

788.44 













































































































Foot-Tons and Tonnes-Metees. 59 



Conversion of Foot-Tons 

into Tonnes-Metres. 


Ft. tn. 

O 

1 

2 

3 

U 

5 

6 

7 

8 

9 


t, m. 

t. in. 

t. m. 

t. m. 

t. m. 

t. m. 

t, m. 

t, m. 

t. in. 

t. m. 

0 

0.0000 

0.3097 

0.6194 

0.9291 

1.2382 

1.5484 

1.8581 

2.1678 

2.4775 

2.7872 

10 

3.0969 

3.3166 

3.7162 

4.0260 

4.3356 

4.6453 

4.9550 

5.2667 

5.5744 

5.8841 

20 

6.1938 

6.4135 

6.8132 

7.1229 

7.4325 

7.7422 

8.0519 

8.3636 

8.6713 

8.9810 

30 

9.2906 

9.6003 

9.9100 

10.219 

10.529 

10.839 

11.149 

11.460 

11.768 

12.078 

40 

12.387 

1 12.697 

13.006 

13.316 

13.626 

13.935 

14.245 

14.557 

14.864 

15.174 

50 

15.484 

15.794 

16.103 

16.413 

16.723 

17.032 

17.342 

17.654 

17.961 

18.271 

(iO 

18.581 

18.891 

19.200 

19.510 

19.820 

20.129 

20.439 

20.751 

21.058 

21.368 

70 

21.678 

i 21.988 

22.297 

22.607 

22.917 

23.226 

23.536 

23.848 

24.155 

24.465 

80 

24.775 

! 24.085 

25.394 

25.704 

26.014 

26.323 

26.633 

26.945 

27.252 

27.562 

90 

27.872 

28.182 

28.491 

28.801 

29.111 

29.420 

29.730 

30.042 

30.349 

30.659 

100 

30.969 

1 31.279 

31.588 

31.898 

32.208 

32.517 

32.827 

33.139 

33.446 

33.756 

Conversion of Tonnes-Metres into Foot-Tons. 

T. M. 

O 

1 

2 

3 

4L 

5 

G 

7 

8 

9 


ft. t„ 

ft. t,. 

ft. t. 

ft. t. 

ft. t. 

ft. t. 

ft, t. 

ft, t. 

ft, t. 

ft, t. 

0 

0.0000 

3.2290 

6.4581 

9.6871 

12.916 

16.145 

19.374 

22.603 

25.832 

29.061 

10 

32.290 

35.519 

38.758 

41.977 

45.206 

48.435 

51.664 

54.893 

58.122 

61.351 

20 

64.581 

67.810 

71.049 

74-268 

77.497 

80.726 

83.955 

87.184 

90.413 

93.642 

30 

96.871 

100.10 

103.34 

106.56 

109.79 

113.01 

116.24 

119.47 

122.70 

125.93 

40 

129.16 

•133.39 

135.63 

138.85 

142.07 

145.30 

148.53 

151.76 

154.99 

158.22 

50 

161.45 

164.68 

167.92 

171.14 

174.36 

177.59 

180.82 

184.05 

187.28 

190.51 

60 

193.74 

196.97 

200.21 

203.43 

206.65 

209.88 

213.11 

216.34 

219.57 

222.80 

70 

226.03 

229.26 

232.50 

235.72 

238.94 

242.17 

245.40 

248.63 

251.86 

255.09 

SO 

258.32 

261.55 

264.79 

268.01 

271.23 

274.40 

277.69 

280.92 

284.15 

287.38 

90 

290.61 

293.84 

297.08 

300.30 

303.52 

306.75 

309.98 

313.21 

316.44 

319.67 

100 

322.90 

326.13 

329.37 

332.59 

335.81 

339.04 

342.27 

345.50 

348.73 

351.96 


English Units 

of Heat into Frencli 

Calories. 


Heat. 

O 

1 

2 

3 

4: 

5 

G 

7 

8 

9 


cal. 

cal. 

cal. 

cal. 

cal. 

cal. 

cal. 

cal. 

cal. 

cal. 

0 

0.0000 

0.2520 

0.5040 

0.7560 

1.0080 

1.2600 

1.5120 

1.7640 

2.0160 

2.2680, 

10 

2.5200 

2.7720 

3.0240 

3.2760 

3.5280 

3.7800 

4.02.20 

4.2840 

4.5360 

4.7880 

20 

5.0399 

5.2919 

5.5439 

5.7959 

6.0478 

6.2699 

6.5419 

6.8039 

7.0559 

7.3079 

30 

7.5600 

7.8120 

8.0640 

8.3160 

8.5680 

8.8200 

9.0720 

9.3340 

9.5760 

9.8280 

40 

10.080 

10.332 

10.584 

10.836 

11.088 

11.340 

11.512 

11.844 

12.096 

12.348 

50 

12.600 

12.852 

13.104 

13.356 

13.608 

13.860 

14.112 

14.364 

14.616 

14.868 

60 

15.120 

15.372 

15.624 

15.876 

16.128 

16.380 

16.632 

16.884 

17.136 

17.388 

70 

17.640 

17.892 

18.144 

18.396 

18.648 

18.900 

19.152 

19.404 

19.056 

19.908 

80 

20.160 

20.412 

20.664 

20.916 

21.168 

21.420 

21.672 

21.924 

22.176 

22.428 

90 

22.680 

22.932 

23.184 

23.436 

23.688 

23.940 

24.192 

24.444 

24.696 

24.948 

100 

25.200 

25.452 

25.704 

25.956 

26.208 

26.460 

26.712 26.964 

27.216 

27.468 


Frencli Calories into English Units 

of Heat.7 


Calories. 

0 

1 

2 

3 

4 

5 

G 

7 

8 

9 


ht. 

lit. 

lit. 

ht. 

ht. 

ht. 

ht. 

ht. 

ht. 

lit. 

0 

0.0000 

3.9683 

7.9366 

11.905 

15.873 

19.842 

22.810 1 

27.778 

31.746 

35.715 

10 

39.6831 

43.651 

47.620 

51.598 

55.520 

59.525 

62.493 

67.461 

71.429 

75.398 

20 

79.366 

83.334 

87.303 

91.271 

95.203 

99.208 

102.24! 

107.14 

111.11 

115.08 

30 

119.05 

123.02 

126.98 

130.95 

134.89 

138.89 

141.86 

146.83 

150.80 

154.77 

40 

158.73 

162.70 

166.66 

170.62 

174.57, 

178.57 

181.54 

186.51 

190.48 

194.45 

50 

198.42 

202.39 

206.35 

210.39 

214.26 

218.26 

221.23 i 

226.20 

230.16 

234.14 

60 

228.10 

232.071 

236.03 

240.00 

243.94 

24S.94 

250.911 

255.88 

259.85 

263.82 

70 

277.78 

281.75; 

285.72 

280.68 

293.621 

297.62 

300.59 

305.56 

309.53 

313.50 

80 

317.46 

321.43, 

325.40 

329.36 

323.29 

337.30 

340.27i 

345.24 

819.20 

353.18 

90 

357.15 

361.121 

365.09 

369.05 

372.98 

376.99 

379.96 

384.93 

388.90 

392.87 

100 

396.83 

400.801 404.77 

408.73 

412.67 

416.67 

419.641 

424.61 

428.58 

432.55 







































































































60 


Abbreviations of Metric Nomenclature. 


ABBREVIATIONS OF METRIC NOMENCLATURE. 

The following abbreviations have been adopted by the International Met¬ 
rical Congress at Paris, and are recommended for general use: 


Length. 


km means kilometre. 
in “ metre. 

dm “ decimetre. 

cm “ centimetre. 

mm “ millimetre. 


Surface. 

1cm! 1 means square kilometres. 
m 3 “ square metre. 

dm 2 “ square decimetre. 

cm' 1 “ square centimetre. 

mnfl “ square millimetre. 

Volume. 

km 3 means cubic kilometre. 
m 3 “ cubic metre. 

dm 3 “ cubic decimetre. 

cm 3 “ cubic centimetre. 

mm 3 “ cubic millimetre. 


Land Measures. 

ha means hectare. 
a “ are. 


Hollow Measures. 

hi means hectolitre. 

I “ litre. 

dl. “ decilitre. 

cl “ centilitre. 

Weights. 

t means tons. 

q “ hundredweight. 

kg “ kilogramme. 

dAv/ “ decagramme. 

g “ gramme. 

d <7 “ decigramme. 

c <7 “ centigramme. 

7«<7 “ milligramme. 


The abbreviations should invariably appear in italic letters, and no stop to 
be used at the right of them except when at the end of a sentence. The ab¬ 
breviations succeed the figures to which they refer, on the same line, and after 
the last decimal when such are used. 

To the above abbreviations the writer proposes to add the following: 

ef. for power or effects in kilogrammetre per second. 
pc. for power puissance de cheval. 
kgm. for work in kilogrammetres. 

Puissance dr Cheval is the correct expression for what the French call 
force de cheval. They do not mean force, but power. Force de cheval is the 
force with which the horse pulls, and not the effect, 75 kilogrammetre per 
second, which is power or puissance. 


It would be advisable to adopt a similar system of abbreviations for Eng¬ 
lish measures—namely, as follows: 


Ijenglh. 

in. for inches. 
ft. “ feet. 
yd. “ yards. 
ch. “ chains. 
ml. “« miles. 

Surface. 

iifi. for square inches. 
ft! 1 . “ square feet. 
i/d 2 . “ square yards. 
ch 2 . “ square chains. 
mP. “ square miles. 


Volume. 

?'w 3 . for cubic inches. 
ft 3 . “ cubic feet. 
yd 3 . “ cubic yards. 


Weights. 

oz. for ounces. 

lbs. “ pounds. 

cwt. “ hundredweights. 

in. “ ton. 






61 


Calculus Differential Formulas* 


FORMULAS. 


DIFFERENTIALS. 

y =*£ 


dy =* dx, 

y ax 2 


d y — 2 a x dx, 

y — x n 


dy — nx* _1 dx, 

Sab x 3 

sas 

9 a & x 2 dx, 

4 a b 2 x* 

s=: 

4 n a 6 2 x n_1 dx, 

a + x 3 


3 x 2 dx, 

(a + 6)x 2 

- 

2 x(a -f &)dx, 

6a 5 4 x 3 — c 


18 a & 4 x 2 dx, 

x + 3 X s — o 


dx + 6 z dz — dv, 


6x 3 -f- 4a* 2 — 3x =**. (18x 2 + 8a x—3)dx, 10 


X V 2 

=■* 

t> dx + 2 xv dv, 

11 

xvz 


1 dx dv dz 

xvz I--1- 

\ X V z 

)l2 

x(x 2 —x& 2 ) 

=f= 

(3 x 2 — b- x) dx, 

13 

X 2 


2 x v dx — x 2 dv 

14 

V 


v 2 ’ 

a 

X 


a dx 

X 3 ’ • ‘ 

15 

a 


nax^' 1 dx 

16 

x» 


x- n ’ 

(a +l/x) 3 


3(a + l/x) 2 dx 

17 


2j/'x 

(a+l/x)** — to 

n _M-il A-1 

(a+l/x) w 1 x w dx,18 

1 


dx 

19 

4(a — x) 


(a — x)n+1’ 

2\/2 ax — x 2 


2 a dx 

20 

X 


9: l/2ax — x 2 


FORMULAS. 


DIFFERENTIALS. 

a* 


a^l a dx, 

21 

d‘l‘x 

a 

dx 
x ’ 

22 

X J'X 


(1 -(- lx)dx, 

23 

he 


(1 — lx)dx 

24 

X" 


X’i+5, ’ 

X 

lx 

=3 

(lx — l)dx 
(Z-X)2 ’ 

25 

ay 


ay x dx — a x 2 dy 

26 




V^ + y- 


V(& + y 2 ) 3 


a — 2 fi x 


2 b 2 xdx 

27 

(a + 6 x) 2 


(a + 6x) 3 ’ 

S 

il 

— 

dx 

2l/x 

28 

(a x -f x 2 )q — «(ax+x 2 )w-i(a + 2x)dx 29 



b xdx 

1/ a 2 -f- b x 2 


1/a 2 + fcx 2 

—- 

30 

d-2(ax 3 ) 


6 a x dx 2 , 

31 

d^a x 3 ) 

— 

6 a dx 3 , 

32 

d 4 (a x 3 ) 


o 6 a xo-idx 4 =* o, 

33 

sin. t; 

= 

+ cos. v dv. 

34 

cos. V 

= 

— sin. v dv 

35 

tan. v 

= 

dv 

cos. 2 y ’ 

36 

cot. V 


dv 

sin. 2 « ’ 

37 

sec. v 

= 

, cos. v dv 
cos. 2 « ’ 

38 

cosec. v 

= 

cos. v dv 
sin. 2 v ’ 

39 

Tan. for any curve t = y \/1 

' \ dy 2 

40 


See page 66. 







































62 


Foot-measures and Pounds. 


Comparison between Foot-measures of Different Nations. 


Linear Ff.f,t. 


English. 

Metre. 

Prussia. 

Saxony. 

Baden. 

Austria. 

Hanover 

Sweden. 

1 

0.3048 

0.9711 

1.0763 

1.0160 

0.9642 

1.04:35 

1.0265 

3.2809 

1 

3.1862 

3.5312 

3.33:33 

3.1634 

3.4235 

3.3678 

1.0297 

0.3138 

1 

1.1083 

1.0462 

0.9929 

1.0745 

1.0572 

0.9291 

0.2832 

0.9023 

1 

0.9440 

0.8959 

0.9695 

0.9538 

0.9843 

0.3000 

0.9559 

1.0594 

1 

0.9490 

1.0271 

1.0164 

1.0371 

0.3161 

1.0072 

1.1163 

1.0537 

I 

1.0822 

1.09(53 

0.9583 

0.2921 

0.9307 

1.0314 

0.9736 

0.9240 

1 

0.9838 

0.9741 

0.2969 

0.9459 

1.0484 

0.9838 

0.9122 

1.0165 

1 

Square Feet. 

1 

0.0929 

0.9431 

1.1584 

1.0322 

0.9297 

1.0888 

1.0537 

10.704 

1 

10.152 

12.469 

11.111 

10.007 

11.721 

11.342 

1.0(503 

0.0985 

1 

1.2283 

1.0945 

0.9858 

1.1545 

1.1130 

0.8603 

0.0802 

0.8141 

1 

0.8911 

0.8026 

0.9400 

0.9097 

0.9088 

0.0900 

0.9137 

1.1222 

1 

0.9007 

1.0549 

1.0:i30 

1.0756 

0.0999 

1.0144 

1.2460 

1.1103 

1 

1.1712 

1.2019 

0.9184 

0.0853 

0.8661 

1.0639 

0.9480 

0.8538 

1 

0.9679 

0.9489 

0.0881 

0.8947 

1.0941 

0.9679 

0.8321 

1.0331 

1 

Cubic Feet. 

1 

0.0283 

0.9159 

1.2468 

1.0487 

0.8964 

1.1362 

1.1018 

35.316 

1 

32.346 

44.032 

37.037 

31.658 

40.126 

38.198 

1.0918 

0.0309 

1 

1.3613 

1.1450 

0.9787 

1.2405 

1.1816 

0.8021 

0.0227 

0.7:346 

1 

0.8411 

0.7190 

0.9113 

0.8677 

0.9535 

0.0270 

0.8733 

1.1889 

1 

0.8548 

1.0834 

1.0501 

1.0756 

0.0999 

1.0144 

1.2460 

1.1103 

1 

1.1712 

1.3176 

0.8801 

0.0249 

0.8061 

1.0973 

0.9230 

0.7890 

1 

0.9522 

0.9243 

0.0262 

0.8483 

1.1444 

0.9522 

0.7590 

1.0501 

1 

Conversion of Pounds of Different Nations. 

Eng. av. 

Kilogram. 

Prussia. 

Austria. 

Spain. 

Ilanover 

Russia. 

Sweden. 

1 

0.4536 

0.9072 

0.8110 

0.9839 

0.9320 

1.1076 

1.0664 

2.2016 

1 

2.00,0 

1.7857 

2.1692 

1.9842 

2.4419 

2.3511 

1.1023 

0.5000 

1 

0.8929 

1.0857 

1.0271 

1.2209 

1.17.55 

1.2346 

0.5600 

1.1200 

1 

1.2132 

1.1490 

1.3G75 

1.3166 

1.0104 

0.4610 

0.9211 

0.8243 

1 

0.9470 

1.1257 

1.0839 

1.0730 

0.4696 

0.9752 

0.8596 

1.0557 

1 

1.1884 

1.1442 

0.9028 

0.4095 

0.8190 

0.7313 

0.8883 

0.8414 

1 

0.9028 

0.9377 

0.4253 

0.8508 

0.7595 

0.9226 

0.8738 

1.0386 

1 


Ancient Measures of Length. 


Scripture. 

Feet. 

Inches. 

Hebrew. 

Feet. 

Inches 

Digit, .... 

• • • 

0.912 

Cubit, .... 

I 

9.868 

Palm = 4 Digits, 

« • • 

3.648 

Sabbatb day’s journey, 

3648 


Span = 3 Palms,. 

• • • 

10.94 

Mile = 4000 Cubits, 

7296 


Cuhit = 2 Spans, 

Fathom = 3.46 Cubits, . 

1 

9.888 

Day’s journey = 33.164 mi. 



7 

3.552 

Sacred Cubit, . 

2 

0.24 

Egyptian. Finger, 
Nahud Cubit, . 

• • • 

1 

.7374 

5.71 

Homan. 

Digit,. 


.7257 

Royal Cubit, , 

1 

8.66 

Uncia (Inch). 

• • • 

.967 

Grecian. 

Digit, .... 
Pons = 16 Digits, . 

• • • 

1 

0.754 

.0875 

I’es (foot) = 12 Uncias, . 
Cubit = 24 Digits, . 

Passus - 3.33 Cubits, 

• • • 

1 

4 

11.60 
5.406 
10.02 

Cubit, .... 

1 

1.598 

Millarium (mile),. . 

4842 

• • • 

Stadium, . 

604 

4.5 

Arabian. Foot, 

1 

1.14 

Mile = 8 Stadiums, 

4835 

. . . 

Babylonian. Foot, 

1 

1.68 





































































Foreign Weights and Measures. 03 


Foreign Measures of Length Compared with American. 

Places. 

Measures. 

Inches. 

Places. 

Measures. 



Inches. 

Amsterdam, 

Foot. . 

1114 

Malta, . . 

Foot, 




11-17 

Antwerp, . 

it 

• • • 

11-24 

Moscow, . 

it 

• 



13-17 

Bavaria, . 

it 

• • • 

11-42 

Naples, . . 

Pal mo. 




10-38 

Berlin, . . 

it 

• • • 

12-19 

Prussia, . 

Foot, . 

• 


• 

12-30 

Bremen, . 

it 

• • • 

11-38 

Persia, . . 

Arish, 




38"27 

Brussels, 

ii 

• • • 

11-45 

Rhineland, 

Foot, . 

• 



12-35 

Chiua, . . 

“ mathematic, 

13-12 


tt 




1079 

t. 

• • 

“ builder’s, 

12-71 

Rome, 

ii 

• 




11-60 

t. 

“ tradesman’s, 

13-32 

Russia, . . 

ii 

• 


• 


13-75 

it 

“ surveyor’s . 

12-58 

Sardinia, 

Pal mo, 

• 


• 

9-78 

Copenhagen, 

it 

• • • 

12-35 

Sicily, . . 

tt 

• 


• 


9-553 

Dresden, 

tt 

• • • 

11-14 

Spain, 

Foot, . 

• 


• 

11-03 

tin gland, . 

tt 

• • • 

12-00 

ii 

Toesas, 


• 


00-72 

Florence, . 

Braccio. 

21-09 

ii 

Palmo, 

• 



8-04 

France, 

Pied de Hoi, . 

12-79 

Strasburg, 

Foot, 


• 


11-39 

t. 

• • 

Metre,. 

39.381 

Sweden, . 

i» 

• 

• 



11-09 

Geneva, . 

Foot, 

19-20 

Turin, 

ii 

• 


• 


12-72 

Genoa, . . 

Pal mo, 

9-72 

Venice,. . 

ii 

• 

• 


• 

13-40 

Hamburg, 

Foot, 

11-29 

Vienna, . 

it 

• 


• 


12-45 

Hanover, . 

tt 

• • • 

11-45 

Zurich, . . 

tt 

• 

• 


• 

11-81 

Leipsic, . 

it 

• • • 

1111 

Utrecht, . 

tt 

• 


• 


1074 

Lisbon, . . 

tt 

• • • 

12-90 

Warsaw, . 

ti 

• 



• 

1403 

it 

Pal mo, 

8-04 







Foreign Hoad Measures Compared with American. 

Places. 

Measures. 

Yards. 

Places. 

Measures. 



Yards. 

Arabia, . . 

Mile, . 

2148 

Hungary, . 

Mile, . 

• 



9113 

Bohemia, . 

tt 

• • 

10137 

Ireland, . 

tt 

• • 


• 


30538 

Chiua, . . 

Li, 

029 

Netherlands, 

tt 

• 


• 

1093 

Denmark, 

Mile, 

8244 

Persia, . . 

Parasang, 


• 


0086 

England, . 

“ statute, 

1700 

Poland, . 

Mile, long, 

• 


• 

8101 

it 

“ geographical. 

2025 

Portugal, . 

League, . 


• 


6760 

Flanders, . 

ii 

0809 

Prussia, . 

Mile, . 

• 


• 

8408 

France, . 

League, marine, 

6075 

Rome, . . 

ii 

• 


• 


2025 

it 

“ common,. 

4801 

Russia, . 

Verst, . 

• 


• 

1107 

(( 

“ post, . 

42(54 

Scotland, . 

Mile, 


• 


1984 

Germany, . 

Mile, long, . 

10126 

Spain,. . 

League, common, 

7410 

Hamburg, 

ii 

• • 

8244 

Sweden, 

Mile, . 

• 


# 

11700 

Hanover, . 

ii 

• • • 

11559 

Switzerland, 

ii 

• 




9153 

Holland,. 

ii 

6395 

Turkey, . . 

Berri, . 

• 


• 

1826 

Foreign Measures of Surface Compared witli American. 


Places. 

Measures. 

Sq. Yds. 

Places. 

Measures. 

Sq. Yds. 

Amsterdam, 

Morgen, 

9722 

Portugal, . 

Geira, 

Morgen, 

6970 

Berlin, . . 

“ great, 

6786 

Prussia, . 

3053 


“ small, 

3054 

Rome, . . 

Pezza, 

Dessetina, 

3158 

Canary Isles, 

Fanegada, 

2422 

Russia, 

13066-6 

England, . 

Acre, . 

4840 

Scotland. . 

Acre.. 

6150 

Geneva, . 

Arpent, 

0179 

Spain, . . 
Sweden, 

Fanegada, 

5500 

Hamburg, . 

Morgen, . . 

11545 

Tunneland, 

5900 

Hanover, . 

ti 

• • 

3100 

Switzerland, 

Faux, . 

7855 

Ireland, . . 

Acre, 

7840 

Vienna, . . 

doth. 

6889 

Naples, . 

Moggia, . 

3998 

Zurich, . 

Common acre, 

3875 0 













































(54 Foreign Weights and Measures. 


Foreign Liquid Measures Compared with American. 

Places. 

Measures. 

Cub. Iii. 

Places. 

Measures. 

Cub. Iu. 

Amsterdam, . . 

Anker, . . . 

2331 

Naples, . . 

Wine Barille, 

2544 

U 

• 

Stoop, . . 

14G 

tt 

Oil Stajo, . 

1133 

Antwerp, . . . 

tt 

194 

Oporto, . . 

Almude, . . 

1555 

Bordeaux, . . 

Barrique, . 

14033 

Rome, 

Wine Barille, 

2.560 

Bremen, . . . 

Stubgeus,. . 

194-5 

tt 

• • 

Oil 

2240 

Canaries, . . 

Arrobas, 

949 

tt 

Boccali, . . 

80 

Constantinople, 

Almud, . . 

319 

Russia, . . 

Weddras, . 

752 

Copenhagen. 

Anker, . . 

2355 

tt 

Kimkas. . . 

94 

Florence, . . . 

Oil Bari lie, . 

1946 

Scotland, . 

Pint, . . . 

103-5 

it 

• • 

Wine “ 

2427 

Sicily, 

Oil Caffiri, . 

662 

France, . . 

Litre, . . . 

Gl-07 

Spain, . . 

Azumbras,. 

22-5 

Geneva, . . . 

Setier, . . 

2760 • 

t» 

Quartillos, 

30-5 

Genoa,.... 

Wine Barille, 

4530 

Sweden, . 

Hi mer, . . 

4794 

tt 

• • • 

Piute, . . . 

90-5 

tt 

Katina, . . 

159-57 

Hamburg, . . 

Stubgeu, . 

221 

Trieste,. . 

Orne,. . . 

4007 

Hanover, . . 

(t 

• • 

231 

Tripoli, . 

Mattari, . . 

1376 

Hungary,. . . 

Kimer, . . 

4474 

Tunis, . . 

Oil “ . . 

1157 

Leghorn, . . 

Oil Barille, . 

1942 

Venice, . 

Seochio, . . 

628 

Lisbon, . . . 

Almude, 

1040 

Vienna, 

Hi mer, . . 

3452 

Malta, . . . 

Caffiri, . . 

1270 

it 

Maas, . . . 

S6-33 

Foreign Dry Measures Compared with American. 

Places. 

Measures. 

Cub. Iu. 

Places. 

Measures. 

Cub. In. 

Alexandria, . . 

Rebel e, . . 

9587 

Malta, . . 

Saltne, . . . 

16930 

4< 

Kislos, . . 

10418 

Marseilles, 

Charge, . . 

9411 

Algiers, . . . 

Tarrie,. . . 

1219 

Milan, . . 

Moggj,. . . 

8444 

Amsterdam, . 

Mudde, . . 

6596 

Naples, . 

Temoli, . . 

3122 

U 

• • 

Sack, . . . 

4947 

Oporto, . . 

Alquiere, . . 

1051 

Antwerp, . . 

Viertel, . . 

4705 

Persia, 

Artaba. . . 

4013 

Azores, . . . 

Alquiere, . . 

731 

Poland,. . 

Zorzec, . . 

3120 

Beilin, . . . 

Scheffel,. . 

3180 

Riga, . . 

Loop. . . . 

3978 

Brenieu, . . . 

it 

• • 

4339 

Rome, . . 

Rtibbio, . . 

16904 

Candia, . . . 

Charge, . . 

9288 

tt 

Quarti, . . 

4226 

Constantinople, 

Kislos, . . . 

2023 

Rotterdam, 

Sadi, . . . 

6361 

Copenhagen, 

'l'oende, . . 

8489 

Russia, '. . 

Chetwert, . 

12448 

Corsica, . . . 

Stajo, . . . 

6014 

Sardinia, 

Starel li, . . 

2988 

Florence, . . . 

Stari, . . 

1449 

Scot laud, . 

Firlot, . . 

2197 

Geneva, . . . 

Coupes, . . 

4739 

Sicily, . 

Sal me gros, . 

21014 

Genoa, .... 

Mina, . . 

7382 

t» 

‘ generale, 

16886 

Greece, . . . 

Mediinni,. . 

2390 

Smyrna, . 

Kislos, . . . 

2141 

Hamburg, . . 

Schelfel, . 

6426 

Spain. . . 

Catrize, . . 

41269 

Hanover, . . 

Mai ter, . . 

6868 

Sweden, . 

Tunna, . . 

8940 

Leghorn, . . . 

Stajo, . . 

1501 

Trieste,. . 

Stari,. . . 

4521 

tt 

Sacco, . . . 

4503 

Tripoli, . 

Caffiri, . . . 

19780 

Lisbon, . . . 

Alquiere, . 

817 

'funis, . . 

It 

• • 

2)855 

ti 

• • • 

Fanega, . 

3268 

Venice, . 

Stajo, . . . 

4945 

Madeira, . . . 

Alquiere, , 

684 

Vienna,. . 

Metzen, . . 

3753 

Malaga, . . . 

Fanaga, . , 

3783 





English Measures of Capacity. 


The Imperial qallon measures 277’274 cubic inches, containing 10 lbs. 

Avoirdvt- 

pois of distilled water, weighed in air, 

at the temperature of 62°, the liarom- 

eter at 30 inches. 





For Grain. 

8 bushels = 1 quarter. 





1 quarter = 10-2694 cubic feet. 



Coal, or heaped measure. 

3 bushels 

= 1 sack. 




12 sacks 

= 1 chaldron 

. 


Imperial bushel = 2218G92 cubic indies. 



* Heaped bushel. 19,j ins. diam., cone G ins. high = 2812-4872 cubic ins. 

1 chaldron 

= 58 - 658 cubic feet, and 

weighs 3136 pounds. 


1 chaldron (Newcastle) = 

5936 pounds. 







































Foreign Weights and Measures. 65 


Foreign Weights Compared with American. 

Places. — . 

Weights. 

Lbs. per 
100 avoir. 

Places. 

Weights. 

Lbs. per 
100 avoir. 

Aleppo,. . . 

Rottoli, . . 

20.46 

Hanover, . . 

Pound, . . 

93.20 

• • 

Oke, . . . 

35.80 

Japan, . . 

Catty, . . 

76.92 

Alexandria, . 

Rottoli, . . 

107. 

Leghorn, . . 

Pound, . . 

133.56 

Algiers, . . 

44 

• • 

84. 

Leipsic, . . 

“ (common) 

97.14 

Amsterdam, . 

Pound,. . . 

91.8 

Lyons, . . . 

“ (silk), . 

98.81 

Antwerp, 

44 

96.75 

Madeira, 

(( 

• • 

143.20 

Barcelona,. . 

a 

112.6 

Mocha, . . 

Maund, . . 

33.33 

Batavia, . . 

Catty, . . 

76.78 

Morea, . . 

Pound, . . 

90.79 

Bengal, . . . 

Seer, . . . 

5o,o i 

Naples, . . 

Rottoli, . . 

50.91 

Berlin, . . 

Pound, . . 

96.8 

Rome, . . 

Pound, . . 

133.69 

Bologna, . . 

44 

125.3 

Rotterdam, . 

44 

• • • 

91.80 

Bremen, . . 
Brunswick, . 

u 

90.93 

Russia, . . 

(4 

110.86 

u 

97.14 

Sicily, . . . 
Smyrna, 

u 

• • • 

142.85 

Cairo, . . . 

Rottoli, . . 

105. 

Oke, . . . 

36.51 

Candia, . . . 

(4 

85.9 

Sumatra, . . 

Catty, . . 

35.56 

China, . . 
Constantinople 

Catty, . . . 

75.45 

Sweden, . . 

Pound, . . 

106.67 

Oke, . . . 

35.55 

44 

“ (miller’s). 

120.68 

Copenhagen, 

Pound, . . 

90.80 

Tangiers, . 

44 

• • • 

94.27 

Corsica, . . . 

44 

• • • 

131.72 

Tripoli, . . 

Rottoli, . . 

89.28 

Cyprus, . . 

Rottoli, . . 

19.07 

Tunis, . . 

44 

90.09 

Damascus, . . 

44 

25.28 

Venice, . . 

Pound(heavy) 

94.74 

Florence, . . 

Pound, . . . 

133.56 

44 

“ (light) 

150. 

Geneva, . . 

“ (heavy), 

82.35 

Vienna, . . 

44 

• • 

81. 

Genoa, . . 

44 44 

92.86 

Warsaw, 

44 

• • • 

112.25 

Hamburgh, . 

44 U 

93.63 






A Uniform System of Metrology much Needed. 

The preceding variety of tables of weights, measures and coins shows 
the great need of a uniform system of metrology throughout the world. 

The French are the first in adopting a uniform decimal system of 
metrology, and an International Decimal Association has been formed 
for the special purpose of advocating the introduction of the French 
system into other countries, which Association has now labored on that 
subject for some twenty years with but slow success. 

The metric system is now adopted all over the continent of Europe, 
and in South and Central America. Among the English-speaking 
nations the metric system is legalized, but not enforced. 

The principal difficulties in the way appear to be prejudices and jeal¬ 
ousy. It must be admitted that the introduction of a new system of 
metrology causes some temporary inconveniences, but the objection is 
only temporary . Some few countries have decimated their old units in 
preference to adopting the French system. 

One difficulty of the decimal system is, that the base 10 does not admit 
of more than one binary division without fraction. See A New System 
of Arithmetic, page 54. 


5 
























■rrrrr-— 


66 


Cotcwlsi's Integral Formulas. 


D IFFERENTIAIiSv IHTEGRAIS. 

X® 

y7x = x-fe Jxdx=~ +C, I 

yiax 3 cPx = 'lajrxZdx; = + 2: 

a?*+i 


Jx n dx 


n + 1 
,1* 


+c; ?■ 


_. < O’* /y> 

f\/x dx Jz™ dx = —- 5 —•+££ 4 


1/* 


r-x T 

- ix ax — 

Zi/x -+C P 

5> 

P 

A V* 

II 

•3 

1 

l~x + C r 

8- 

b 

A x ~ 

a 

1 

II 

1 


T 

J SIR. 

1 

Of? 

£ 

II 

1 

2p°’ 

Sr 

jfC 03.. 


4» 


z s -|- 


6 \ _ tfz 4 

lax = 


/ 


2i/x / 4 

ndx 


-b]/x-\-Cp 


x 

h dx 
■\-x 
•>rt x 3 dx 


P 

J a 

J b + ax 3 


— « l-x + C t 10 

= &i*(o + ar>+C; 11 
= £*(& -f- ctx^)> -f- C T 12 

/ ir2' 

faxdx + §xhlx-b\lx —+s*-&*c+(7,13* 

z 

/(rt 2 + 6 s ) = *(tf + ft*)+c; 14 

( n % 4%8 \ 

——ar+-g- 1+C r 15 

y3(ax—x 2 ) 2 ^—2 x)cfo = (ax —x®)*+ C, 16 
'n^'^dx) 


A 


= j/a s 1 -x"+(7, 17 
18 


i/a -f- x» 

_ ,y±£ +c , 

J a 2 — a: 2 a — * 


/l/« s +a: 5 <f* =2 1/« 2 +z 2 +7^'(+ 


l/a a -i-z 2 ), 19 
/]/a +b rdx — ~b(l/a -fbx) 3 4 - C\ 20 


©IFFERKSTTALS. INTEGRALS. 
dx 


A 


p/a 2 +z 2 


= l-(x+l/dAf&j, 21 


fZmx-dx — TubZ—muP, 22 
a■ 

b m 

jfmxdx = g (IP —<s-X 23 


ao 

A~a 


dx 

a®+z 2 


Tf 


24 


2a* 

V, *25 


a 

•/ 

& 


e h a 

y=y+y 

nab 


Jsin.xdx — — cosjr+(7, 27 

jFcos..xdx — simt+C?, 28 

_/tan.x<fx = — fcos^+C; 29 

fc-ot.xdx = — f-sin.r+C’, 30 

f dX — = 7-tan.-^-+3 31 

y sm^B 2 ^ 

-*«.£+£) + <* 32 
J cos-.x \4 2 / ’ 


/sin-r co&jc (£tr = — &Ln 5 .x -f Ct 83 

z 


A 


00 

sin. 6 x 


x 

CO 

cos.6 ar 


/ cos. 

x 
0 

A 
A 


dx = 


dx = 


T 

2 * 

OO, 


34 

35 


j ^_^-=cireIearcof which 7=tan. 


- = circle arc of which 

T/2 x — x 2 x — sin. versus. 37 

fffZadx % ~ffiaxdx~=fZtvxklx— ax^y C\ 
AA 2 (« + b) <ix 2 = (a +&)x 2 + C t 39 


/A 2 v 2 (7-c 2 -f 8 r x (fx dv-\-2 x 2 dv 2 =x v 3 ,40 
See page 101. 






































Geometry. 


67 


GEOMETRY. 

DEFINITIONS. 

Demonstration is a course of reasoning by which a truth is established. It 
ounsLsts of, 

Thesis, the truth to be established, and, 

Hypothesis, something assumed and to be demonstrated. 

Axiom is that which is self-evident and requires no demonstration. 

Theorem is something to be proved by demonstration. 

Postulate is something to be done, but is self-evident and requires no demon¬ 
stration. 

Problem is a question proposed and requires solution. 

Proposition is either a Theorem or a Problem. 

Corolary is an obvious conseqence deduced from something that has gone 
before. 

Scolium is a remark on preceding propositions, commonly demonstrated hy 
algebraical formulas. 

Lemma is something premised for a following demonstration. 

Geometrical Quantities! 

Point has position, but no magnitude. 

A Line lias length, without breadth or thickness. 

A Straight Line is the shortest distance between two points. 

Curved line is a length which in every point changes its direction. 

Superficies, Surface, Area, is that which has length and breadth, but no 
thickness. 

Plane surface is a surface which coincides with a straight line in every direc¬ 
tion. 

Curved surface is a plane which coincides with a curved line. 

A Solid has length, breadth and thickness. 

Circle. 

Circle, Cirumference, Periphery, is a curved line drawn on a plane surface, and 
bounded at a common distance from one point in the plane, (centre.) 

Radius is a line* drawn from the centre in a circle to the periphery. 

Diameter is aline drawn through the centre to the periphery, or the longest 
line in a circle. 

Chord is any line extending its both ends to the periphery of a circle, and doer 
not go through the centre. 

Arc is a part of a periphery. 

Circle plane, is a plane surface bounded within a circumference. 

Sector is a part of a circle-plane bounded within an arc aud two radii. 

Segment is a part of a circle plane bounded within a chord and an arc. 

Zone is a part of a circle included between two parallel chords. 

Lune is the space between the intersecting arcs of two eccentric circles. 

Oval is a round figure having one long and one short diameter at right angles 
to one another. 

Semicircle is a half circle. 

Quadrant is a quarter of a circle. 

Angles* 

Angle is the opening or inclination of two lines which meet in one point. 

If two radii being drawn from the extremities of a circle arc, to the centre; 
'the arc, is a measure of the angle at the centre. 

Right angle is when the opening is a quarter of a circle. 

Acute angle is less than a right angle. 

Obtuse angle is greater than a right angle. 

• Line by itself means a straight line. 















Constructions. 



1 . 

To divide a given line A B into two equal 
parts; and to erect a perpendicular through 
the middle. 

With the end A and B as centres, draw the 
dotted circle arcs with a radius greater than 
half the line. Through the crossings of the 
arcs draw the perpendicular CD, which divides 
the liue into two equal parts. 



£ 


2 . 


From a given point Con the line A B , to erect 
a perpendicular CD. 

With Cas a centre, draw the dotted circle arcs 
at A and B equal distances from C. With A 
and B as centres, draw the dotted circle arcs at 
D. From the crossing D draw the required 
perpendicular, D C. 



3. 


From a given point C at a distance from the 
line A B, to draw a perpend'cu'ar t<> the line. 

With Cas a centre, draw the dotted circle arc 
so that it cuts the line at A and B. With A and 
B as centres, draw the dotted cross arcs at D 
with equal radii. Draw the required perpen¬ 
dicular through C and D. 


Ck 


t 

/ 

I 

I 

I 

\ 

\ 

\ 


\J> 


XE 


-B 


4. 

At the end A of a given line A B , to erect a 
perpendicular A C. With the point D as a centre 
at. a distance from the line, and with A D as 
radius, draw the dotted circle arc so that it 
cuts the line at E; through E and Z), draw the 
diameter E C ; then join Caud A, which will be 
the required perpendicular. 




5. 



Through a given point C at a distance from the 
line A B, to draw a line CD parallel to A B. 

With C as a centre, draw the dotted arc E D ; 
with E as a centre, draw through C the dotted 
arc FC. With the radius F’Cand I? as a centre, 
draw the cross arc at D. Join C with the cross 
at D, which will be the required parallel line. 



6 . 

On a given line A B and at the point B, to 
construct an angle equal to the angle CDE. 

With D as a centre, draw the dotted arc C E\ 
and with the same radius and B as a centre, 
draw the arc G F\ then make G F equal to CE\ 
then join B F, which will form the required 
angle, FB G—CD E. 






































Constructions. 


69 



7. 


To divide the angle A B C into two equal parts. 
With C as a centre, draw the dotted arc DE\ 
with D and E as centres, draw the cross arcs at 
A 1 with equal radii. Join CF, which divides the 
angle into the required parts. 

Angles A C F= F C £ = i{A C B). 




8 . 


To divide an angle into two equal parts, when 
the lines do not extend to a meeting point. 

Draw the lines CD and CE parallel, and at 
equal distances from the lines A B and F G. 
With C as a centre, draw the dotted arc B G\ 
and with B and G as centres, draw the cross 
arcs H. Join CE, which divides the angle into 
the required equal parts. 



9. 


To construct a parallelogram, with the given 
sides A and B and angle C. 

Draw the base line D E , and make the angle 
FD E= C; lines D E= B and D F— A ; com¬ 
plete the parallelogram by cross arcs at G, and 
the problem is thus solved. 


10 . 


c 



To divide the line A B in the same proportion 
of parts as A C. 

Join C*andi?, and through the given divisions 
1, 2, and 3 draw lines parallel with CB, which 
solves the problem. 




11 . 


To find the centre of a circle which will pass 
through three given points A, B, and C. 

With B as a centre, draw the arc D EFG; 
and with the same radius and A as a centre, 
draw the cross arcs D and F\ also with C as a 
centre, draw the cross arcs E and G. Join D 
and E. and also F and G, and the crossing 0 is 
the required centre of the circle. 


12 . 

To construct a square upon a given line A B. 

With A B as radius and A and B as centres, 
draw the circle arcs A E D and B E C. Divide 
the arc B E in two equal parts at F, and with- 
E F as radius, and E as centre, draw the circle 
C FD. Join A and C, B and D , C'and D, which 
completes the required square. 

























70 


Constructions. 



13. 

i 

Through a given point A in a circumference, 
to draw a tangent to the circle. 

Through the given point A and centre C 
draw the line B C. With A as a centre, draw 
the circle arcs B and F; with B and F as cen¬ 
tres, draw the cross arcs D and E\ then join D 
and E, which is the required tangent. 


14. 


From a given point A outside of a circum¬ 
ference, to draw a tangent to the circle. 

Join A and C, and upon AC as a diameter 
draw the half circle A B C, wlnfch cuts the given 
circle at B. Join A and B, which is the re¬ 
quired tangent. 


15. 




To draw a circle with a given radius JR, that 
will be tangent to the circle A B C at C. 

Through the given point C, draw the diameter 
A G’extended beyond J)-, from C’setoti'the given 
radius II to D\ then D is the centre of the re¬ 
quired circle, which is tangent to the given 
circle at C. 



16 . 

To draw a circle with a given radius Ii, that 
will be tangent to two given circles. 

Join the centres A and B of the given circles. 
Add the given radius B to each of the radii of 
the given circle, and draw the cross arcs C, 
which is the centre of the circle required to 
be tangent to the other two. 



17. 

To draw a tangent to two circles of different, 
diameters. 

Join the centres C and c of the given circles, 
and extend the line to 7); draw the radii A C 
and ac parallel with one another. Join A a, 
and extend the line to D. On CD as a diam¬ 
eter, draw the half circle Ce D ; on c D as a 
diameter, draw the half circle cf D; then the 
crossings e and / are the tangential points of 
the circles. 



I 

TV 


18. 

To draw a tangent, between two circles. 

Join the centres Cand c of the given circles; 
draw the dotted circle arcs, and join the cross¬ 
ing m, n, which line cuts the centre line at a. 
With a C as a diameter, draw the half circle 
a f C ; and with a c as a diameter, draw the half 
circle cea; then the crossings e and / are the 
tangential points of the circles. 











Constructions. 


71 



20 . 

To find the centre and radios of a circle that 
•will be tangent to the given circle A B at C, and 
the line D E. 

Through the given point C, draw the tangent 
G F\ bisect the angle FG E\ then o is the centre 
of the required circle that will be tangent to 
A B at C, and. the line D E. 


21 . 

To find the centre and radius of a circle that 
will be tangent to the given line Ai?at C, and 
the circle I) E. 

Through the point C, draw the line E F at 
right angles to A B ; set off from C the radius 
r = CF of the given circle. Join G and F. 
With G and F as centres, draw the arc crosses 
m and n. Join run, and where it crosses the 
line E F is the centre for the required circles. 


22 . 

To find the centre and radius of a circle that 
will be tangent to the given line A B at C, and 
the circle 1) E. 

From C, erect the perpendicular C G ; set off 
the given radius r from C to H. With H as a 
centre and r as radius, draw the cross arcs on 
the circle. Through the cross arcs draw the 
line/6?; then G is the centre of the circle arc 
FIG\ which tangents the line at C and the 
circle at F. 


23 . 

Between two given lines, to draw two circles 
that will be tangent to each other and the lines. 

Draw the centre line A B between the given 
lines; assume D to be the tangenting point of 
the circles; draw D C at right angles to A B. 
With C as centre and CD as radius, draw the 
circle EDF. From E, draw Em at right angles 
to E F; and from F, draw Fn at right angles to 
FE\ then m and n are the centres for the re¬ 
quired circles. 


To draw a circular arc that will be tangent to 
two given lines AB and CD inclined to one 
another, one tangential point E being given. 

Draw the centre line G F. From //draw EF 
at right angles to A B; then F is the centre of 
the circle required. 


19 . 

With a given radius r, to draw a circle that 
will be tangent to the given line A B and the 
given circle CD. 

Add the given radius r to the radius R of the 
circle, and draw the are cd. Draw the Line ce ] 
parallel with and at a distance r from the line 
A B. Then the crossing c is the centre of the 
required circle that will be tangent to the given 
line and circle. 












72 


Constructions. 







To draw a circle that will be tangent to two 
lines and pass through a given point C on the 
line FC, which bisects the angle of the lines. 

Through C draw A B at right angles to CF\ 
bisect the angles DAB and EB A, and thecross- 
ing on CF is the centre of the required circle. 


26. 

To draw a cyma, or two circular arcs that will 
be tangent to each other, and two parallel lines 
at given points A and B. 

Join A and B\ divide A B into four equal 
parts and erect perpendiculars. Draw A m at 
right angles from A , and Bn at right angles 
from B\ then m and n are the centres of the 
circle arcs of the required cyma. 


27. 


To draw a talon, or two circular arcs, that will 
be tangent to each other, and meet two parallel 
lines at right angles in the given points .-t and B. 

Join .4 and B\ divided B into four equal parts 
and erect perpendiculars; then m and n are the 
centres of the circle arcs of the required talon. 


28. 

To plot a circular arc without recourse to its 
centre, its chord A B and height h being given. 

With the chord as radius, and A and B as cen¬ 
tres, draw the dotted circular arcs A C and B D. 
Through the point Odraw the lines A Oo and 
BOo. Make the arcs Co =Ao and Do = Bo. 
Divide these arcs into any desired number of 
equal parts, and number them as shown on the 
illustration. Join A and B with the divisions, 
and the crossings of equal numbers are points 

in the circle arc. 

_ 


To find the centre and radius of a circle that 
will be tangent to the three sides in a triangle. 

Bisect two of the angles in the triangle, and 
the crossing C is the centre of the required 
circle. 


30. 

To inscribe an equilateral triangle in a circle. 
With the radius of the circle and centre C 
draw the arc D FE\ with the same radius, and 
D and E as centres, set off the points A and B. 
Join A and B, B and C, C and J., which will be 
the required triangle. 



















Constructions. 


73 



31 . 


To inscribe a square in a given circle. 

Draw the diameter A B, and through the cen¬ 
tre erect the perpendicular CD, and complete 
the square as shown in the illustration. 



32 . 

To describe a square about, a given circle. 

Draw the diameters AB and CD at right 
angles to one another; with the radius of the 
circle, and A,B, C, and D as centres, draw the 
four dotted half circles which cross one another 
in the corners of the square, and thus solve the 
problem. 


V 



33 . 


To inscribe a pentagon in a given circle. 

Draw the diameter A B, and from the centre 
C erect the perpendicular CD. Bisect the radius 
A Cat E: with E as centre, and DE as radius, 
draw the arc D E, and the straight. lineD-Fis the 
length of the side of the pentagon. 




34 . 

To construct a pentagon on a.given line A B. 

From B erect BC perpendicular to and half 
the length of A B ; join A and C prolonged to 
Z>; vvith C as a centre and CB as radius,draw 
the arc B D \ then the chord B D is the radius 
of the circle circumscribing the pentagon. With 
A and B as centres, and B D as radius, draw the 
cross 0 in the centre. 


35 . 

To construct a pentagon on a given line A B 
without resort to its centre. 

From B erect Bo perpendicular and equal to 
A B\ with C as centre and do as radius, draw 
the arc Do ; then A D is the diagonal of the 
pentagon. With A D as radius and A as centre, 
draw the arc D E\ and with B as centre and A B 
as radius, finish the cross E, and thus complete 
the pentagon. 

36 . 


To construct a hexagon in a given circle. 

The radius of the circle is equal to the s-ide 
of the hexagon. 






















Constructions. 


74 


(f) 

37. 

To construct a Heptagon. 

The appotem a in a hexagon is the length of 
the side of a heptagon. 

Set off A B equal to the radius of the circle; 
draw a from the centre C at right angles to 
A B\ then a is the required side of the hep¬ 
tagon. m 

\ / f% / 

'V#w 

38. 

To construct an octagon on the given line A B. 

Prolong A B to C. With B as centre and A B 
as radius, draw the circle A FD EC', from B, 
draw B I at right angles to A B\ divide the 
angles A B 1) and D B C each into two equal 
parts: then B E is one side of the octagon. With 
A and E as centres, draw the arcs H K E and 
A KI, which determine the points H and i, and 
thus complete the octagon as shown in the illus¬ 
tration. 

TTTv 
\ • ,-- a ' 

z/V 

- - -' ^ V \ s **•... 

i \ / l 

i \j ! / j 

39. 

To cut off the corners of a square, so as to 
make it a regular octagon. 

With the corners as centres, draw circle arcs 
through the centre of the square to the side, 
which determines the cut-off. 

A A /A a/^ 

40. 

The area of a regular polygon is equal to the 
area of a triangle whose base is equal to the 

00 IKn 

sum of all the sides, and the height a equal to 
the appotem of the polygon. 

VIV ^—%-•*- 

The reason of this is that the area of two or 


more triangles ABC and ADC having a com¬ 
mon or equal base b and equal height h are 
alike. 

\* 

aV~ T^x 

\ X «s/ * / \ 

\ / '7 ' / A \ 
v '''>A ', 

41. 

To construct any regular polygon on a given 
line A B without resort to its centre. 

Extend A B to C, and, with B as centre, draw 
the half circle A D B. Divide the half circle 
into as many parts as the number of sides in 
the polygon, and complete the construction as 
shown in the illustration. 

jCV> ! \#V, 

Ah i/A 

\ c -< - 

\ \ 1 ' 3 
\ll 

"-’i' 

42. 

To construct an isometric ellipse by compasses 
and six circle arcs. 

Divide 0 A and 0 B each into three equal 
parts; draw the quadrant A C. From C, draw 
the lino Cc through the point 1. Through the 
points 2 draw de at. an angle of 45° with the 
major axis. Then 2 is the centre for the ends 
of the ellipse; e is the centre for the arc dc: 
and C is the centre for the arc cf. 



































Constructions. 


75 



To construct an Hyperbola by plotting, 
Having given the transverse axis B C, ver¬ 
texes .4 a, and foci//'. Set off any desired num¬ 
ber of parts on the axis below the focus, and 
number them 1, 2, 3, 4, 5, etc. Take the dis¬ 
tance al as radius, and, with/' as centre, strike 
the cross 1 with /'l = al. With the distance 
J 1, and the locus / as centre, strike the cross 1 
with the radius FI = A 1, and the cross 1 is a 
point in the hyperbola. 

~44. 

To draic an Hyperbola by a pencil and a string, 
Having given the transverse axis B C, foci f' 
and/, and the vertexes A and a. Take a rule B, 
and fix to it a string at e; fix the other end of 
the string at the focus /. The length of the 
string should be such that, when the rule R is 
in the position/' C, the loop of the string should 
reacli to A ; then move the rule on the focus /', 
and a pencil at P, stretching string, will trace 
the hyperbola. 


To construct a Parabola by plotting, 
Having given the axes, vertex, and focus of 
the parabola. Divide the transverse axis into 
any desired number of parts 1, 2, 3, etc., and 
draw ordinates through the divisions; take the 
distance A 1, and set it off on the 1st ordinate 
from the focus / to a, so that A 1 =f a. Repeat 
the same operation with the other ordinates— 
that is, set oif the distance A 5 from / to e, so 
that A 5=/e; and so the parabola is constructed. 



46. 

To draw a Parabola by a pencil and a string, 

Having given the two axes, vertex, and focus 
of the parabola. Take a square c d e, and fix to 
it a string at e; fix the other end of the string 
at the focus/. The length of the string should 
be such that when the square is in the position 
of the axis A f, the string should reach to the 
vertex A. Move the square along B B, and the 
pencil P will describe the parabola. 



47. 

The Tractrix or Shield’s anti-friction curve. 

R represents the radius of the shaft, and 
C 1, 2, 3, etc., is the centre line of the shaft. 
From o, set off the small distance oa\ and set 
off a 1 = R. Set off the same small distance 
from a to b, and make b 2 = R. Continue in the 
same way with the other points, and the anti¬ 
friction curve is thus constructed. 



48. 

Isometric Perspective. 

This kind of perspective admits of scale 
measurements the same as any ordinary draw¬ 
ing, and gives a clear representation of the 
object. It is easily learned. All horizontal 
rectangular lines are drawn at an angle of 30°. 

All circles are ellipses drawn as shown on the 
preceding page (No. 42). 











































76 


Constructions. 



49 


To construct an ellipse. 

With o as a centre, draw two concentric cir¬ 
cles with diameters equal to the long and short 
axes of the desired ellipse. Draw from o any 
number of radii, A, B, &c. Draw the line Bb' 
parallel to n and bb 1 parallel to m, then b' is a 
point in the desired ellipse. 


50 . To draw an ellipse with a string. 


Having given the two axes, set off from c 
half the great axis at a and b , which are the 
two foci in the ellipse. Take an endless string 
as long as the three sides in the triangle a, b, c, 
fix two pins or nails in the foci, one in a and 
one in b, lay the string round n and b, stretch 
it with a pencil d, which then will describe the 
desired ellipse. 



51 


To draw an oval by circular arcs. 


Divide the long axis into three equal parts, 
draw the two circles and where they intersect 
one another are the centres for the tangent 
arcs of the oval as shown by the figures. 



52 To draw an oval, by circular arcs. 
Given the two axes, set off the short axis 


from A to b, divide b B into three equal parts, 
set off' two of these parts from o towards c 
and c which are the centres for the ends of 
the oval. Make equilateral triangles on c c , 
when ee will be the centres for the sides of the 
oval. If the long axis is more than twice the 
short one, this construction will not make a 


good oval. 


53 


To construct an ellipse. 


— 7 

m 

v 

s x 

N \ 

\ 

* 

r 


Given the two axes, set off half the long axis 
from c to//, which will be the two focuses in 
the ellipse. Divide the long axis into any num¬ 
ber of parts, say a to be a division point. Take 
A a as radius and f as centre and describe a 
circle arc about b , take a B as radius and / as 
centre describe another circle arc about b, then 
the intersection b is a point in the ellipse, and 
so the whole ellipse can be constructed. 



54 


To draw an ellipse that will be tangent to 
two parallel lines in A and B. 


Draw a semicircle on A B, draw ordinates 
in the circle at right angle to A B, the corre¬ 
sponding and equal ordinates for the ellipse 
to be drawn parallel to the lines, and thus the 
elliptic curve is obtained as shown by the 


■ figure. 

















































Constructions. 


77 



55 To construct a cycloid. 

The circumference C=314 D. Divide the 
rolling circle and base line C into a number 
of equal parts, draw through the division 
point the ordinate^ and abscissas, make a a/ 
=V b b'=2' e , c c'=3 J\ then o' b' and d are 
points in the cycloid. In the Epicycloid and 
Hypocycloid the abscissas are circles and the 
ordinates are radii to one common centre. 





56 Evolute of a circle. 

Given the pitch p, the angle v, and radius r. 
Divide the angle v into a number of equal 
parts, draw the radii and tangents for each 
part, divide the pitch p into an equal number 
of equal parts, then the first tangent will be 
one part, second two parts, third three parts, 
&c., and so the Evolute is traced. 


57 To construct a spiral with compasses 
and four centres. 

Given the pitch of the spiral, construct a 
square about the centre, with the four sides 
together equal to the pitch. Prolong the 
sides in one direction as shown by the figure, 
the corners are the centres for each arc of the 
external angles. 


58 To construct a Parabola. 

Given the vertex A, axis x , and a point P. 
Draw A B at right angle to x, and B P parallel 
to x, divide A B and B P into an equal num¬ 
ber of equal parts. From the vertex A draw 
lines to the divisions on B P, from the divi¬ 
sions on A B draw the ordinates parallel to x , 
the corresponding intersections are points in 
the parabola. 




59 To construct a Parabola. 

Given the axis of ordinates B, and vertex A. 

Take A as a centre and describe a semicircle 
from B which gives the focus of the parabola at 
f. Draw any ordinate y at right angle to the 
abscissa A x, take a as radius and the focus/ 
as a centre, then intersect the ordinate y, by 
a circle-arc in P which will be a point in the 
parabola. In the same manner the whole 
Parabola is constructed. 

60 

To draw an arithmetic spiral. 

Given the pitch p and angle v, divide them 
into an equal number of equal parts say 6. 
make 01=01, 0 2=02, 03=0 3, 0 4=04, 06=06, 
and 0 6=the pitch p ; then join the points 1, 2, 
3, 4, 6, and 6, which will form the spiral re¬ 
quired. 






























































78 


The Circle. 


Periphery or Circum¬ 
ference. 

n d = 3.14 d. 


THE CIRCLE. 

Notation. 


d = diameter of the circle, 
r = radius of the circle. 
p —z periphery or circumfer¬ 
ence. 

a = area of a circle or part 
thereof. 


c = chord of a segment, length 
of. 

h = height of a segment. 
s = side of a regular polygon. 
v = centre angle. 
w = polygon angle. 


b = length of a circle-arc. 

All measures must be expressed by the same unit. 

Formulas for tlie Circle. 


V 

V 


2tt r = 6.28r. 


p = 2 ;/ r a = 

3.54]/ ci. 

_2a _4a 


Diameter and Radius. 

d = r- = JN. 

ir 3.14 

= JL _ P 

27r 


6.28 


d 


2aI- = 1.128|/o. 




0.564]/ a. 


Area of the Circle. 

7T (jP 

a = —— = 0.7 85d?. 
4 

a = 7r r 1 = 3.147' 2 . 






F 


a 


4t r 12.56 

pr p d 

T ~ T’ 


7T = 3.141592653589793238402043383279502884197169399 


2tt=6.283185 

i7r=0.785398 

1 =0.318310 

7r 

3tt=9.424778 

$*■=1.047197 

2 =0.636619 

4tt=12.566370 

|tt=1.570796 

8 =0.954929 

7T 

5ttz=1 5.707963 

1tt=0.392699 

1=1.273239 

6tt=:1 8.849556 

$7t=0.523599 

#1 

6 =1.909859 

7tt=21.991148 

T yr=0.261799 

7T 



1=2.546478 

8tt=25.132741 

|7t=2. 094394 

7T 

12 =3.819718 

9t=28.274334 

¥ i^7r=0.008726 

7T 


860 =114.5915 

7T 

tt 2 =9.869650 


l/ 7T—1.772453 
a/—=0.564189 

\ 7T 

-x/—=1.253314 

\2 




-=0.797884 

7T 


Log. 7T= 

0.49714087 































Longimetrt, 


79 



















































80 


Longimetet 



|07 


V — V, ID = to. 
to -f v — 180°, to > v. 





68 


D = B + C , A' + B'+ C=180°, 
B = D-C y A + B + C = 180°, 
A' = A, B' = 5'. 



69 


A + I? + C - 180°, 
A' = A, B' =B. 



70 


E+C = A + D = 180°, 
D = B + C, 

E = A + B. 


k ^ ' 

. 

b 

9 

O' 

ab 


71 


(a + £)* “ a* + 2o5 + 


72 


■KT-—■ 

r'f 
i\ / 

(a>~b) z 

,/\ 

i _ i 


(a — &)* — a" 1 — 2a 5 + b*. 






































Longimetry 


81 


* j 

L-' 

"V _ 

6 ’ 



73 


(a + b) (a — b) = a* — 4». 


i 
























































82 


Longimetry. 









































LONGIMETRr. 


83 











































84 


Longimetry. 


Hi' 

To find the length of a Solid Spiral. 

nr* l r 

l = nm = , n= — = » 

P 7ir P 

p nr* r P= Pitch. 

1 n * w. = wo. turns. 

(U 

7o yZ/icZ Z/ie length of an Open Spiral. 

1 = n n { R + r)> 

‘-p(R' 



93 Periphery of an Ellipse. 

VJ 

LV 

_p = 2v'l> 3 + l'4G74d\ 


4 



O 

\ 

A. 

4- 

3 


^"4 

-li 


94 


To construct a screw or Helix. 



95 


To rectify a Circumference. 

R = 0-555355 d = 1-1107 r = 0-7071 S. 
S = 0-785398 d = 1-57079 r = 1-4142 R. 
d = 1 -27322 S = 1 -79740 R--=2r. 



96 


To square a Circle. 


f R = 0-626657 d = 1-253314 r = 0-7071 S. 
S = 0-886226 d = 1 -77245 r = 1-4142 R. 
d = 1-12838 S = 1-5367 R = 2 r. 





















































Polyhedrons. 


85 



Tetrahedron . 

r = 0-20413 s. 

R = 0-61237 s. 
a = 1*73205 s a . 
c = 0-11785 s 3 . 




Hexahedron. 

r = 0-50000 s. 
i? = 0-86602 s. 
a = 6-00000 sa. 
c = 1-00000 s». 


Octahedron. 

r = 0-40721 s. 
R = 0-70710 s. 
a = 3-46410 s*. 
c = 0-47140 s* 


Dodecahedron. 

r = 1-11350 s. 
R = 1-40122 s. 
a = 20-6457 §*. 
c = 7-66312 s*. 


Icosahedron. 

r = 0-7558 s* 
R = 0-9510 s. 
a = 8-66025 s’, 
c = 2-18169 s\ 


r = Radius of an inscribed Sphere. 

R = Radius of circumscribed Sphere, 
a = Area of the Polyhedrons. 

C = Cubic contents of the Polyhedrons. 
S = Side or edge of the Polyhedrons. 










































































































86 


PlANEMETRY. 



102 


Square . 

a = s’ = 46\ 


a = 0-5 d 2 


< CL > 

103 Redangle. 

a = a b } 

a = A Vd'-b\ 

9 


104 Triangle . 

A A .. , 
a = 2 - = ib A, 

-V *»-T' 

"‘Al 


b 

105 Triangle. 

a =- 46 h t 



Quadrangle. 
a = 4A(a + b). 


107 Quadrangle. 


a = 4( a [A + h'] + bh' + c A). 









































PLAKEMETRr. 


87 



108 Circle Plane. 

a => Tt r a = 0-785 (I*, 

n = ~= 0.0796 P\ 


109 Circle Ring. 

a = 77(P a — r a ) = rc(P + r)(P—r), 

a = 0-785(2)°-<P). 


110 Sector. 

a = ibr, 

: 

77 r 2 V r-iV ■ 

a ~ 360 ~ 114-5* 

<. 

111 Segment. 

a = — c (j— A)], 

n r a v c , t x 

a ~ 360 + 2 r ^ 

/% : 

\ 

r 

y 

112 Quadrant. 

a = 0-785 r° = 0-3927 e*. 

> 

2 


113 

a = 0-215 r* = 0*1075 e*. 




































ss 


Planemetet. 




Irregular Figure, 
a = b(h + h’ + h"). 

Ellipsoid. 

a = 8-88 r vQ^M 7 / 7 *, 
a = 2-22 d VD' + d\ 

Cylinder. 

a = 2 Tirh = n d h, 
i a a 

tl - rr- ' - —-• 

2 nr 7i d 


119 



Simpson’s Rule for Irregular Curves. 

Divide A B into any even number of parts of a 
length d. Erect the ordinates lt 0 , h lt h 2 , h 3 , and h A 
at the divisions. Then 

a — ~ (^o + + 2 /i 2 + 4/> a + /t 4 ). 

This is more accurate than 116. 
































































121 


Forus. 


a = 4 ti* R r = 39-44 R r , 


a = 9-86 D d. 


122 


123 


Sphere Sector. 


» -^r~(4 h + c). 


Circle Zone. 



n = 


= 2tfr A=^(c* + 4A 2 ) 



124 


Cone. 


a = tc R s, 

• • tz R VR' + h\ 



125 


a? = 


d h 


D- d 

it s 


Cone. 

R 


s + 


d $ 


D-d 


a = 


V> + d). 


v = 


180 D 180 {D-d). 
















































00 


Stereometry. 



128 Sphere Sector. 


c = | Tt r* h — 2*0944 r" h, 
C = % it r*(r + y/ r* — i c*). 


129 Section of Sphere. 




C, ^ 7t A°(r — J A), 

c* + 4A a 

c - * h \ -nr ~ ‘ *)* 


130 Cone. 


Ttr* h , 

c = — 0 — = 1*047 r h , 


c = 0*2618 <TA. 









































r ■* 


Stereometry. 



Wedge Frustum. 


h s, .. 
C - y( a + b). 


1 132 Cylinder. 

c = Tt r a h = 0.785 d 5 /t, 

c=^-=0 0796 / A. 

4 7t r 

133 Ellipsoid. 

c = 0-424 tt* K r* = 4-1847 jR r*. 
c = 0 053 t? D d* — 0-5231 D d* 


134 Paraboloid. 


= 1*5707 r a A. 


Pyramid 


c = 






















































Stereometry 


92 



138- Cask. 

c = 1-0453 1(0-4 />’+ 0-2 D d + 015<C), 

Gallon = + 2 DJ + 1-5 <f )• 



139 Cylinder Sections. 

C = it r*(l + T — | r), 

c = n r\l + V) — 2-1 r*. 


140 



Circular Spindle. 


c = ^r(* c s — 0-2 t/[c+§V c a +c^] >/d‘* + c*) 


i 

Example 1. Fig. 104. The base of a Triangle is b = 8 feet, 3 inches, and the 
height, h — 5 feet, 6 inches. What is the area a = ? 

bh 8-25X5-5 00 , . 

a = — —-^- = 22-6875 square feet. 

A A 

Example 2. Fig. 110. A Circle Sector having an angler = 39° and the radiu* 
r = 67 J inches. What is the area of the sector a = ? 

TT r'v 3*14 X 67-75* X 39° 

— = 1562T square feet. 


a = 


360 


360 


Example 3. Fig. 123. A Spherical Zone having its diameter c = 18J inches 
and height li = 7 j inches. What is the convex surface of the Zone ? 


a 


= —- ^c 2 + 4/t-j = ^18'5 2 + 4 X 7‘75 2 ^ = 614 - 87 square inches. 


Example 4. Fig. 100. Require the radius R of a Sphere that will circumscribe 
a Dodecahedron with the side s = 9 inches. 

R — 1-3G428 X 9 = 12 27852 inches, the answer. 

Example 5. Fig. 131. A Frustrum of a Cone having its bottom diameter D — 13 
inches, (he top diameter d = 5^ inches, and the height h = 25 inches. What is 
the cubic contents c = ? 

c rt h{n* + Dd + d 2 )=-- 0-2618 X 25 (l3* + 13 X 5'25 + 5'25*)= 20995 
cubic inches. 

Example 6. Fig. 138. A Cask having its hung diameter D — 36 inches, head 
diameter d — 28 inches, and length l — 56 inches, (inside measurement) how 
many gallons of liquid can be contained iu the cask ? (The gallon — 231 cub. in.) 

Gallons = 4 X 30* + 2 X 36 X 23 + 1*5 X 28«)= 214. 


























































Geometry.—Table of Polygons. 


93 


Example 7. Fig. 62. Required the length of the circular arc b, when the angle 
v = 42°, and the radius r = 4 feet, 3 inches ? 

nrv 3T4X 4-25X42 


6 = 


= 3-113 feet. 


180 180 

Example 8. Fig. 64. Required the radius of a circular arc, whose chord is 9 
feet, 4 inches, and height, /* = 1 foot, 8 inches ? 


r = 


c 2 +4/i2 9-332+4X1-66 0 98-0711 

8 h 


= 7-384 feet. 


8X1-66 13-28 

Example 9. Fig. 80. The three sictes in a triangle being, a = 6-42, 
and c = 8-66 feet. How high is the triangle over the base b l 


77ft 


d = 


a 2 -f& 2 —c 2 6-42H7-75 2 — 8'66 2 


26 


2X8-66 


= 1-5175 feet, 


the height h — }/a 2 — d r i — j/6.42 2 —1-5175 2 = 6-24 feet, the answer 

'Example 10. Fig. 89. The radius of a working beam is, r = 8-36 feet, the stroke 
S — 5-5 feet. IIow much is the vibration V — ? 


Vibration, 


V=r— = 8-36 — S^J 8-36 2 — 5 -^ 


21 

=0-471 feet = 5-65 inches = 5-—, the answer. 


TABLE OP POLYGONS. 


Number 
of sides 
in the 
Polygon. 


Trigon. 

Tetragon. 

Pentagon. 

Hexagon. 

Heptagon. 

Octagon. 

Nonagon. 

Decagon. 

Undecagon. 

Dodecagon. 




1-732 
1-4112 
1-1755 
1-0000 
0-86 77 
0-7653 
0-68 40 
0-6180 
0-5634 
0-5176 
0-4450 
0-4158 
0-3900 
0-3472 
0-3130 
0-2610 


Aral 
= Jk S*. 



0-4330 

1-0000 

1- 7205 

2- 5980 

3- 6339 

4- 8284 
6-1820 
7-6942 
9-3656 

11-196 

15-334 

17-642 

20-128 

25-534 

40-634 

45-593 


Apotem 

— AS. 



0-5000 

0-7071 

0-8090 

0-8660 

0-9009 

0-9238 

0-9396 

0-9510 

0-9595 

0-9659 

0-9762 

0-9781 

0-9807 

0-9848 

0-9877 

0-9914 



3-4641 

2-0000 

1-4536 

1-1547 

0-9631 

0-82S4 

0-7279 

0-6498 

0-5872 

0-5359 

0-4562 

0--f250 

0-4068 

0-3526 

0-3166 

0-2632 


5-1961 

4-0000 

3-6327 

3-4640 

3-3710 

3-3130 

3-2750 

3-2490 

3-2290 

3-2152 

3-1935 

3-1882 

3-1824 

3-1737 

3-1676 

3-1596 


Explanation of tiie Tabic for Polygons. 

The number of sides in the polygon is noted in the first column. 
k — tabular coefficient, to be multiplied as noted on the top of the columns. 
Example 1. IIow long is the side of an inscribed Pentagon, when the radius 
of the circle is 3 feet, and 4 inches? (4 inches = 0-333 feet.) 

3-333Xl'1755 = 3-9179 feet, the answer. 

Example 2. What is the area of a Heptagon when one of its sides is 13-75 in chos 
13-75 a XS-C339=C87-02 square inches. 



































































94 


Circumference and Area of Circles. 


Diam¬ 

eter. 

Circum. 

o 

Area. 

1 

3-1416 

0-7854 

2 

6-2832 

3-1416 

3 

9-4248 

7*0686 

4 

12-566 

12-5664 

5 

15-708 

19-6350 

6 

18-850 

28-2743 

7 

21-991 

38*4845 

8 

25*133 

50-2655 

9 

28-274 

63-6173 

10 

31-416 

78-54 

11 

34*558 

95-03 

12 

37-699 

113-10 

13 

40-S41 

132-73 

14 

43-982 

153-94 

15 

47-124 

176-71 

16 

50-265 

201-06 

17 

53-407 

226-98 

18 

56-549 

254-47 

19 

59-690 

283-53 

20 

62-832 

314-16 

21 

65-973 

346-36 

22 

69-115 

380-13 

23 

72-257 

415-4S 

24 

75-398 

452-39 

25 

78-540 

490-87 

26 

81-681 

530-93 

27 

84-823 

572-56 

28 

87-965 

615-75 

29 

91-106 

660*52 

30 

94-248 

706-86 

31 

97-389 

754-77 

32 

100-53 

804-25 

33 

103-67 

855*30 

34 

106-81 

907-92 

35 

109-96 

962-11 

36 

113-10 

1017-88 

37 

116-24 

1075-21 

38 

119-38 

1134-11 

39 

122-52 

1194*59 

40 

125-66 

1256-63 

41 

128-81 

1320-25 

42 

131*95 

1385-44 

43 

135-09 

1452-20 

44 

’ 138-23 

1520-52 

45 

141-37 

1590-43 

46 

144-51 

1661-90 

47 

147-65 

1734*94 

48 

150-80 

1809-55 

49 

153.94 

1885*74 

50 

157*08 

1963*5 



Circum. 

Area. 

Diam¬ 

eter. 

O 


51 

160-22 

2042-8 

52 

163-36 

2123.7 

53 

166-50 

2206.2 

54 

169-65 

2290-2 

55 

172-79 

2375-8 

56 

175*93 

2463-0 

57 

179-07 

2551-8 

58 

182*21 

2642-1 

59 

185*35 

2734-0 

60 

188-50 

2827-4 

61 

191-64 

2922*5 

62 

194*78 

3019-1 

63 

197-92 

3117-2 

64 

201-06 

3217-0 

65 

204-20 

3318-3 

66 

207-35 

3421-2 

67 

210-49 

3525-7 

68 

213-63 

3631-7 

69 

216*77 

3739*3 

70 

219*91 

3848-5 

71 

223*05 

3959*2 

72 

226-19 

4071.5 

73 

229-34 

4185-4 

74 

232-48 

4300*8 

75 

235.62 

4417-9 

76 

238*76 

4536*5 

77 

241-90 

4656-6 

78 

245*04 

4778*4 

79 

24S-19 

4901-7 

80 

251*33 

5026-6 

SI 

254*47 

5153-0 

82 

257*61 

5281-0 

S3 

260*75 

5410-6 

84 

263-89 

5541-8 

85 

267.04 

5674-5 

86 

270-18 

5808-8 

87 

273*32 

5944-7 

88 

276-46 

6082-1 

89 

279-60 

6221*1 

90 

2S2-74 

6361*7 

91 

285-88 

6503-9 

92 

289-03 

6647*6 

93 

292-17 

6792-9 

94 

295*31 

6939-8 

95 

29S-45 

7088*2 

96 

301-59 

7238*2 

97 

304*73 

73S9-8 

98 

307-88 

7543*0 

99 

311*02 

7697-7 

100 

314-1*6 

7854-0 



Circum. 

Area. 

Diam¬ 

eter. 

o 

® ' 

101 

317-30 

8011-9 

102 

320-44 

8171-3 

103 

323*58 

8332*3 

104 

326*73 

8494-9 

105 

329-87 

8659-0 

106 

333-01 

8824*7 

107 

336*15 

8992-0 

108 

339-29 

9160-9 

109 

342-43 

9331*3 

110 

345-58 

9503*3 

111 

348-72 

9676-9 

112 

351*86 

9852.0 

113 

355.00 

10028-S 

114 

35S-14 

10207-0 

115 

361-28 

10386-9 

116 

364-42 

10568.3 

117 

367-57 

10751*3 

118 

370-71 

10935*9 

119 

373*85 

11122-0 

120 

376-99 

11310 

121 

380.13 

11499 

122 

383*27 

11690 

123 

386-42 

11882 

124 

389-56 

12076 

125 

392-70 

12272 

126 

395*84 

12469 

127 

398-98 

12668 

128 

402-12 

12868 

129 

405-27 

13070 

130 

408-41 

13273 

131 

411-55 

13478 

132 

414-69 

13685 

133 

417-83 

13893 

134 

420*97 

14103 

135 

424-12 

14314 

136 

427.26 

14527 

137 

430-40 

14741 

138 

433*54 

14957 

139 

436*68 

15175 

140 

439-82 

15394 

Ml 

442-96 

15615 

142 

446-11 

15837 

143 

449-25 

16061 

144 

452-39 

162S6 

145 

455-53 

16513 

146 

458-67 

16742 

147 

461*81 

16972 

148 

464-96 

17203 

149 

468-10 

17437 

150 

471*24 

17671 






















Diart 

eter, 

151 

152 

153 

151 

155 

150 

157 

158 

159 

100 

161 

162 

163 

164 

165 

166 

167 

168 

169 

170 

171 

172 

173 

174 

175 

176 

177 

178 

179 

180 

181 

1S2 

183 

184 

185 

186 

187 

188 

189 

190 

191 

192 

193 

194 

195 

196 

197 

198 

199 

200 


Circumference and Area of Circles. 


96 


Area. 


Circum. 

Area. 


Circum. 

Area. 


Diam- 



Diam- 

f \ 


lisp 

eter. 

o 

ijlp 

eter. 

o 

ISP 

17908 

201 

631-46 

31731 

251 

788-54 

49481 

18146 

202 

634-60 

32047 

252 

791-68 

49876 

18385 

203 

637-74 

32365 

253 

794-82 

50273 

18627 

204 

640-89 

32685 

254 

797-96 

50671 

18869 

205 

644-03 

33006 

255 

801-11 

51071 

19113 

206 

647-17 

33329 

256 

804-25 

51472 

19359 

207 

650-31 

33654 

257 

807-39 

51875 

19607 

208 

653-45 

33979 

258 

810-53 

52279 

19856 

209 

656-59 

34307 

259 

813-67 

52685 

20106 

210 

659-73 

34636 

260 

816-81 

53093 

20358 

211 

662-88 

34967 

261 

819-96 

53502 

20612 

212 

666-02 

35299 

262 

823-10 

53913 

20867 

213 

669-16 

35633 

263 

826-24 

54325 

21124 

214 

672-30 

35968 

264 

829-38 

54739 

21382 

215 

675-44 

36305 

265 

832-52 

55155 

21642 

216 

678-58 

36644 

266 

835-66 

55572 

21904 

217 

681-73 

369S4 

267 

838-SI 

55990 

22167 

218 

684-87 

37325 

268 

841-95 

56410 

22432 

219 

688-01 

37668 

269 

845-09 

56832 ' 

22698 

220 

691-15 

38013 

270 

848-23 

57256 

22966 

221 

694-29 

38360 

271 

851-37 

57680 

23235 

222 

697-43 

38708 

272 

854-51 

58107 

23506 

223 

700-58 

39057 

273 

857*66 

58535 

23779 

224 

703-72 

39408 

274 

860-80 

58965 

24053 

225 

706-86 

39761 

275 

863-94 

59396 

24328 

226 

710-00 

40115 

276 

867-08 

59S28 

24606 

227 

713-14 

40471 

277 

870-22 

60263 

24885 

22S 

716-28 

40828 

278 

873-36 

60699 

25165 

229 

719-42 

41187 

279 

876-50 

61136 

25447 

230 

722-57 

41548 

280 

879-65 

61575 

25730 

231 

725-71 

41910 

281 

882-79 

62016 

26016 

232 

728-85 

42273 

282 

885-93 

62458 

26302 

233 

731-99 

42638 

283 

889-07 

62902 

26590 

234 

735-13 

43005 

284 

892-21' 

63347 

26880 

235 

738-27 

43374 

285 

895-35 

63794 

27172 

236 

741-42 

43744 

286 

898-50 

64242 

27465 

237 

744-56 

44115 

287 

901-64 

64692 

27759 

238 

747-70 

44488 

288 

904-78 

65144 

28055 

239 

750-84 

44863 

2S9 

907-92 

65597 

28353 

240 

753-98 

45239 

290 

911-06 

66052 

28652 

241 

757-12 

45617 

291 

914-20 

66508 

28953 

242 

760-27 

45996 

292 

917-35 

66966 

29255 

243 

763-41 

46377 

293 

920-49 

67426 

29559 

244 

766-55 

46759 

294 

923-63 

67887 

29865 

245 

769-69 

47144 

295 

926-77 

68349 

30172 

246 

772-83 

47529 

296 

929-91 

6S813 

30481 

247 

775-97 

47916 

297 

933-05 

69279 

30791 

248 

779-12 

48305 

298 

936-19 

69747 

31103 

249 

782-26 

48695 

299 

939-34 

70215 

31416 

250 

785-40 

49087 

300 

942-48 

70686 

































96 Circumference and Area of Circles. 



Circum. 

Area. 


Circum. 

Area. 


Circum. 

Diam¬ 

eter. 

o 

UP 

Diam¬ 

eter. 

O 

fjP 

Diam¬ 

eter. 

o 

301 

945-62 

71158 

351 

1102-70 

96762 

401 

1259-78 

302 

948-76 

71631 

352 

1105-84 

97314 

402 

1262-92 

303 

951-90 

72107 

353 

1108-98 

97868 

403 

1266-06 

304 

955-04 

72583 

354 

1112-12 

9S423 

404 

1269-20 

305 

958-19 

73062 

355 

1115-27 

98980 

405 

1272-35 

306 

961-33 

73542 

356 

1118-41 

99538 

406 

1275-49 

307 

964-47 

74023 

357 

1121-55 

100098 

407 

1278-63 

308 

967-61 

74506 

358 

1124-69 

100660 

408 

1281-77 

309 

970-75 

74991 

359 

1127*83 

101223 

409 

1284-91 

310 

973-89 

75477 

360 

1130-97 

101788 

410 

1288-05 

311 

977-04 

75964 

361 

1134-11 

102354 

411 

1291-19 

312 

980-18 

76454 

362 

1137-26 

102922 

412 

1294-34 

313 

983-32 

76945 

363 

1140-40 

103491 

413 

1297-48 

314 

986-46 

77437 

364 

1143-54 

104062 

414 

1300-62 

315 

9S9-60 

77931 

365 

1146-68 

104635 

415 

1303-76 

316 

992-74 

78427 

366 

1149-82 

105209 

416 

1306-90 

317 

995-88 

78924 

367 

1152-96 

105785 

417 

1310-04 

318 

999-03 

79423 

368 

115611 

106362 

418 

1313-19 

319 

1002-17 

79923 

369 

1159-25 

106941 

419 

1316-33 

320 

1005-31 

80425 

370 

1162-39 

107521 

420 

1319-47 

321 

1008-45 

80928 

371 

1165-53 

108103 

421 

1322-61 

322 

1011-59 

81433 

372 

1168-67 

1086S7 

422 

1325-75 

323 

1014-73 

81940 

373 

1171-81 

109272 

423 

1328-89 

324 

1017-88 

82448 

374 

1174-96 

109858 

424 

1332-04 

325 

1021-02 

82958 

375 

1178-10 

110447 

425 

1335-18 

326 

1024-16 

83469 

376 

1181-24 

111036 

426 

1338-32 

327 

1027-30 

S3982 

377 

1184-38 

111628 

427 

1341 -46 

328 

1030-44 

84496 

378 

1187-52 

112221 

428 

1344-60 

329 

1033-58 

85012 

379 

1190-66 

112815 

429 

1347-74 

330 

1036-73 

85530 

380 

1193-81 

113411 

430 

1350-88 

331 

1039-87 

86049 

381 

1196-95 

114009 

431 

1354-03 

332 

1043-01 

86570 

382 

1200-09 

114608 

432 

1357-17 

333 

1046-15 

87092 

383 

1203-23 

115209 

433 

1360-31 

334 

1049-29 

87616 

3S4 

1206-37 

115812 

434 

1363-45 

335 

1052-43 

88141 

385 

1209-51 

116416 

435 

1366-59 

336 

1055-58 

88668 

386 

1212-65 

117021 

436 

1369-73 

337 

1058-72 

89197 

387 

1215-80 

117628 

437 

1372-88 

338 

1061-86 

89727 

388 

1218-94 

118237 

438 

1376-02 

339 

1065-00 

90259 

389 

1222-08 

118847 

439 

1379-16 

340 

1068-14 

90792 

390 

1225-22 

119459 

440 

1382-30 

341 

1071-28 

91327 

391 

1228-36 

120072 

441 

1385-44 

342 

1074-42 

91863 

392 

1231-50 

120687 

442 

1388-58 

343 

1077-57 

92401 

393 

1234-65 

121304 

443 

1391-73 

344 

1080-71 

92941 

394 

1237-79 

121922 

444 

1394-87 

345 

1083-85 

93482 

395 

1240-93 

122542 

445 

1398-01 

346 

1086-99 

94025 

396 

1244-07 

123163 

446 

1401-15 

347 

1090-13 

94569 

397 

1247-21 

123786 

447 

1404-29 

348 

1093-27 

95115 

398 

1250-35 

124410 

448 

1407-43 

349 

1096-42 

95662 

399 

1253-50 

125036 

449 

1410-58 

350 

1099-56 

96211 

400 

1256-64 

125664 

450 

1413-72 



126293 

126923 

127556 

128190 

128S25 ' 

129462 

130100 

130741 

131382 

132025 

132670 

133317 

133965 

134614 

135265 

135918 

136572 

137228 

137885 

138544 

139205 

139867 

140531 

141196 

141863 

142531 

143201 

143S72 

144545 

145220 

145896 

146574 

147254 

147934 

148617 

149301 

149987 

150674 

151363 

152053 

152745 

153439 

154134 

154830 

155528 

156228 

156930 

157633 \ 

158337 

159043 
























Circumference and Area of Circles. 97 



Circum. 

Area. 


Circum. 

Area. 


Circum. 

Area. 

Diam- 

Oi 


Diam- 



Diam- 



eter. 

w 

IjjP 

eter. 

U 

llP 

eter. 

(J 

lip 

451 

1416-86 

159751 

501 

1573-94 

197136 

551 

1731-02 

238448 

452 

1420-00 

160460 

502 

1577-08 

197923 

552 

1734-16 

239314 

453 

1423-14 

161171 

503 

1580-22 

198713 

553 

1737-30 

240182 

454 

1426-28 

161883 

504 

1583-36 

199504 

554 

1740-44 

241051 

455 

1429-42 

162597 

505 

1586-50 

200296 

555 

1743-58 

241922 

456 

1432-57 

163313 

506 

1589-65 

201090 

556 

1746-73 

242795 

457 

1435-71 

164030 

507 

1592-79 

201886 

557 

1749-87 

243669 

458 

1438-85 

164748 

508 

1595-93 

202683 

558 

1753-01 

244545 

459 

1441-99 

165468 

509 

1599-07 

203482 

559 

1756-15 

245422 

460 

1445-13 

166190 

510 

1602-21 

204282 

560 

1759-29 

246301 

461 

1448-27 

166914 

511 

1605-35 

205084 

561 

1762-43 

247181 

462 

1451-42 

167639 

512 

1608-50 

205887 

562 

1765-58 

248063 

463 

1454-56 

168365 

513 

1611-64 

206692 

563 

1768-72 

248947 

464 

1457-70 

169093 

514 

1614-78 

207499 

564 

1771-86 

249832 

465 

1460-84 

169823 

515 

1617-92 

208307 

565 

1775-00 

250719 

466 

1463-98 

170554 

516 

1621-06 

209117 

566 

1778-14 

251607 

467 

1467-12 

171287 

517 

1624-20 

209928 

567 

1781-28 

252497 

468 

1470-27 

172021 

518 

1627-35 

210741 

568 

1784-42 

253388 

469 

1473-41 

172757 

519 

1630-49 

211556 

569 

1787-57 

254281 

470 

1476-55 

173494 

520 

1633-63 

212372 

570 

1790-71 

255176 

471 

1479-69 

174234 

521 

1636-77 

213189 

571 

1793-85 

256072 

472 

1482-83 

174974 

522 

1639-91 

214008 

572 

1796-99 

256970 

473 

1485-97 

175716 

523 

1643-05 

214829 

573 

1800-13 

257869 

474 

1489-11 

176460 

524 

1646-20 

215651 

574 

1803-27 

258770 

475 

1492-26 

177205 

525 

1649-34 

216475 

575 

1806-42 

259672 

476 

1495-40 

177952 

526 

1652-48 

217301 

576 

1809-56 

260576 

477 

1498-54 

178701 

527 

1655-62 

21812S 

577 

1812-70 

261482 

478 

1501-68 

179451 

528 

1658-76 

218956 

578 

1815-84 

262389 

479 

1504-82 

180203 

529 

1661-90 

219787 

579 

1818-98 

263298 

480 

1507-96 

180956 

530 

1665-04 

220618 

580 

1822-12 

264208 

481 

1511-11 

181711 

531 

1668-19 

221452 

581 

1825-27 

265120 

482 

1514-25 

182467 

532 

1671-33 

222287 

5S2 

1828-41 

266033 

483 

1517-39 

183225 

533 

1674-47 

223123 

583 

1831-55 

266948 

484 

1520-53 

1S3984 

534 

1677-61 

223961 

584 

1834-69 

267865 

485 

1523-67 

184745 

535 

1680-75 

224801 

585 

1837-83 

268783 

486 

1526-81 

185508 

536 

1683-89 

225642 

586 

1840-97 

269702 

487 

1529-96 

186272 

537 

1687-04 

226484 

587 

1844-11 

270624 

488 

1533-10 

187038 

538 

1690-18 

227329 

588 

1847-26 

271547 

489 

1536-24 

187805 

539' 

1693-32 

228175 

589 

1850-40 

272471 

490 

1539-38 

188574 

540 

1696-46 

229022 

590 

1853-54 

273397 

491 

1542-52 

189345 

541 

1699-60 

229871 

591 

1856-68 

274325 

492 

1545-66 

190117 

542 

1702-74 

230722 

592 

1859-82 

275254 

493 

1548-81 

190S90 

543 

1705-88 

231574 

593 

1862-96 

276184 

494 

1551-95 

191665 

544 

1709-03 

232428 

594 

1866-11 

277117 

495 

1555-09 

192442 

545 

1712-17 

233283 

595 

1869-25 

278051 

496 

1558-23 

193221 

546 

1715-31 

234140 

596 

1872-39 

278986 

497 

1561-37 

194000 

547 

1718-45 

234998 

597 

1875-53 

279923 

498 

1564-51 

194782 

548 

1721-59 

235858 

598 

1878-67 

280862 

499 

1567-65 

195565 

549 

1724-73 

236720 

599 

1881-81 

281802 

500 

1570-80 

196350 | 

550 

1727-88 

237583 

600 

1884-96 

282743 


7 
























98 


Circumference and Area of Circles. 



Circum. 

Area. 


Circum. 

Area. 


Circum. 

Area. 

Diam¬ 

eter. 

O 

# 

Diam¬ 

eter. 

o 

# 

Diam¬ 

eter. 

O 

HP 

601 

1888-10 

283687 

651 

2045-18 

332853 

701 

2202-26 

385945 

602 

1891-24 

284631 

652 

2048-32 

333876 

702 

2205-40 

387047 

603 

1894-38 

285578 

653 

2051-46 

334901 

703 

2208-54 

38S151 

604 

1897*52 

286526 

654 

2054-60 

335927 

704 

2211-68 

389256 

605 

1900-66 

287475 

655 

2057-74 

336955 

705 

2214-82 

390363 

606 

1903-81 

288426 

656 

2060-S8 

337985 

706 

2217-96 

391471 

607 

1906-95 

289379 

657 

2064-03 

339016 

707 

2221-11 

392580 

608 

1910-09 

290333 

658 

2067-17 

340049 

708 

2224-25 

393692 

609 

1913-23 

291289 

659 

2070-31 

341083 

709 

2227-39 

394805 

610 

1916-37 

292247 

660 

2073-45 

342119 

710 

2230-53 

395919 

611 

1919-51 

293206 

661 

2076-59 

343157 

711 

2233-67 

397035 

612 

1922-65 

294166 

662 

2079-73 

344196 

712 

2236-81 

398153 

613 

1925-80 

295128 

663 

2082-88 

345237 

713 

2239-96 

399272 

614 

1928-94 

296092 

664 

20S6-02 

346279 

714 

2243-10 

400393 

615 

1932-08 

297057 

665 

2089-16 

347323 

715 

2246-24 

401515 

616 

1935-22 

298024 

666 

2092-30 

348368 

716 

2249-38 

402639 

617 

1938-36 

298992 

667 

2095-44 

349415 

717 

2252-52 

403765 

618 

1941-50 

299962 

668 

2098-58 

350464 

718 

2255*66 

404892 

619 

1944-65 

300934 

669 

2101-73 

351514 

719 

2258-81 

406020 

620 

1947-79 

301907 

670 

2104-87 

352565 

720 

2261-95 

407150 

621 

1950-93 

302882 

671 

2108 01 

353618 

721 

2265-09 

4082S2 

622 

1954-07 

303858 

672 

2111-15 

354673 

722 

2268-23 

409416 

623 

1957-21 

304836 

673 

2114-29 

355730 

723 

2271-37 

410550 

624 

1960-35 

305815 

674 

2117-43 

356788 

724 

2274-51 

411687 

625 

1963-50 

306796 

675 

2120-58 

357847 

725 

2277-65 

412825 

626 

1966-64 

307779 

676 

2123-72 

358908 

726 

2280-80 

413965 

627 

1969-78 

308763 

677 

2126-86 

359971 

727 

2283-94 

415106 

628 

1972-92 

309748 

678 

2130-00 

361035 

728 

2287-08 

416248 

629 

1976-06 

310736 

679 

2133-14 

362101 

729 

2290-22 

417393 

630 

1979-20 

311725 

680 

2136-28 

363168 

730 

2293-36 

418539 

631 

1982-35 

312715 

681 

2139-42 

364237 

731 

2296-50 

419686 

632 

1985-49 

313707 

682 

2142-57 

365308 

732 

2299-65 

420835 

633 

1988-63 

314700 

683 

2145-71 

366380 

733 

2302-79 

421986 

634 

1991-77 

315696 

684 

2148-85 

367453 

734 

2305-93 

423139 

635 

1994-91 

316692 

685 

2151-99 

368528 

735 

2309-07 

424292 

636 

1998-05 

317690 

686 

2155-13 

369605 

736 

2312*21 

425447 

637 

2001-19 

318690 

687 

2158-27 

370684 

737 

2315-35 

426604 

638 

2004-34 

319692 

688 

2161-42 

371764 

738 

2318-50 

427762 

639 

2007-48 

320695 

689 

2164-56 

372*845 

739 

2321-64 

428922 

640 

2010-62 

321699 

690 

2167-70 

373928 

740 

2324-78 

430084 

641 

2013-67 

322705 

691 

2170*84 

375013 

741 

2327*92 

431247 

642 

2016-90 

323713 

692 

2173-98 

376099 

742 

2331-06 

432412 

643 

2020-04 

324722 

693 

2177-12 

377187 

743 

2334*30 

433578 

644 

2023*19 

325733 

694 

2180-27 

378276 

744 

2337-34 

434746 

645 

2026-33 

326745 

695 

2183-41 

379367 

745 

2340-49 

435916 

646 

2029-47 

327759 

696 

2186-55 

380459 

746 

2343-63 

437087 

647 

2032-61 

328775 

697 

2189-69 

381554 

747 

2346-77 

43S259 

648 

2035-75 

329792 

698 

2192-83 

382649 

748 

2349-91 

439433 

649 

2038-89 

330810 

699 

2195-97 

383746 

749 

2353-05 

440609 

650 

2042-04 

331831 

700 

2199-11 

384845 

750 

2356-19 

441786 















Circumference and Area of Circees. 


99 



Circum. 

Area. 


Circum. 

Area. 


Circum. 

Diam¬ 


f"' 

Diam¬ 

A 


Diam¬ 


eter. 

u 

ISP 

eter. 

kJ 

eter. 


751 

2359-34 

442965 

SOI 

2516-42 

503912 

851 

2673-50 

752 

2362*48 

444146 

802 

2519-56 

505171' 

852 

2676-64 

753 

2365-62 

445328 

803 

2522-70 

506432 

853 

2679-78 

754 

2368-76 

446511 

804 

2525-84 

507694 

854 

2682-92 

755 

2371-90 

447697 

805 

2528-98 

508958 

855 

2686-06 

756 

2375-04 

4488S3 

806 

2532-12 

510223 

856 

2689-20 

757 

2378-19 

450072 

807 

2535-27 

511490 

857 

2692-34 

758 

2381-33 

451262 

808 

2538-41 

512758 

858 

2695-49 

759 

2384-47 

452453 

809 

2541-55 

514028 

859 

2698-63 

760 

2387-61 

453646 

810 

2544-69 

515300 

860 

2701-77 

761 

2390-75 

454841 

811 

2547-83 

516573 

861 

2704-91 

762 

2393-89 

456037 

812 

2550-97 

517848 

862 

2708-05 

763 

2397-04 

457234 

813 

2554-11 

519124 

863 

2711-19 

764 

2400-18 

458434 

814 

2557-26 

520402 

864 

2714-34 

765 

2403-32 

459635 

815 

2560-40 

521681 

865 

2717-48 

766 

2406-46 

460837 

816 

2563*54 

522962 

866 

2720-62 

767 

2409-60 

462041 

817 

2566-68 

524245 

867 

2723-76 

768 

2412-74 

463247 

818 

2569-82 

525529 

868 

2726-90 

769 

2415-88 

464454 

819 

2572-96 

526814 

869 

2730-04 

770 

2419-03 

465663 

820 

2576-11 

528102 

870 

2733-19 

771 

2422-17 

466873 

821 

2579-25 

529391 

871 

2736-33 

772 

2425-31 

468085 

822 

2582-39 

530681 

872 

2739-47 

773 

2428-45 

469298 

823 

2585-53 

531973 

873 

2742-61 

774 

2431-59 

470513 

824 

2588-67 

533267 

874 

2745-75 

775 

2434-73 

471730 

825 

2591-81 

534562 

875 

2748-89 

776 

2437-88 

472948 

826 

2594-96 

53585S 

876 

2752-04 

777 

2441-02 

474168 

827 

2598-10 

537157 

877 

2755-18 

778 

2444-16 

475389 

828 

2601-24 

538456 

87S 

2758-32 

779 

2447-30 

476612 

829 

2604-38 

539758 

879 

2761-46 

780 

2450-44 

477836 

830 

2607-52 

541061 

880 

2764-60 

781 

2453-58 

479062 

831 

2610-66 

542365 

881 

2767-74 

782 

2456-73 

480290 

832 

2613-81 

543671 

882 

2770-88 

783 

2459-87 

481519 

833 

2616-95 

544979 

883 

2774-03 

784 

2463-01 

482750 

834 

2620-09 

546288 

884 

2777-17 

785 

2466-15 

483982 

835 

2623-23 

547599 

885 

2780-31 

786 

2469-29 

485216 

836 

2626-37 

548912 

886 

2783-45 

787 

2472-43 

486451 

837 

2629-51 

550226 

887 

2786-59 

788 

2475-58 

487688 

S38 

2632-65 

551541 

888 

2789-73 

789 

2478-72 

488927 

839 

2635-80 

552858 

S89 

2792-88 

790 

2481-86 

490167 

840 

2638-94 

554177 

890 

2796-02 

791 

2485-00 

491409 

841 

2642-08 

555497 

891 

2799-16 

792 

2488-14 

492652 

842 

2645-22 

556819 

892 

2802-30 

793 

2491-28 

493897 

843 

2648-36 

558142 

893 

2805-44 

794 

2494-42 

495143 

844 

2651-50 

559467 

894 

2808-58 

795 

2497-57 

496391 

845 

2654-65 

560794 

895 

2811-73 

796 

2500-71 

497641 

846 

2657-79 

562122 

896 

2814-87 

797 

2503-85 

498892 

847 

2660-93 

563452 

897 

2818-01 

79S 

2506-99 

500145 

848 

2664-07 

564783 

898 

2821-15 

799 

2510-13 

501399 

849 

2667-21 

566116 

899 

2824-29 

800 

2513-27 

502655 

850 

2670-35 

567450 

900 

2827-43 


Area. 



508786 

570124 

571463 

572803 

574146 

575400 

576835 

578182 

579530 

580880 

582232 

583585 

584940 

586297 

587655 

589014 

590375 

591738 

593102 

594468 

595835 

597204 

598575 

599947 

601320 

602696 

604073 

605451 

606831 

608212 

609595 

610980 

612366 

613754 

615143 

616534 

617927 

619321 

620717 

622114 

623513 

624913 

626315 

627718 

629124 

630530 

631938 

633348 

634760 

636173 





















100 


Circumference and Area of Circles 



Circum. 

Area. 


Circum. 

Area. 


Circum. 

Are;u 

Diam¬ 

eter. 

o 

|j| 

Diam¬ 

eter. 

o 

mi 

Diam¬ 

eter. 

o 

m 

901 

2830-58 

637587 

934 

2934-25 

685147 

967 

3037-92 

734417 

902 

2833*72 

639003 

035 

2937-39 

686615 

968 

3041-06 

735937 

903 

2836*86 

640421 

936 

2940-53 

688084 

969 

3044-20 

737458 

904 

2840-00 

641840 

937 

2943-67 

689555 

970 

3047-34 

738981 

905 

2843-14 

643261 

938 

2946-81 

691028 

971 

3050-49 

740506 

900 

2846-28 

644683 

939 

2949-96 

692502 

972 

3053-63 

742032 

907 

2849-42 

646107 

940 

2953-10 

69397S 

973 

3056-77 

743559 

908 

2852-57 

647533 

941 

2956-24 

695455 

974 

3059-91 

745088 

909 

2855-71 

648960 

942 

2959-38 

696934 

975 

3063-05 

746619 

910 

2858-85 

650388 

943 

2962-52 

698415 

976 

3066-19 

748151 

911 

2861-99 

651S18 

944 

2965-66 

699897 

977 

3069*34 

749685 

912 

2865-13 

653250 

945 

2968-81 

701380 

978 

3072-48 

751221 

913 

2868-27 

654684 

946 

2971-95 

702865 

979 

3075-62 

752758 

914 

2871*42 

656118 

947 

2975-09 

704352 

980 

3078-76 

754296 

915 

2874-56 

657555 

948 

2978-23 

705840 

981 

3081-90 

755837 

916 

2877-70 

658993 

949 

29S1-37 

707330 

982 

3085-04 

757378 

917 

2880-84 

660433 

950 

2984-51 

708822 

983 

3088-19 

758922 

918 

2883-98 

661874 

951 

2987-65 

710315 

984 

3091-33 

760466 

919 

2887-12 

663317 

952 

2990-80 

711809 

985 

3094-47 

762013 

920 

2890-27 

664761 

953 

2993-94 

713307 

986 

3097-61 

763561 

921 

2893-41 

666207 

954 

2997-08 

714803 

987 

3100-75 

765111 

922 

2896-55 

667654 

955 

3000-22 

716303 

988 

3103-89 

766662 

923 

2899-69 

669103 

956 

3003-36 

717804 

989 

3107-04 

768215 

924 

2902-83 

670554 

957 

3006-50 

719306 

990 

3110-18 

769769 

925 

2905-97 

672006 

958 

3009-65 

720810 

991 

3113-32 

771325 

926 

2909-11 

673460 

959 

3012-79 

722316 

992 

3116-46 

772882 

927 

2912-26 

674915 

960 

3015-93 

723823 

993 

3119-60 

774441 

928 

2915-40 

676372 

961 

3019-07 

725332 

994 

3122-74 

776002 

929 

2918-54 

677831 

962 

3022-21 

726842 

995 

3125-88 

777564 

930 

2921-68 

679291 

963 

3025-35 

728354 

996 

3129-03 

779128 

931 

2924-82 

680752 

964 

3028-50 

729867 

997 

3132-17 

780693 

932 

2927-96 

682216 

965 

3031-64 

7313S2 

998 

3135-31 

782260 

933 

2931-11 

683680 

966 

3034-78 

732899 

999 

1 3138-45 

783828 


Explanation of the Preceding Table. 

When the diameter is expressed in more or less units than in the table, add or 
subtract so many figures more in the circumference; add or subtract twice as 
many in the area. 

Examples. 

Diameter. Circumference. Area. 

9370 2943G.7 68955500 

93.7 294.367 6895.55 

9.37 29.4367 68.9555 

0.937 2.94367 0.689555 

























Maxima and Minima 


101 


Two variable quantities x and y depended on one another, to find the 
value of one, when the other is a maxima or minima. 


*1 

y\ 


is a maxima or minima when its 
first differential coefficient 

<Py 



When the second d • coef. is positive, y is a minimum, and when 

negative y is a maximum. The variables may have both maximums and 
minimums, as formulas will indicate. 

Example 1. Find the value of x when y is a maximum or minimum, in 

the formula y = x 3 — 12x + 22? dy = ( 3x 2 —12 )dx, —= 3x 2 — 12 = o. 


d-y 


dx 


Of which x — 1/*$ = 2 the answer. = 6x, which is positive, conse¬ 

quently y = 2 s — 12X2 + 22 = 6, a minimum, when x = 2. 

Example 2. It is required to cut out the strongest possible beam of height 
h and breadth b, from a log of diameter Z>, fig. 221, page 174? The strength 
of a beam is in proportion to bh 2 which is to be a maximum. 

J) 2 = b 2 + h 2 , h 2 — I) 2 — b 2 , b h 2 = b (D 2 — h 2 ), d(bh 2 ) = (D 2 — 3 b 2 )db. 
d (bh 2 ) 

db 


= D 2 — 3b 2 —o, of wliicli the breadth 6 = Z>]/^= 0.577 Z>, and height 


h = ~\/D 2 — Y — Z>l/0-6666 — 0-8164 D, the answer. The second d 4 coef. 
(]?(}) // 2 ^ 

—-—-- = — 6 5, which is negative, and therefore b h 2 is a maximum 
d b 2 

when b — 0"577 D. 

Example 3. It is required to know the proportion of height h and 
diameter D of a cylinder, having the greatest cubic content v, with the 

smallest surface "z including top and bottom ? z = ~~ 2 ~+ u Z> h = —^—b nDIt, 

2 

which is to be a minimum. Let v = 1 and D = 1, then 2 = — + n k, and 


dz*±(w- -~^ = n -when A =-/-!—m 2 842>,tlie a 


d 2 z 4X2 

The second d 4 coef. —nr = H-rr— 

d/i* n* 

h = 1-1284 D. 


answer. 


, which is positive, and z a minimum when 


Mnclaurin’s Theorem. 

Maclauxin's Theorem, explains how to develop into a series a function with 
one variable, as 


u 


t S , S / du \ , S, 2 / d2u \ | S 3 / 
^ + 1 \ dx ) + 2 ( dx 2 ) + 2 X 3 \ 


d 3 u \ x n / d n u 

~ + 2 X 3X ...«\ 


^etc. 


2X3 \ dx 3 J 

where the factor in the parenthesis is that which it assumes when x — o. 

The function u = —-—- developed into a series will be 

a + x 

i = L_ ** etc- 

a + x a a 2 a 8 a 4 aa+i’ 

Taylor’s Theorem. 

Taylor's Theorem , explains how to develop into a series a function of the 
sum or difference of two variable as u = x±y. 

d n u y* 


_, . du 

T(x±y)=u±- zyz -y 


d 2 u y 2 i dhi 
~dtf ' 2 ± ~drf 


y z , 

2X3 ^ 


dx * ' dx 2 2 ~ dx 3 2X3 ' dx* 
wherew represents the value of the function when y = o. 


2X3 ... Xn 



















102 


Circumferences and Areas of Circles. 



35 — 

1 *0981 

•00076 

5 — 

15-70 

19-635 

.n — 

34.55 

95-033 

Vs 

•1963 

.00306 


16-10 

20-629 


34-95 

97-205 

A -• 

•3926 

•01227 

4 - 

16-49 

21-647 

A - 

35-34 

99-402 

3 

1 «! 

•5890 

•02761 


16-88 

22-690 


35-73 

101-62 

A-- 

•7854 

•04908 

A— 

17-27 

23-758 

A- 

36-12 

103-86 

A 

•9817 

•07669 


17-67 

24-850 


36-52 

106-13 

i 

1*178 

•1104 

S - 

18-06 

25-967 

1 - 

36-91 

108-43 

7 

I B 

1-374 

•1503 


18-45 

27-108 


37*30 

110-75 

A- 

1-570 

•1963 

6_ 

18-84 

28-274 

12— 

37-69 

113-09 

i a o 

1-767 

•2485 


19-24 

29-464 


38-09 

115-46 

1 - 

1-963 

•3067 

A - 

19-63 

30-679 

A - 

38-48 

117-85 

u 

2-159 

•3712 


20-02 

31-919 


38-87 

120-27 

z — 

2-356 

•4417 

A— 

20-42 

33-183 

A - 

39-27 

122-71 

H 

2-552 

•5184 


20-81 

34-471 


39-66 

125-18 

A - 

2.748 

•6013 

Z - 

21-20 

35-784 

Z 

40-05 

127-67 

i* 

2-945 

•6902 


21-57 

37-122 


40-44 

130-19 

i _ 

3-141 

•7854 

7 

21-99 

38-484 

13- 

40-84 

132-73 


3-534 

•9940 


22-38 

39-871 


41-23 

135-29 

i -- 

J 3-927 

1-227 

A - 

22-77 

41-282 

A - 

41-62 

137-SS 


J 4-319 

1-484 


23-16 

42-718 

_ 

42-01 

140-50 

A- 

-4-712 

1-767 

A- 

■23-56 

23-95 

44-178 

A- 

42-41 

143-13 


5-105 

2-073 


45-663 


42-80 

145-80 

1 - 

5-497 

2-405 

Z - 

24-34 

47-173 

Z - 

43-19 

148-48 


5-890 

2-761 


-24-74 

48-707 


43-58 

151-20 

a— 

6-283 

3-141 

8 — 

-25-13 

50-265 

14— 

43-98 

153-93 


6-675 

3-546 


•25-52 

51-848 


44-37 

156-69 

i -- 

7-068 

3-976 

A - 

-25-91 

53-456 

A - 

44-76 

159-48 


7-461 

4-430 


26-31 

55-0S8 


45-16 

162-29 

A- 

7-854 

4-908 

A— 

•26-70 

56-745 

A- 

45-55 

165-13 


8-246 

5-411 


-27-09 

58-426 


45-94 

167-98 

Z - 

8-639 

5-939 

Z - 

27-48 

60-132 

Z - 

46-33 

170-87 


j 9-032 

6-491 


27-88 

61-862 


46-73 

173-78 

3_ 

9-424 

7-068 

9— 

28-27 

63-617 

15 — 

47-12 

176-71 


9-817 

7-669 


28-66 

65-396 


47-51 

179-67 

A - 

10-21 

8-295 

1 - 

29-05 

67-200 

A - 

47*90 

182-65 


10-60 

8-946 


29-45 

69-029 


48-30 

185-66 

A- 

10- 99 

11- 38 

9- 621 

10- 320 

A- 

29- 84 

30- 23 

70-882 

72-759 

A- 

48- 69 

49- OS 

188-69 

191-74 

Z ~ 

11-78 

11-044 

5 - 

30-63 

74-662 

Z - 

49-48 

194-82 


12-17 

11-793 


■31-02 

76-588 


49-87 

197-93 

4 — 

-12-56 

12-566 

10- 

■31-41 

78-539 

16— 

50-26 

201-06 


12-95 

13-364 


-31-80 

80-515 


50-65 

204-21 

A - 

13-35 

14-186 

i - 

32-20 

82-516 

A - 

51-05 

207-39 


13-74 

15-033 


•32-59 

84-540 


51-44 

210-59 

A-- 

14-13 

15-904 

A- 

•32-98 

86-590 

A- 

51-83 

213-82 


14-52 

16-800 

• 

-33-37 

88-664 


52-22 

217-07 

a • 

14-92 

17-720 

Z - 

33-77 

90-762 

z - 

52-62 

220-35 


15.31 

18-665 


-34-16 

92-885 


53-01 

223-65 

• 



































CIRCUMFERENCES AND AREAS OF CIRCLES. 



Circ. 

Area. 


Circ. 

Area 


Circ. 

Area. ) 

Diame¬ 

ter. 

o 

HI 

Diame¬ 

ter. 

o 

HI 

Diame¬ 

ter. 

c 


17-r 

i 53*40 

226*98 

23 —| 

72*25 

415*47 

29 - r 

91*10 

660*52 


53*79 

230*33 


72*64 

420*00 


91*49 

666*22 

i -■ 

54*19 

233*70 

i '■ 

73*04 

424*55 

i - 

-91*89 

671*95 


54*58 

237*10 


73*43 

429*13 


-92*28 

677*71 

i— 

54*97 

240*52 

*— 

73*82 

433*73 

*—[ 

92*67 

683*49 


55*37 

243*97 


74*21 

438*30 


93*06 

689*29 

1 - 

55*76 

247*45 

2 - 

74*61 

443*01 

1 " 

93*46 

695*12 


56*16 

250*94 


75* 

447*69 


93*85 

700*98 

18— 

56*54 

254*46 

24—■ 

75*39 

452*39 

30— 

94*24 

706*86 


56*94 

258*01 


75*79 

457*11 


94*64 

712*76 

i - 

57*33 

261*58 

i - 

76*18 

461*86 

i - 

95*03 

718*69 


i57*72 

265*18 


76*57 

466*63 


95*42 

724*64 

1 

2- 

58*11 

268*80 

i— 

76*96 

471*43 


95*81 

730*61 


- 58*51 

272*44 


77*36 

476*25 


96*21 

736*61 

s - 

- 58*90 

276*11 

2 - 

77*75 

481*10 

2 

96*60 

742*64 

-1 59*29 

279*81 


-78*14 

485*97 


96*99 

748*69 

19- 

-59*69 

283*52 

25- 

-78*54 

490*87 

si— 

97*38 

754*76 


-'60*08 

287*27 


78*93 

495*79 


97*78 

760*86 

i - 

60*47 

291*03 

i ' 

79*32 

500*74 

i ' 

98*17 

766*99 

- 60*86 

294*83 


-*79*71 

505*71 


98*56 

773*14 

h- 

- 61*26 

298*64 

i- 

-80*10 

510*70 


98*96 

779*31 

61*65 

302*48 


-80*50 

515*72 

99*35 

785*51 

2 - 

- 62*04 

306*35 

2 - 

-80*89 

520*70 

s - 

99*74 

791*73 

62*43 

310*24 


-81*28 

525*83 


100*1 

797*97 

20 - 

- 62*83 

314*16 

26- 

-81*68 

530*93 

32~- 

-100*5 

804*24 


-63*22 

318 09 


82*07 

536*04 


100*9 

810*54 

i - 

63*61 

322*06 

i - 

-82*46 

541*18 

i - 

101*3 

816*86 


-64*01 

326*05 


-82*85 

546*35 


101*7 

823*21 

i- 

-'64.40 

330*06 

i- 

-83*25 

551*54 

i~ 

102*1 

829*57 


-j 64*79 

334*10 


-83*64 

556*76 


102*4 

835*97 

2 - 

-'65*18 

338*16 

2 - 

-184*03 

562*00 

2 - 

102*8 

842*39 

- 65*58 

342*25 


-84*43 

567*26 


103*2 

848*83 

21 - 

- 65*97 

346*36 

27— 

-84*82 

572*55 

33- 

103*6 

855*30 


-j 66*36 

350*49 


85*21 

577*87 


104* 

861*79 

i - 

-166*75 

354*65 

i ~ 

-85*60 

583*20 

i “ 

104*4 

868*30 ■ 


67*15 

358*84 


-86* 

588*57 


-j 104*8 

874*84 

i— 

- 67*54 

363*05 

i- 

-86*39 

593*95 

i- 

105*2 

881*41 


67*93 

367*28 


•86*78 

599*37 

105*6 

888*00 

1 - 

- 68*32 

371*54 

2 - 

-87*17 

604*80 

s ~ 

106* 

894*61 


- 68*72 

375*82 


-87*57 

610*26 


106*4 

901*25 

22- 

-69*11 

380*13 

28- 

-87*96 

615*75 

34- 

- 106*8 . 

907*92 


-69*50 

384*46 


-88*35 

621*26 


- 107*2 

914*61 

i - 

-69*90 

388*82 

i - 

i88*75 

626*79 

i - 

107*5 

921*32 


-70*29 

393*20 


489*14 

632*35 


107*9 

928*06 

h- 

-j 70*68 

397*60 

i- 

-89*53 

637*94 


10S-3 

934*82 


-71*07 

402*03 


- 89*92 

643*54 


10S-7 

941*60 

2 ■ 

-71*47 

406*49 

5 J 

-90*32 

649*1S 

2 - 

- 109*1 

948*41 


-71*86 

410*97 

-90*71 

654*83 

( 

-109*5 

955*25 






































104 


Circumferences and areas of Circles, 



Circ. 

Area. 


Circ. 

1 Area. 


Diame- 



Diame- 


|lBl 

Diame- 

ter. 

vj 

K j 

ter. 


ter. 

35 —i 

n 109-9 

962-11 

41- 

128.8 

1320-2 

47 — r 


- 110-3 

968-99 


-129-1 

1328-3 

- 

i - 

- 110-7 

975-90 

i - 

- 129-5 

1336-4 

i - 


- 111-1 

982-84 


- 129-9 

11344-5 



- 111-5 

989-80 

i~| 

130-3 

1352-6 

£-[ 


- 111-9 

996-78 


- 130-7 

1360-8 


2 - 

112-3 

1003*7 


131-1 

1369-0 

2 - 


- 112-7 

1010-8 


- 131-5 

1377-2 


36— 

- 113- 

1017-8 

42— 

- 131-9 

1385*4 

48— 


- 113-4 

1024-9 


1- 132-3 

1393-7 

- 

i ~ 

- 113-8 

1032-0 

i - 

- 132-7 

1401-9 

i - 


-i 114*2 

1039-1 


- 133-1 

1410-2 



- 114-6 

1046-3 

£- 

' 133-5 

1418-6 

i , 


- 115- 

1053-5 


- 133-9 

1426-9 



115-4 

1060-7 

5 - 

134-3 

1435-3 

2 


- 115-8 

1067-9 


- 134-6 

1443-7 


37- 

J 116-2 

1075-2 

43- 

- 135- 

1452-2 

49 - ■ 


J 116-6 

1082-4 


- 135-4 

1460-6 


i - 

-1 117* 

1089-7 

i - 

H 135-8 

1469-1 

i -- 


117*4 

1097-1 


J 136-2 

1477-6 


i — 

J 117-8 

1104-4 


1136-6 

1486-1 

i ~ 


J 118-2 

11 11*8 


137- 

1494-7 

$ j 

-118-6 

1119-2 

1 “ 

- 137-4 

1503-3 

1 


118-9 

1126-6 


-137-8 

1511-9 


38- 

-j 119-3 

1134-1 

44— 

-138-2 

1520-5 

50-J 


119-7 

1141-5 


138-6 

1529-1 


i - 

120-1 

1149-0 

i " 

139- 

1537*8 

i ~ 


120-5 

1156-6 


139-4 

1546-5 


i- 

120-9 

1164-1 

i- 

-139-8 

1555"2 

i H 


121-3 

1171-7 


140-1 

1564-0 

- 

5 - 

J 121*7 

1179-3 

s -- 

140-5 

15-72-8 

s -- 


122-1 

1186-9 


140-9 

1581*6 

- 

39 - 

122-5 

1194-5 

45—1 

141-3 ; 

1590-4 

61 —- 


122-9 

1202-2 


141-7 

1599-2 

- 

i - 

123-3 

1209-9 

i - 

142-1 

1608-1 

i “j 1 

- 


123-7 

1217-6 


142-5 

1617-0 

i- 

124- 

1225-4 

*— 

142-9 

1625-9 

i -1 


124-4 

1233-1 


143-3 

1634-9 

1 - 

124-8 

1240-9 

2 - 

143-7 

1643-8 

2 * l 


125-2 

1248-7 


144-1 

1652-8 

- 

40 — 

125-6 

1256-6 

46-- 

144-5 

1661*9 

62 -- 


126- 

1261-5 


144-9 

1670-9 

- 1 

i ~ 

126-4 

1272-3 

k - 

145-2 

1680-0 

i -1 


126-8 

1280-3 


145-6 

1689-1 

- 

i-t 

127-2 

1288-2 

i- - 

146- 

1698*2 

1 


127-6 

1296-2 


146-4 

1707-3 

- 1 

i 

128- 

1304-2 

2 -- 

146-8 

1716-5 

| -1 
* 1 


128-4 

1312*2 


147-2 

1725-7 







—l 


Circ. 




1734-9 
1744*1 
1753-4 
1762-7 
1772-0 
1781-3 
1790-7 
1800-1 
f 3 809-5 
1818-9 
1828-4 
1837-9 
1847-4 
1856-9 
1866-5 
1876-1 
1S85-7 
1895-3 
j 1905-0 
1914-7 
1924-4 
1934-1 
1943-9 
1953-6 
11963-5 
j 1973-3 
11983-1 
j 1993-0 
2002-9 
2012-8 
2022-8 
2032-8 
2042-8 
2052-8 
2062-9 
2072-9 
2083-0 
2093-2 
2103-3 
2113-5 
2123-7 
2133-9 
2144-1 
2154-4 
2164-7 
2175-0 
2185-4 
12195*7 























































Circumferences and Areas of Circles. 




Circ. 

Area. 


Circ. 

Area. 


Circ. 

Area. 

Dlame- 



Diame- 



Diame- 



ter. 


vy 

qp 

ter. 


fip 

ter. 

u 

isp 

53—i 


166-5 

2206-1 

59- 

185-3 

2733-9 

65- 

204-2 

3318-3 



166-8 

2216-6 


-185-7 

2745-5 


204-5 

3331-0 

i ~ 


167-2 

2227-0 

i ' 

186-1 

2757-1 

l - 

-204-9 

3343-8 



167-6 

2237-5 


186-5 

2768-8 


-205-3 

3356-7 

i- 


168- 

2248-0 

i~ 

186-9 

2780-5 

i-{ 

205-7 

3369-5 


- 

168-4 

2258-5 


187-3 

2792-2 


-206*1 

3382-4 

f -| 


168-8 

2269-0 

2 - 

187-7 

2803-9 

1 - 

206-5 

3395-3 



169-2 

2279-6 


188-1 

2815-6 


206-9 

3408-2 

54— 


169-6 

2290-2 

60— 

188-4 

2827-4 

66— 

207-3 

3421-2 


- 

170- 

2300-8 


188*8 

2839-2 


207-7 

3434-1 

i ~ 


170-4 

2311-4 

i ' 

189*2 

2851-0 

i - 

208-1 

3447-1 



170-8 

2322-1 


1S9-6 

2862-8 


208-5 

3460-1 



171-2 

2332-8 


190- 

2874-7 

4- 

208-9 

3473-2 



171-6 

2343-5 


190-4 

2886-6 


209-3 

3486-3 



172- 

2354-2 

2 - 

190-8 

2898-5 

2 

209-7 

3499-3 


- 

172-3 

2365-0 


191-2 

2910-5 


210- 

3512-5 

55- 


172-7 

2375-8 

61- 

191-6 

2922-4 

67- 

210-4 

3525-6 



173-1 

2386-6 


1192* 

2934-4 


210-8 

3538-8 

i - 


173-5 

2397-4 

i ' 

192-4 

2946-4 

2 -- 

211-2 

3552-0 


- 

173-9 

2408-3 


J 192*8 

2958-5 


211*6 

3565-2 

i- 


174-3 

2419-2 

i- 

il93*2 

2970-5 

4 — 

1 212* 

3578-4 



174-7 

2430-1 


193-6 

2982-6 


212-4 

3591-7 

2 J 


175-1 

2441-0 

1 - 

J 193-9 

2994-7 

2 - 

212-8 

3605-0 



175-5 

2452-0 


194-3 

3006-9 


213-2 

3618-3 

56 — 


175-9 

2463-0 

62 — 

-194-7 

3019-0 

68— f 

-213-6 

3631-6 



176-3 

2474-0 


195-1 

3031-2 


214- 

3645-0 

i - 


176-7 

2485-0 

i - 

195-5 

3043-4 

i - 

-214*4 

3658-4 



177-1 

2496-1 


- 195-9 

3055-7 


-214*8 

3671-8 

i- 


177-5 

2507-1 


-196*3 

3067-9 

4- 

-215*1 

3685-2 



177-8 

2518-2 


196-7 

30S0-2 


215-5 

3698-7 

2 -J 


178-2 

2529-4 

2 - 

197-1 

3092-5 

i - 

215-9 

3712-2 



178-6 

2540-5 


197*5 

3104-8 


216-3 

3725-7 

57- 


179- 

2551-7 

63 — l 

197*9 

3117-2 

69-- 

216-7 

3739-2 



179-4 

2562-9 


198-3 

3129-6 


217-1 

3752-8 

i - 


179-8 

2574-1 

i - 

198-7 

3142-0 

2 - 

217-5 

3766-4 


- 

180-2 

2585-4 


199- 

3154-4 


217-9 

3780-0 

i- 


180-6 

2596-7 

i- 

199-4 

3166-9 

4- 

218-3 

3793-6 



181- 

2608-0 


199-8 

3179-4 


218-7 

3807-3 

2 - 

- 

181-4 

2619-3 

2 “ 

-200-2 

3191-9 

2 - 

219-1 

3821-0 



181-8 

2630-7 


-1200-6 

3204-4 


-219-5 

3834-7 

58— 

- 

182-2 

2642-0 

64- 

-201- 

3216-9 

70- 

-219-9 

3848-4 



182-6 

2653-4 


201-4 

3229-5 


220-3 

3862-2 

i ' 

- 

182-9 

2664-9 

i - 

201-8 

3242-1 

4 - 

220-6 

3875-9 


- 

183-3 

2676-3 


202*2 

3254-8 


221- 

3889 8 

i- 


183-7 

2687-8 


202-6 

3267-4 

4- 

221-4 

3903-6 


- 

1S4-1 

2699-3 


203- 

3280-1 


221-8 

3917-4 

2 * 


184-5 

2710-8 

2 - 

203-4 

3292-8 

2 * 

222-2 

3931-3 



184-9 

2722-4 


203-8 

3305-5 


222-6 

3945-2 













































106 


Circumferences and Areas of Circles. 



Circ. 

Area. 


Circ. 

Area. 


Circ. 

| Area. 

Diame¬ 

ter. 

0 

m 

Diame¬ 

ter. 

o 

H 

Diame¬ 

ter. 

O 


71 -n 

223* 

3959-2 

77—i 

241-9 

4656-6 

83-r 

>260-7 

5410-6 1 

- 

223-4 

3973-1 


242-2 

4671-7 


261-1 

5426-9 

i -- 

223-8 

3987-1 

i -■ 

242-6 

4686-9 

i " 

261-5 

5443-2 


224-2 

4001-1 


243- 

4702-1 


261-9 

5459*6 

4-- 

224-6 

4015-1 

i~ 

243-4 

4717-3 

i-j 

262-3 

5476-0 


225- 

4029-2 


243-8 

4732-5 


262-7 

5492-4 

a - 1 

225-4 

4043-2 

z - 

244-2 

4747*7 

z - 

263-1 

5508-8 


225-8 

4067-3 

- 

244-6 

4763-0 

- 

263-5 

5525-3 

72— 

226-1 

4071*5 

78-- 

245- 

4778-3 

84— 

263-8 

5541-7 

- 

226-5 

4085-6 


245-4 

4793-7 

- 

264-2 

5558-2 

i - 

226-y 

4099-8 

i - 

245*8 

4S09-0 

i - 

264-6 

5574-8 


227-3 

4114-0 


246-2 

4824-4 


265- 

5591*3 

1 

2 — 

227-7 

4128-2 

i- 

246-6 

4839-8 


265-4 

5607-9 


228-1 

4142-5 


247- 

4855-2 


265-8 

5624-5 

a - 

228-5 

4156-7 

z - 

247-4 

4870-7 

z 

266-2 

5641-1 


22S-9 

4171-0 


247-7 

4886-1 


266-6 

5657-8 

73 — 

229-3 

4185-3 

79-- 

248-1 

4901-6 

85- 

267- 

5674-5 


J 229-7 

4199-7 


248-5 

4917-2 


267-4 

5691-2 

i - 

230-1 

4214-1 

i - 

1248-9 

4932-7 

i ' 

267-8 

5707-9 


230-5 

4228-5 


249-3 

4948-3 


268-2 

5724-6 

i- 

230-9 

4242-9 

i— 

1 249-7 

4963-9 


269-6 

5741-4 


231-3 

4257-3 


250-1 

4979-5 


268-9 

5758-2 

z - 

231-6 

4271-8 

1 - 

-250-5 

4995-1 

z - 

269-3 

5775-0 


J 232* 

4286-3 


250-9 

5010-8 


269-7 

5791-9 

74- 

232-4 

4300-8 

80 — 

-251-3 

5026-5 

86—r 

270-1 

5808-8 


-1232-8 

4315-3 


251*7 

5042-2 


270-5 

5825-7 

* - 

-j 233-2 

4329-9 

i - 

-252-1 

5058-0 

i - 

270-9 

5842-6 


-{233*6 

4344-5 


-252-5 

5073-7 


271-3 

5859*5 

4‘- 

J234- 

4359-1 

* 4- 

-252-8 

5089-5 

i- 

271-7 

5876-5 


H 234-4 

4373-8 


-253-2 

5105-4 


272-1 

5893-5 

1 - 

- 1 234-8 

4388-4 

z - 

-253-6 

5121-2 

z - 

272-5 

5910-5 


-235-2 

4403-1 


-254- 

5137-1 


272-9 

5927-6 

75 - 

235-6 

4417-S 

81-4 

-254-4 

5153-0 

87-- 

273-3 

5944-6 


236- 

4432-6 


254-8 

5168-9 


273-7 

5961-7 

i - 

1236*4 

4447-3 

i - 

255-2 

5184-8 

i “ 

274-1 

5978-9 


236-7 

4462-1 


255-6 

5200-8 


274-4 

| 5996-0 

4— 

237-1 

4476-9 

4- 

-256- 

5216-8 


274-8 

6013-2 

a - 

237-5 

4491-8 


256-4 

5232-8 

275-2 

6030-4 

237-9 

4506-6 

z *- 

-256-8 

15248-8 

1 ^ 

- 275-6 

6047-6 


- 238-3 

4521-5 


-257-2 

5264-9 


- 276- 

'6064-8 

-76- 

- 238-7 

4536-4 

82- 

-257-6 

5281-0 

88 - 

- 276-4 

6082-1 


-239-1 

4551-4 


- 258- 

5297-1 


- 276-8 

6099-4 

1 - 

239-5 

4566-3 

4 - 

-i 258-3 

5313-2 

i " 

- 277-2 

6116-7 


J 239-9 

4581-3 


-1258-7 

5329-4 


* 277-6 

6134-0 

4— 

n 240-3 

4596-3 

4- 

-259-1 

5345-6 

i- 

- 278- 

6151-4 


- 240-7 

4611-3 


- 259-5 

5361-8 


- 278-4 

6168-8 

i J 

- 24M 

4626-4 

z - 

- 259-9 

5378-0 

z - 

" 278-8 

6186-2 


-241-5 

j 4641-5 


- 260-3 

j 5394-3 

J 

f 279-2 

6203-6 











































107 


Circumferences and Areas of Circles. 



Circ. 

| Area. 


Circ. 

Are: 1 . 


Circ. 

Area. 

Diame¬ 

ter. 

o 

iii 

Diame¬ 

ter. 

o 

I® 

Diame¬ 

ter. 

o 

|j||i 

89 

n 279*6 

6221*1 

93- 

292-1 

6792*9 

97- 

r 304*7 

7389*8 


- 279*9 

6238*6 


-292*5 

6S11-1 


- 305*1 

7408*8 

X _ 

* 

- 280*3 

6256*1 

£ - 

-292*9 

6S29-4 

£ - 

-305*5 

7427*9 


- 280*7 

6273*6 

- 293*3 

I 6847*8 

- 305*9 

7447*0 


1- 281*1 

6291*2 

ft- 

- 293*7 

6866*1 

ft- 

306*3 

7460*2 


- 281*5 

6308*8 


- 294*1 

6884*5 

- 306*6 

7485*3 

2 - 

281*9 

6326*4 

1 - 

- 29-1*5 

6902*9 

£ - 

- 307* 

7504*5 


- 282*3 

6344 0 


- 294*9 

6921*3 


307*4 

7523*7 

90- 

- 282*7 

6361*7 

94— 

-295*3 

6939*7 

98- 

- 307*8 

7542*9 


- 283*1 

6379*4 


- 295*7 

6958*2 


308*2 

7562*2 

£ - 

- 283*5 

6397*1 

£ - 

-296* 

6976*7 

£ - 

308*6 

7581*5 


- 283*9 

6414*8 


296-4 

6995*2 


309*0 

7600*8 


- 2S4-3 

6432-6 

ft- 

296*8 

7013-S 

ft 

309*4 

7620*1 


- 284*7 

6450*4 


-297-2 

7032*3 


309*8 

7639*4 

2 - 

285*1 

6468*2 

1 - 

297-6 

7050*9 

2 

310*2 

7658*8 


- 285*4 

6486 0 


298* 

7069*5 


310*6 

7678*2 

91 — 

- 285*8 

-i 286*2 

6503*8 

95 — 

2J8-4 

7088*2 

99- 

311*0 

7697*7 


6521*7 


298-8 

7106*9 


311*4 

7717*1 

£ - 

J 286*6 ! 

6539*6 

£ -- 

299*2 

7125*5 

£ - 

J 311*8 

7736*6 


-'287* 

6557*6 


-1299-6 

7144*3 


312*1 

7756*1 

i-T 

i 1 287*4 

6575*5 

ft-- 

-300* 

7163*0 

£ 

■1312-5 

7775*6 


-287*8 

6593*5 


300*4 ! 

7181*8 

312*9 

7795*2 

2 -L 

-j 288*2 

6611*5 

2 - 

300*8 

7200*5 

2 -- 

313*3 

7814*7 


288*6 

6629*5 


301-2 

7219-4 


313 7 

7S34-3 

92- 

289* 

6647*6 

96—r 

301*5 

7238*2 

100—C 

314*1 

7853*9 


-289*4 

6665*7 


301*9 

7257*1 


314*5 

7853*6 

£ - 

289*8 

66S3-S 

£ -- 

302*3 

7275*9 

£ -- 

314*9 

7893*3 


290 2 

6701-9 


302*7 

7294*9 


315*3 

7913*1 

ft— 

290*5 

6720-0 

ft— 

303*1 

7313*8 

ft- - 

315*7 

7932*7 

» 

290*9 

673S-2 


303*5 

7332*8 


316*0 

7942*4 

ft - 

291*3 

6756*4 

*T] 

303*9 

7351*7 

2 -- 

316*4 

7972*2 

l-i 291-7 

6776*4 

304*3 

7370*7 


316-8 

7991*9 


EXPLANATION OF THE TABLE FOR SEGMENTS, &c. 

The chord divided by the height is the gauge in the Table, the quotient in the 
first column. 

k = tabular coefficient, always to be multiplied by the chord. 

To find tile angle of an arc of a circle* 

RULE. Divide the base (chord) of the arc by its height, (sine verse ) and find 
the quotient in the first column. The corresponding number in tho second 
I column is the angle of the arc in degrees of the circle. 

To find the radius of an arc of a circle* 

RULE. Divide the chord of the arc by its height, and find the quotient in 
the first column. The corresponding number in the third column, multiplied 
by the chord, is the radius of the arc. 























































108 lABLE FOR SEGMENTS &C., OF A CIRCLE. 


Chord div. 
ly height. 

Centre 
Angle n. 

i Radius 

1 r =- kc. 

Cir. Arc. 
b =* * e. 

Area Seg. 
a = A ca. 

Surface 
a = k c a 

Solidity 

C = Jk c*. 

CborA 
c= A r. 




* 

r i 
* 

1/ 







x> 



'V 


458-08 

1 

57-296 

1-0000 

•01091 

•78539 

•00085 

•01744 

229-18 

2 

28-649 

1-0000 

•00218 

•78549 

•00172 

•03490 

152-77 

3 

19-101 

1-0000 

•00327 

•78462 

•00255 

•05234 

114-57 

4 

14-327 

1-0000 

•00436 

•78574 

•00310 

•06978 

84-747 

5 

11-462 

1-0001 

.00647 

•78586 

•00401 

•08722 

76-375 

6 

9-5530 

1-0003 

•00741 

•78599 

•00514 

•10466 

65-943 

7 

8-1902 

1-0004 

•00910 

•78621 

•00592 

•12208 

57-273 

8 

7-1678 

1-0006 

•010S9 

•7S630 

•00686 

•13950 

50-902 

9 

6-3 

728 

1-0008 

•01254 

•7S665 

•00772 

•15690 

45-807 

10 

5-7368 

1-0011 

•01407 

•78695 

•00857 

•17430 

41-203 

11 

5-2167 

1-0013 

•01552 

•78730 

•00964 

•19168 

38-133 

12 

4-7834 

1-0016 

•01695 

•78725 

•01031 

•20904 

35-221 

13 

4-4168 

1-0019 

•01841 

•78794 

•01114 

•22640 

32-742 

14 

4-1027 

1-0023 

•02000 

•78832 

•01199 

•24372 

30-514 

15 

3-8307 

1-0027 

•02157 

•788S9 

•01288 

•26104 

28-601 

16 

3-5927 

1-0029 

•02269 

•78909 

•01375 

•27834 

26-915 

17 

3-3827 

1-0034 

•02434 

•78969 

•01462 

•29560 

25-412 

18 

3-1962 

1-0039 

•02592 

•79028 

•01542 

•312S6 

24-068 

19 

3-0293 

1-0044 

•02744 

•79084 

•01635 

•33008 

22-860 

20 

2-8793 

1-0048 

•02878 

•79140 

•01722 

•34728 

21-760 

21 

2-7440 

1-0054 

•03040 

•79234 

•01802 

•36446 

20-777 

22 

2-6222 

1-0059 

•03178 

•79300 

•01S97 

•3S160 

19-S62 

23 

2-5080 

1-0066 

•03343 

•79340 

•019S4 

•39S72 

19-028 

24 

2-4050 

1-0072 

•03493 

•79416 

•02072 

•41582 

18-261 

25 

2-3101 

1-0078 

•03639 

•79486 

•02159 

•432S6 

17-553 

26 

2-2233 

1-0084 

•03784 

•79530 

•02248 

•44990 

16-970 

27 

2-1418 

1-0091 

•03970 

•79639 

•02315 

•46688 

16-288 

28 

2-0673 

1-0101 

•04115 

•79748 

•02424 

•48384 

15-721 

29 

1-9969 

1-0105 

•04230 

•79811 

•02511 

•50076 

15-191 

30 

1-9319 

1-0113 

•04385 

•79907 

•02600 

•51762 

14-970 

31 

1-87 

10 

1-0121 

•04476 

•78530 

•02692 

•53446 

14-230 

32 

1-S140 

1-0129 

•01710 

•80098 

•02778 

•55126 

13-796 

33 

1-7605 

1-0138 

•04842 

•801S1 

•02866 

•56S02 

13-382 

34 

1-7102 

1-0146 

•04989 

•S0300 

•02956 

.58479 

12-994 

35 

1-6628 

1-0155 

•05137 

•80405 

•03046 

•60140 

12-733 

36 

1-6184 

1-0167 

•05311 

•80531 

•03137 

•61802 

12-473 

37 

1-5758 

1-0174 

•05401 

•80622 

•03226 

•63460 

11-931 

38 

1-5358 

1-0184 

•05628 

•80713 

•03328 

•65112 

11-621 

39 

1-49 

79 

1-0194 

•05755 

•S0850 

•03418 

•66760 

11-342 

40 

1-4619 

1-0204 

•05899 

•80987 

•03506 

•68404 

11-060 

41 

1-4266 

1-0207 

•06001 

•81046 

•03589 

•70040 

10-791 

42 

1-3952 

1-0226 

•06196 

•81240 

•03680 

•71672 

10-534 

43 

1-3643 

1-0237 

•06359 

•81377 

•03773 

•73300 

10-289 

44 

1-3347 

1-0248 

•06574 

•81505 

•03S64 

•74920 

10-043 

45 

1-3066 

1-0260 

•06628 

•81756 

•03890 

•76536 

9-8303 

46 

1-2797 

1-0272 

•06S26 1 

.81795 

•04050 

•78146 

9-6153 

47 

1-2539 1 

1-0290 

•06998 

•81939 

•04143 

•79748 

9-4092 

48 j 

1-2289 j 

1-0297 

■09138 

•82064 

•04247 

•81346 

i 


















































Table for Segments Ac., of a Circle. 


109 


Chord di* 

Centre 

Rad ius 

Cir. Arc. 

Area Sflg. 

Surface 

Solidity 

Chord 

by height. 

Angle v. 

r — h c. 

b = kc. 

a = k «•. 

a = A c a - 

C = k c*. 

e = k r. 









V' 


"P 


- 

'“v / 

N/ 

V"- 


9-2113 

49 

1-2057 

1-0309 

•07290 

•82244 

•04330 

•82938 

9-0214 

50 

1-1831 

1-0323 

•07453 

•82384 

•04424 

•84522 

8-8387 

51 

1-1614 

1-0336 

•07611 

•82562 

•04519 

•86102 

8-6629 

52 

1-1406 

1-0349 

•07758 

•82729 

•04614 

•87674 

8-4462 

53 

1-1206 

1-0364 

•07959 

•83363 

•04685 

•89238 

8-3306 

54 

1-1014 

1-0378 

•08083 

•83072 

•04805 

•90798 

8 1733 

55 

1-0S28 

1*0393 

•08246 

•83249 

•04901 

•92348 

8 0215 

56 

1-0650 

1-0407 

•08400 

•83422 

•05002 

•93S94 

7-8750 

57 

1-0478 

1-0422 

•08579 

•83602 

•05098 

•95430 

7-7334 

58 

1-0313 

1-0431 

•08680 

•S3796 

•05191 

•96960 - 

7-5895 

59 

1-0154 

1-0454 

•08S91 

•84064 

•05299 

•98484 

7-4565 

60 

1-0000 

1*0470 

•09106 

•84266 

•05400 

1-0000 

7-3358 

61 

•98515 

1-0486 

•09209 

•843S0 

•054f6 

1-0150 

7-211S 

62 

•97080 

1-0503 

•09375 

•84581 

•05583 

1-0300 

7-0914 

63 

•95694 

1-0520 

•09540 

•84791 

•05684 

1-0450 

6-9748 

64 

•94352 

1-0537 

•09697 

•84996 

•05784 

1-0598 

6-8616 

65 

•93058 

1-0555 

•09865 

•85215 

•05885 

1-0746 

6-7512 

66 

•91804 

1-0573 

•10036 

•85441 

•059S7 

1-0892 

6 6453 

67 

•90590 

1-0591 

•10201 

•85640 

•06088 

1-1038 

6-5469 

68 

•89415 

1-0610 

•10367 

•85815 

•06181 

1-1184 

6-4902 

69 

•88276 

1-0629 

•10520 

•85464 

•06201 

1-1328 

6-3431 

70 

•87172 

1-064S 

•10710 

•86350 

•06396 

1-1471 

6-2400 

71 

•86102 

1-0668 

•10887 

•86699 

•06515 

1-1614 

6-1553 

72 

•85065 

1-06S7 

•11046. 

•86834 

•06604 

1-1755 

6-0652 

73 

•84058 

1-0708 

•11225 

•S70S1 

•06709 

1-1896 

5-9773 

74 

•S3 08 2 

1-0728 

•11385 

•87935 

•06815 

1-2036 

5-8918 

75 

•82134 

1-0749 

•11563 

•87590 

•06921 

1-2175 

5-8084 

76 

•81213 

1-0770 

•11736 

•87853 

•07037 

1-2313 

5-7271 

77 

•80319 

1-0792 

•11910 

•8S120 

•07136 

1-2450 

5-6478 

78 

•79449 

1-0814 

•12072 

•88389 

•07244 

1-2580 

5-5704 

79 

•78606 

1-0836 

•12281 

•S8677 

•07352 

1-2721 

5-, 949 

80 

•77786 

1-0859 

•12441 

•88949 

•07462 

1-2855 

5-4254 

81 

•76988 

1-0882 

•12660 

•89161 

•07512 

1-2989 

5-3492 

82 

•76212 

1-0905 

•12793 

•89520 

•07683 

1-3121 

5-2705 

83 

•75458 

1-0920 

•12958 , 

•S9958 

•07819 

1-3252 

5-2101 

84 

•74724 

1-0953 

•13157 ! 

•90095 

•07907 

1-3383 

5*1429 

85 

•74009 

1-0977 

•13330 

•90420 

•07960 

1-3512 

5-0772 

86 | 

•73314 

1-1012 

•13546 

•90734 

•08102 

1-3639 

5-0134 

87 

•72637 

1-1027 

•13704 

•91036 

•08340 

1-3767 

4-9501 

88 

•71978 

1-1054 

•13893 

•91363 

•08436 

1-3893 

4-8886 

89 

•71336 

1-1079 

•14078 : 

•91696 

•08530 

1-4818 

4-8216 

90 

•70710 

1-1105 

•14279 

•92210 

•08621 

1-4142 

4-7694 

91 

•70101 

1-1132 

•14449 

•92352 

•0S716 1 

1-4265 

4-7117 

92 

•69508 

1-1159 

•14643 

•92476 

•0S798 j 

1-4387 

4-6615 

93 

•68930 

1-1186 | 

•14817 

•92914 

•0S932 

1-4507 

4-5999 

94 

•68366 

1-1211 

•15009 

•933S5 

•09076 j 

1-4627 

1*5453 

95 

•67817 

1-1242 

•15211 

•93746 

•09197 

1-4745 

4-4845 

96 

•672S2 

1-1271 

•15375 

•94272 

•09348 

1-4863 




























































Table *or Sj=omekt5 &c.. op a Circle. 


110 


Chord dir j 

Centre 

Rad iuj j 

Cir. Arc. 1 

Area Seg. ! 

Surface I 

Solidity 

Chord 

bv height. | 

Angle v. 

r = k e. | 

b = he. ' 

CL = k 

a = ft c a . 

C = A c*. 

e = h r. 



jr y 






\ * 

\ / 

V 


* 

V / 

'% / 

X / 

X / 


V 

X'' 

4-4398 

97 

•66760 

1-1300 

•15600 

•94470 

•09442 

1-4979 

4-3859 

98 

•66250 

1-1329 

•15801 

•94852 

•09567 

1-5094 

4-3383 

99 

•65754 

1-1359 

•15995 

•95236 

•09693 

1-5208 

4-2862 

100 

•65270 

1-1382 

•16180 

•95682 j 

•09831 

1-5321 

4-2406 

101 

•64798 

1-1420 

.16393 

•96011 

•09856 

1-5432 

4-1930 

102 

•64338 

1-1451 

•16610 

•96412 

•10076 

1-5543 

4-1570 

103 

•63889 

1-1483 

•16925 

•96568 

•10557 

1-5652 

4-1006 

104 

•63450 

1-1515 

•17001 

•97246 

•10273 

1-5760 

4-0555 

105 

•63023 

1-1547 

•17204 

•97643 

•10471 

1-5867 

4-0113 

106 

•62607 

1-15S0 

•17414 

•98067 

•10601 

1-5973 

3-9679 

107 

•62200 

1-1614 

•17619 

•98495 

•10735 

1-6077 

3-9252 

108 

•61803 

M648 

•17832 

•98931 

•10870 

1-6180 

3-8832 

109 

•61416 

1-1682 

•18041 

•99376 

•11007 

1-62S2 

3*8419 

110 

•61039 

1-1716 

•18257 

•98827 

•11149 

1*6388 

3-S013 

111 

•60670 

1-1752 

•18472 

1-0028 

•112S4 

1-6482 

3-7612 

112 

•60325 

1-1790 

•18696 

1-0077 

•11426 

1-6581 

3-7221 

113 

•59960 

1-1823 

•18900 

1-0122 

•11566 

1-6677 

3-6837 

114 

•59618 

1-1859 

•19117 

1-0169 

•11709 

1-6773 

3-6454 

115 

•592S4 

1-1897 

•19339 

1-0218 

•11853 

1-6867 

3-6086 

116 

•58959 

1-1934 

•19559 

1-0266 

•11995 

1-6961 

3-5712 

117 

•58641 

1-1972 

•19787 

1-0317 

•12145 

1-7053 

3-5349 

118 

•58331 

1-2011 

•20009 

1-0368 

•12294 

1-7143 

3-4992 

119 

•58030 

1-2050 

•20227 

1-0417 

•12444 

1-7232 

3-4641 

120 

•57735 

1-20S9 

•20453 

1-0472 

•12596 

1-7320 

3-4296 

121 

•57450 

1-2130 

•20678 

1-0525 

•12748 

1-7407 

3-3953 

122 

•57168 

1-2177 

•20945 

1-0578 

•12903 

1-7492 

3-3616 

123 

•56895 

1-2213 

•21175 

1-0634 

•13060 

1-7576 

3-3285 

124 

•56628 

1-2253 

•21399 

1-0690 

•13218 

1-7659 

3-2910 

125 

•56370 

1-2295 

•21538 

1-0753 

•13391 

1-7740 

3-2637 

126 

•56116 

1-2338 

•21859 

1-0803 

•I355S 

1-7820 

3-2319 

127 

•55870 

1-2381 

•22121 

1-0862 

•13701 

1-7898 

3-2006 

128 

•55630 

1-2425 

•22370 

1-0921 

•13866 

1-7976 

3-1716 

129 

•55396 

1-2470 

•22617 

1-0974 

•14028 

1-8051 

31393 

130 

•55169 

1-2515 

•22865 

1-1040 

•14202 

1-8126 

3-1093 

131 

•54947 

1-2561 

•23113 

1-1104 

•14371 

1-8199 

3-0805 

132 

•54732 

1-2607 

•23372 

1-1164 

•14537 

1-8271 

3-0555 

133 

•54522 

1-2654 

•23603 

1-1212 

•14676 

1-8341 

3-0216 

134 

•54318 

1-2701 

•23892 

M295 

•14894 

1-8410 

2-9777 

135 

•54120 

1-2749 

•24198 

1-1420 

•15209 

1*8477 

2-9651 

136 

•53927 

1-2798 

•24364 

1-1428 

•15252 

1-8543 

2-9374 

137 

•53740 

1-2847 

•24676 

M495 

•15422 

1-8608 

2-9115 

138 

•53557 

1-2897 

•24938 

1-1558 

•15605 

1 1-8671 

2-8S29 

139 

•53380 

1-2948 

•25222 

1-1634 

•15807 

1-8733 

2-8562 

140 

•53209 

1-2999 

•25485 

1-1705 

•15996 

1-8794 

2-8299 

141 

•53042 

1-3051 

•25759 

1-1777 

•16201 

1-8853 

2-8038 

142 

•52881 

1-3065 

•25936 

1-1851 

•16381 

1-8910 

2-7781 

143 

•52724 

1-3157 

•26320 

1-1925 

•16577 

1-8966 

2-7527 

144 

•52573 

1-3211 

•26604 

1-2000 

•16776 

1-9021 






























































Ill 


Table for Segments &c., op a Circle. 


Chord dir. 
by height. 

Centre 
Angle e. 

Rad ius 
r = he. 

Cir. Arc. 
b = h c. 

Area Seg. 
a = h c a . 

Surface 
a = h c 2 - 

Solidity 

C = k 

| Chord 
e = k r. 



^■"71 

s 

< 

r 

N 





— 

\ V 

V 


N 

V 

N 

r ,> 

V 

‘ / 




7- 1 

•» /' 

2-7276 

145 

•52426 

1-3265 

•26SS9 

1-2077 

•16965 

1-9074 

2-7002 

146 

•52284 

1-3320 

•27196 

1-2166 

•17209 

1-9126 

2-6816 

147 

•52147 

1-3377 

•27449 

1-2219 

•17205 

1-9176 

2-6583 

148 

•52015 

1-3433 

•27772 

1-2318 

•17605 

1-9225 

2-6301 

149 

•51887 

1-3491 

•28168 

1-2396 

•17809 

1-9272 

2-6064 

150 

•51764 

1-3549 

•28369 

1-2476 

•18023 

1-9318 

2-5830 

151 

•51645 

1-3608 

•28674 

1-2563 

•18666 

1 9363 

2-5598 

152 

•51530 

1-3668 

•28983 

1-2648 

•18751 

1-9406 

2-5239 

153 

•51420 

1-3729 

•29397 

1-2801 

•18845 

1-9447 

2-5143 

154 

•51315 

1-3790 

•29607 

1-2824 

*18913 

‘ 1-9487 

2-4919 

155 

•51214 

1-3852 

•29928 

1-2914 

•19147 

1 -9526 

2-4699 

156 

•51117 

1-3919 

•30259 

1-3004 

•19374 

1-9563 

2-4478 

157 

•51014 

1-3973 

•30560 

1-3094 

•19607 

1-9598 

2-4262 

158 

•50936 

1-4043 

•30905 

1-3191 

•20029 

1-9632 

2-4047 

159 

•50851 

1-4109 

•31239 

1-3287 

•20095 

1-9663 

2-3835 

160 

•50771 

1-4175 

•31575 

1-3368 

•20342 

1-9696 

2-3613 

161 

•50695 

1-4243 

•31931 

1-3490 

•20609 

1-9725 

2-3417 

162 

•50623 

1-4311 

•32263 

1-3583 

•20847 

1-9753 

2-3211 

163 

•50555 

1-4380 

•32618 

1-3682 

•21105 

1-9780 

2-3004 

164 

•50491 

1-4450 

•32969 

1-3791 

•21371 

1-9805 

2*2805 

165 

•50431 

1-4520 

•33327 

1-3895 

•21634 

1-9829 

2-2605 

166 

•50374 

1-4592 

•33684 

1-4021 

•21904 

1-9851 

2-2408 

167 

•50323 

1-4665 

•34048 

1-4111 

•22177 

1-9871 

2-2212 

168 

•50275 

1-4739 

•34422 

1-4222 

•21946 

1-9890 

2-2013 

169 

•50231 

1-4813 

•34802 

1-4344 

•22766 

1-9908 

2-1826 

170 

•50191 

1-4889 

•35230 

1-4476 

•23028 

1-9924 

2-1636 

171 

•50154 

1-4966 

•35563 

1-4565 

•23266 

1-9938 

2-1447 

172 

•50122 

1-5044 

•35953 

1-4684 ; 

•23650 

1-9951 

2*12x71 

173 

•50093 

1-5123 

•36337 

1-4797 ! 

•23900 

1-9962 

2-1075 

174 

•50068 

1-5202 

•36747 

1-4927 

•24225 

1-9972 

2-0892 

175 

•50047 

1-5283 

•37152 

1-5052 

•24537 

1-9981 

2-0710 

176 

•50030 

1-5365 

•37562 

1-5179 

•24856 

1-9988 

2-0530 

177 

•50017 

1-5448 

•37974 

1-5308 

•25179 

1-9993 

2-0352 

178 

•50007 

1-5533 

•38401 

1-5439 

•25531 

1-9996 

2-0175 

179 

•50002 

1-5618 

•38828 

1-5573 

•25840 

1-9999 

2-0000 

180 

•50000 

1-5707 

•39269 

1-5708 

•26179 

2-0000 


To find the length of an arc of a circle* 

RULE. Divide the chord of the arc by its height, and find the quotient in 
the first column. The corresponding number in the fourth column multiplied 
by the chord is the length of the arc. 


To find the area of a segment of a circle* 

RULE. Divide the chord of the segment by its height, and find the quotient 
in the first column. The corresponding number in the fifth column multiplied 
by the square of the chord, is the area of the segment. 

















































112 


Table of Squares, Cubes, Square and Cube Roots. 








Number. 

Squares. 

Cubes. 

s/ Roots. 

</ Roots. 

Reciprocals. 

1 

1 

1 

1-0000000 

1-0000000 

l.oooooooc 

2 

4 

8 

1-4142136 

1*2599210 

*500000000 

3 

9 

27 

1-7320508 

1*4422496 

•333333333 

4 

16 

64 

2-0000000 

1*5874011 

•250000000 

5 

25 

125 

2-2360680 

1*7099759 

•200000000 

0 

36 

216 

2*4494897 

1-8171206 

•166666667 

7 

49 

343 

2-6457513 

1*9129312 

*142857143 

8 

64 

512 

2-S284271 

2-0000000 

*125000000 

9 

81 

729 

3-0000000 

2*0800837 

*111111111 

10 

100 

1000 

3-1622777 

2*1544347 

•100000000 

11 

121 

1331 

3-3166248 

2-2239801 

•090909091 

12 

144 

1728 

3*4641016 

2-2894286 

*083333333 

13 

169 

2197 

3-6055513 

2*3513347 

*076923077 

14 

196 

2744 

3-7416574 

2-4101422 

*071428571 

15 

225 

3375 

3*8729833 

2*4662121 

*066666667 

16 

256 

4096 

4-0000000 

2-5198421 

*062500000 

17 

289 

4913 

4-1231056 

2*5712816 

.058823529 

18 

324 

5832 

4-2426407 

2-6207414 

*055555556 

19 

361 

6859 

4*3588989 

2-6684016 

*052631579 

20 

400 

8000 

4*4721360 

2-7144177 

•050000000 

21 

441 

9261 

4-5825757 

2-7589243 

•047619048 

22 

484 

10648 

4-6904158 

2*8020393 

*045454545 

23 

529 

12167 

4-7958315 

2-8438670 

*043478261 

24 

576 

13824 

4-8989795 

2-S844991 

*041666667 

25 

625 

15625 

5-0000000 

2-9240177 

•040000000 

26 

676 

17576 

5-0990195 

2-9624960 

*038461538 

27 

729 

19683 

5-1961524 

3-0000000 

*037037037 

28 

784 

21952 

5*2915026 

3-0365889 

*035714286 

29 

841 

24389 

5*3851648 

3-0723168 

*034482759 

30 

900 

27000 

5*4772256 

3-1072325 

*033333333 

31 

961 

29791 

5-5677644 

3-1413806 

*032258065 

32 

1024 

32768 

5-6568542 

3-1748021 

•031250000 

33 

1089 

35937 

5-7445626 

3*2075343 

*030303030 

34 

1156 

39304 

5-8309519 

3-2396118 

*029411765 

35 

1225 

42875 

5-9160798 

3-2710663 

*028571429 

36 

1296 

46656 

6-0000000 

3-3019272 

•027777778 

37 

1369 

50653 

6-0S27625 

3*3322218 

•027027027 

38 

1444 

54872 

6-1644140 

3*3619754 

•026315789 

39 

1521 

59319 

6-2449980 

3-3912114 

•025641026 

40 

1600 

64000 

6-3245553 

3-4199519 

•025000000 

41 

1681 

68921 

6-4031242 

3-4482172 

•024390244 

42 

1764 

74088 

6*4807407 

3-4760266 

•023809524 

43 

1849 

79507 

6*5574385 

3-5033981 

•023255814 

44 

1936 

85184 

6-6332496 

3*5303483 

•022727273 

45 

2025 

91125 

6-7082039 

3*5568933 

•022222222 

<16 

2116 

97336 

6-7823300 

3-5830479 

•021739130 

47 

2209 

103823 

6-8556546 

3-6088261 

•021276600 

48 

2304 

110592 

6-9282032 

3-6342411 

•020833333 

49 

2401 

117649 

7-0000000 

3-6593057 

•020408163 

50 

2500 

125000 

7*0710678 

3-6840314 

•020000000 

51 

2601 

132651 

7-1414284 

3-7084298 

•019607843 

52 

2704 

140608 

7-2111026 

3-7325111 

•019230769 











Table of Squares, Cubes, Square and Cube Roots, 


113 


Number. 

l 

Squares. 

Cubes. 

V Roots. 

3/ 

V Roots. 

Reciprocals. 

53 

2809 

148877 

7-2801099 

3-7562858 

018867925 

54 

2916 

157464 

7-3484692 

3-7797631 

•018518519 

55 

3025 

166375 

7*4161985 

3-8029525 

•018181818 

56 

3136 

175616 

7-4833148 

3-8258624 

•017857143 

57 

3249 

185193 

7-549S344 

3-8485011 

•017543860 

58 

3364 

195112 

7-6157731 

3-8708766 

•017241379 

59 

3481 

205379 

7-6811457 

3-8929965 

•016949153 

60 

3600 

216000 

7-7459667 

3-9148676 

•016666667 

61 

3721 

226981 

7-8102497 

3-9304972 

•016393443 

62 

3844 

238328 

7-8740079 

3-9578915 

•016129032 

63 

3969 

250047 

7-9372539 

3-9790571 

•015873016 

64 

4096 

262144 

8-0000000 

4-0000000 

•015625000 

65 

4225 

274625 

8-0622577 

4-0207256 

•015384615 

66 

4356 

287496 

8-1240384 

4-0412401 

•015151515 

67 

4489 

300763 

8-1853528 

4-0615480 

•014925373 

68 

4624 

314432 

8-2462113 

4-0816551 

•014705882 

69 

4761 

328509 

8-3066239 

4-1015661 

•014492754 

70 

4900 

343000 

8-3666003 

4-1212853 

•014285714 

71 

5041 

357911 

8-4261498 

4-1408178 

•014084517 

72 

5184 

373248 

8-4852814 

4-1601676 

•013888889 

73 

5329 

389017 

8-5440037 

4-1793390 

•01369S630 

74 

5476 

405224 

8-6023253 

4-1983364 

•013513514 

75 

5625 

421875 

8-6602540 

4-2171633 

•013333333 

76 

5776 

438976 

8-7177979 

4-2358236 

•013157895 

77 

5929 

456533 

8-7749644 

4-2543210 

•012987013 

78 

6084 

474552 

8-8317609 

4-2726586 

•012820513 

79 

6241 

493039 

8-8881944 

4-2908404 

•012658228 

80 

6400 

512000 

8-9442719 

4-3088695 

•012500000 

81 

3561 

531441 

9-0000000 

4-3267487 

•012345679 

82 

6724 

551368 

9-0553851 

4-3444815 

•012195122 

83 

6889 

571787 

9-1104336 

4-3620707 

•012048193 

84 

7056 

592704 

9-1651514 

4-3795191 

•011904762 

85 

7225 

614125 

9-2195445 

4-3968296 

•011764706 

86 

7396 

636056 

9-2736185 

4-4140049 

•011627907 

87 

7569 

658503 

9-3273791 

4-4310476 

•011494253 

88 

7744 

681472 

9-3808315 

4-4479692 

•011363636 

89 

7921 

704969 

9-4339S11 

4-4647451 

•011235955 

90 

8100 

729000 

9-4868330 

4-4814047 

•011111111 

91 

8281 

753571 

9-5393920 

4-4979414 

•010989011 

92 

8464 

778688 

9-5916630 

4-5143574 

•010869565 

93 

8649 

804357 

9-6436508 

4-5306549 

•010752688 

94 

8836 

830584 

9-6953597 

4-5468359 

•010638298 

95 

9025 

857375 

9-7467943 

4-5629026 

•010526316 

96 

9216 

884736 

9-7979590 

4-5788570 

•010416667 

97 

i)409 

912673 

9-S488578 

4-5947009 

•010309278 

98 

9604 

941192 

9-8994949 

4-6104363 

•010204082 

99 

9801 

970299 

9-9498744 

4-6260650 

•010101010 

100 

10000 

1000000 

10-0000000 

4-6415888 

•010000000 

101 

10201 

1030301 

10-0498756 

4-6570095 

•009900990 

102 

10404 

1061208 

10-0995049 

4-6723287 

009803922 

103 

10609 

1092727 

10-1488916 

4-6875482 

•009708738 

104 

10816 1 

1124864 

10-1980390 

4-7025694 

•009615385 


8 































Table of Squares, Cubes, Square and Cube Room 





- 



Number. 

Squares. 

Cubes. 

•J Roots. 

V Roots. 

Reciprocals. 

105 

11025 

1157625 

10-2469508 

4-7176940 

•009523810 

106 

11236 

1191016 

10-2956301 

4-7326235 

•009433962 

107 

11449 

1225043 

10-3440804 

4-7474594 

•009345794 

108 

11664 

1259712 

10-3923048 

4-7622032 

•009259259 

100 

11881 

1295029 

10-4403065 

4-7708562 

•009174312 

110 

12100 

1331000 

10-4880885 

4-7914199 

•009090909 

111 

12321 

1367631 

10-5356538 

4-8058995 

•009009009 

112 

12544 

1404928 

10-5830052 

4-8202845 

•00S928571 

113 

12769 

1442897 

10-6301458 

4-8345881 

•008849558 

114 

12996 

1481544 

10-6770783 

4-8488076 

•008771930 

115 

13225 

1520875 

10-7238053 

4-8629442 

•008695652 

116 

13456 

1560896 

10-7703296 

4-8769990 

•008620690 

117 

13689 

1601613 

10-8166538 

4-8909732 

•008547009 

118 

13924 

1643032 

10-8627805 

4-9048681 

•008474576 

119 

14161 

1685159 

10-9087121 

4-9186847 

•008403361 

120 

14400 

1728000 

10-9544512 

4-9324242 

•008333333 

121 

14641 

1771561 

11-0000000 

4-9460874 

•008264463 

122 

14884 

1815S48 

11-0453610 

4-9596757 

•008196721 

123 

15129 

1860867 

11-0905365 

4-973189S 

•008130081 

124 

15376 

1906624 

11-1355287 

4-9S66310 

•008064516 

125 

15625 

1953125 

11-1803399 

5-0000000 

•008000000 

126 

15876 

2000376 

11-2249722 

5-0132979 

•007936508 

127 

16129 

2048383 

11-2694277 

5-0265257 

•007874016 

128 

16384 

2097152 

11-3137085 

5-0396842 

•007812500 

129 

16641 

2146689 

11-3578167 

5-0527743 

•007751938 

130 

16900 

2197000 

11-4017543 

5-0657970 

•007692308 

131 

17161 

2248091 

11-4455231 

5-0787531 

•007633588 

132 

17424 

2299968 

11-4891253 

5-0916434 

•007575758 

133 

17689 

2352637 

11-5325626 

5-1044687 

•007518797 

134 

17956 

2406104 

11-5758369 

5-1172299 

•007462687 

135 

1S225 

2460375 

11-6189500 

5-1299278 

•007407407 

136 

18496 

2515456 

1T6619038 

5-1425632 

•007352941 

137 

18769 

2571353 

11-7046999 

5-1551367 

•007299270 

138 

19044 

262S072 

11-7473401 

5-1676493 

•007246377 

139 

19321 

2685619 

11-7898261 

5-1801015 

•007194245 

140 

19600 

2744000 

11-8321596 

5-1924941 

•007142857 

141 

19S81 

2803221 

U-S743421 

5-2048279 

•007092199 

142 

20164 

2863288 

11-9163753 

5-2171034 

•007042254 

143 

20449 

2924207 

11-9582607 

5-2293215 

•006993007 

144 

20736 

2985984 

12-0000000 

5-2414828 

•006944444 

145 

21025 

3048625 

12-0415946 

5-2535879 

•006896552 

146 

21316 

3112136 

12-0830460 

5-2656374 

•006849315 

147 

21609 

3176523 

12-1243557 

5-2776321 

•006802721 

148 

21904 

3241792 

12-1655251 

5-2895725 

•006756757 

149 

22201 

3307949 

12-2065556 

5-3014592 

•006711409 

150 

22500 

3375000 

12-2474487 

5-3132928 

•006666667 

151 

22801 

3442951 

12-2882057 

5-3250740 

•006622517 

152 

23104 

3511008 

12-3288280 

5-3368033 

•006578947 

153 

23409 

3581577 

12-3693169 

5-3484S12 

•006535948 

154 

23716 

3652264 

12-4096736 

5-3601084 

•006493506 

155 

24025 

3723875 

12-4498996 

5-3716854 

•006451613 

156 

24336 

3796416 

12-489996U 

l 5-3832126 

•006410256 
























Table of Squares, Cubes, Square and Cube Roots. 115 


Number. 

Squares. 

Cubes. 

\/ Roots. 

4/ Roots. 

Reciprocals. 

157 

24649 

3869893 

12*5299641 

5*3946907 

•006369427 

158 

24964 

3944312 

12*5698051 

5*4061202 

•006329114 

159 

25281 

4019679 

12*6095202 

5*4175015 

*006289308 

ICO 

25600 

4096000 

12*6491106 

5*4288352 

•006250000 

161 

25921 

4173281 

12*6885775 

5*4401218 

•006211180 

162 

26244 

4251528 

12*7279221 

5*4513618 

•006172840 

163 

26569 

4330747 

12*7671453 

5*4625556 

•006134969 

164 

26896 

4410944 

12*8062485 

5*4737037 

*006097561 

165 

27225 

4492125 

12*8452326 

5*4848066 

•006060606 

166 

27556 

4574296 

12*8840987 

5*4958647 

*006024096 

167 

27889 

4657463 

12*9228480 

5*5068784 

•005988024 

168 

28224 

4741632 

12*9614814 

5*5178484 

•005952381 

169 

28561 

4826809 

13-0000000 

5*5287748 

*005917160 

170 

28900 

4913000 

13*0384048 

5*5396583 

*005882353 

171 

29241 

5000211 

13*0766968 

5*5504991 

*005847953 

172 

29584 

5088448 

13*1148770 

5*5612978 

*005813953 

173 

29929 

5177717 

13*1529464 

5*5720546 

*005780347 

174 

30276 

5268024 

13*1909060 

• 5*5827702 

•005747126 

175 

30625 

5359375 

13*2287506 

5*5934447 

•005714286 

176 

• 30976 

5451776 

13*2664992 

5*6040787 

•005681818 

177 

31329 

5545233 

13*3041347 

5*6146724 

•005649718 

178 

31684 

5639752 

13*3416641 

5*6252263 

•005617978 

179 

32041 

5735339 

13*3790882 

5*6357408 

•005586592 

180 

32400 

5832000 

13*4164079 

5*6462162 

•005555556 

181 

32761 

5929741 

13*4536240 

5*6560528 

•005524862 

182 

33124 

6028568 

. 13*4907376 

5*6670511 

•005494505 

183 

33489 

6128487 

13*5277493 

5*6774114 

•005464481 

184 

33856 

6229504 

13*5646600 

5*6877340 

•005434783 

185 

34225 

6331625 

13*6014705 

5*6980192 

•005405405 

186 

34596 

6434856 

13*6381817 

5*7082675 

•005376344 

187 

34969 

6539203 

13*6747943 

5*7184791 

•005347594 

188 

35344 

6644672 

13*7113092 

5*7286543 

•005319149 

1S9 

35721 

6751269 

13*7477271 

5*7387936 

•005291005 

190 

36100 

6859000 

13*7840488 

5*7488971 

•005263158 

191 

36481 

6987871 

13*8202750 

5*7589652 

•005235602 

192 

36864 

7077888 

13*8564065 

5*7689982 

*005208333 

193 

37249 

7189517 

13*8924400 

5*7789966 

•005181347 

194 

37636 

7301384 

13*9283883. 

5*7889604 

•005154639 

195 

3S025 

7414875 

13*9642400 

5*7988900 

•005128205 

196 

38416 

7529536 

14*0000000 

5-S0S7857 

•005102041 

| 197 

38809 

7645373 

14*0356688 

5*8186479 

•005076142 

| 198 

39204 

7762392 

14*0712473 

5*8284867 

*005050505 

199 

39601 

7880599 

14*1067360 

5*8382725 

*005025126 

200 

40000 

8000000 

14*1421356 

5*8480355 

*005000000 

201 

40401 

8120601 

14*1774469 

5*8577660 

*004975124 

202 

40804 

8242408 

14*2126704 

5*8674673 

*004950495 

203 

41209 

8365427 

14*2478068 

5*8771307 

*004926108 

204 

41616 

8489664 

14-2S28569 

5*8867653 

*004901961 

205 

42025 

8615125 

14*3178211 

5*8963685 

*004878049 

206 

42436 

8741810 

14*3527001 

5*9059406 

•004S543C9 

207 

42849 

8869743 

14*3874946 

5*9154817 

*004830918 

208 

43264 

8998912 

14*4222051 

5*9249921 

*004807692 

1 


















116 Table of Squares, Cubes, Square axd Cube Roots. 


Number. 

Squares 

Cubes. 

y/ Roots. 

y/ Roots. 

j Reciprocals. 

209 

43681 

9129329 

14-4568323 

5-9344721 

•004784689 

210 

44100 

9261000 

14-4913767 

5-9439220 

•004761905 

211 

44521 

9393931 

14-5258390 

5-9533418 

•004739336 

212 

44944 

9528128 

14-5602198 

5-9627320 

•004716981 

213 

45369 

9663597 

14-5945195 

5-9720926 

•004694836 

214 

45796 

9800344 

14-6287388 

6-9814240 

•004672S97 

' 215 

46225 

9938375 

14-6628783 

5-9907264 

•004651163 

216 

46656 

10077696 

14-6969385 

6-0000000 

•004629630 

217 

47089 

10218313 

14-7309199 

6-0092450 

•00460S295 

218 

47524 

10360232 

14-7648231 

6-0184617 

•004587156 

219 

47961 

10503459 

14-79S6486 

6-0276502 

•004566210 

220 

48400 

10648000 

14-8323970 

6-0368107 

•004545455 

221 

48841 

10793861 

14-S660687 

6-0459435 

•004524887 

222 

49284 

10941048 

14-8996644 

6-05504S9 

•004504505 

223 

49729 

11089567 

14-9331845 

6-0641270 

•004484305 

224 

50176 

11239424 

14-9666295 

6-0731779 

•004464286 

225 

50625 

113.90625 

15-0000000 

6-0824020 

•004444444 

226 

51076 

11543176 

15-0332964 

6-0991994 

•004424779 

227 

51529 

11697083 

15-0665192 

6-1001702 

•004405286 

228 

51984 

11852352 

15-0996689 

6-1091147 

•004385965 

229 

52441 

12008989 

15-1327460 

6-1180332 

•004366812 

230 

52900 

12167000 

15-1657509 

6-1269257 

•004347826 

231 

53361 

12326391 

15-1986842 

6-1357924 

•004329004 

232 

53824 

12487168 

15-2315462 

6-1446337 

•004310345 

233 

54289 

12649337 

15-2643375 

6-1534495 

•004291845 

234 

54756 

12812904 

15-2970585 

6-1622401 

•004273504 

235 

55225 

12977875 

15-3297097 

6-1710058 

•004255319 

236 

55696 

13144256 

15-3622915 

6-1797466 

•004237288 

237 

56169 

13312053 

15-3948043 

6-1884628 

•004219409 

238 

56644 

13481272 

15-4272486 

6-1971544 

•004201681 

239 

57121 

13651919 

15-4596248 

6-205S218 

•004184100 

240 

57600 

13824000 

15-4919334 

6-2144650 

•004166667 

241 

58081 

13997521 

15-5241747 

6-2230843 

•004149378 

242 

58564 

14172488 

15-5563492 

6-2316797 

•004132231 

243 

59049 

14348907 

15-5884573 

6-2402515 

•004115226 

244 

59536 

14526784 

15-6204994 

6-2487998 

•004098361 

245 

60025 

14706125 

15-6524758 

6-2573248 

•004081633 

246 

60516 

14886936 

15-6843871 

6-265S266 

•004065041 

247 

61009 

15069223 

15-7162336 

6-2743054 

•004048583 

248 

61504 

15252992 

15-7480157 

6-2827613 

•004032258 

249 

62001 

15438249 

15-7797338 

6-2911946 

•004016064 

250 

62500 

15625000 

15-8113883 

6-2996053 

•004000000 

251 

63001 

15813251 

15-8429795 

6-3079935 

•003984064 

252 

63504 

16003008 

15-8745079 

6-3163596 

•00396S254 

253 

64009 

16194277 

15-9059737 

6-3247035 

•003952569 

254 

64516 

16387064 

15-9373775 

6-3330256 

•00393700S 

255 

65025 

16581375 

15-9687194 

6-3413257 

•003921569 

256 

65536 

16777216 

16-0000000 

6-3496042 

•003906250 

257 

66049 

16974593 

16-0312195 

6-3578611 

•003891051 

258 

66564 

17173512 

16-0623784 

6-3660968 

•003875969 

25 9 

67081 

17373979 

16-0934769 

6-3743111 

•003861004 

260 

67600. 

17576000 

16-1245155 

. 6-3825043 

•003846164 




















i able of Squares. Cubes, Square and Cube Roots. 117 







, • 1 

Number. 

Squares. 

Cubes. 

V Roots. 

\/ Roots. 

Reciprocals. 

261 

68121 

17779581 

16-1554944 

6-3906765 

•003831418 

282 

68644 

17984728 

16-1864141 

6-3988279 

•003816794 

263 

69169 

18191447 

16-2172747 

6-4069585 

•003802281 

264 

69696 

18399744 

16-2480768 

6-4150687 

*003787879 

265 

70225 

18609625 

16-2788206 

6-4231583 

•003773585 

266 

70756 

1SS21096 

16-3095064 

6-4312276 

•003759398 

267 

71289 

19034163 

16-3401346 

6-4392767 

•003745318 

268 

71824 

1924S832 

16-3707055 

6-4473057 

•003731343 

269 

72361 

19465109 

16-4012195 

6-4553148 

•003717472 

270 

72900 

19683000 

16.4316767 

6-4633041 

*003703704 

271 

73441 

19902511 

16-4620776 

6-4712736 

•003690037 

272 

73984 

20123643 

16-4924225 

6-4792236 

•003676471 

273 

74529 

20346417 

16-5227116 

6-4871541 

•003663004 

274 

75076 

20570824 

16-5529454 

6-4950653 

•003649635 

275 

75625 

20796875 

16-5831240 

6-5029572 

•003636364 

276 

76176 

21024576 

16-6132477 

6-5108300 

•003623188 

277 

76729 

21253933 

16-6433170 

6-5186839 

•003610108 

278 

77284 

21484952 

16-6783320 

6-5265189 

•003597122 

279 

77841 

2 1 717639 

16-7032931 

6-5343351 

•003584229 

280 

78400 

21952000 

16-7332005 ' 

6-5421326 

•003571429 

281 

78961 

22188041 

16-7630546 

6-5499116 

•003558719 

282 

79524 

22425768 

16-7928556 

6-5576722 

•003546099 

283 

80089 

22665187 

16-8226038 

6-5654144 

•003533569 

284 

80656 

22906304 

16-8522995 

6-5731385 

•003521127 

285 

81225 

23149125 

16*8819430 

6-5808443 

•003508772 

286 

81796 

23393656 

16-9115345 

6-58S5323 

•003496503 

287 

82369 

23639903 

16-9410743 

6-5962023 

•003484321 

288 

82944 

23887S72 

16-9705627 

6-6038545 

•003472222 

289 

83521 

24137569 

17-0000000 

6-6114890 

•003460208 

290 

84100 

24389000 

17-0293864 

6-6191060 

•003448276 

291 

84681 

24642171 

17-0587221 

6-6267054 

•003436426 

292 

85264 

24897088 

17-0880075 

6-6342874 

•003424658 

293 

85849 

25153757 

17-1172428 

6-6418522 

•003412969 

294 

86436 

25412184 

17-1464282 

6-6493998 

•003401361 

295 

87025 

25672375 

17-1755640 

6-6569302 

•003389831 

296 

87616 

25934836 

17-2046505 

6-6644437 

•003378378 

297 

88209 

26198073 

17-2336879 

6-6719403 

•003367003 

298 

88804 

26463592 

17-2626765 

6.6794200 

•003355705 

299 

89401 

26730899 

17-2916165 

6.6868831 

•003344482 

300 

90000 

27000000 

17-3205081 

6.6943295 

•003333333 

301 

90601 

27270901 

17*3493516 

6.7017593 

*003322259 

302 

91204 

27543608 

17-3781472 

6*7091729 

•003311258 

303 

91809 

2781 SI 27 

17-4068952 

6-7165700 

•003301330 

304 

92416 

28094464 

17-4355958 

6-7239508 

•003289474 

305 

93025 

28372625 

17-4642492 

6-7313155 

■0032786S9 

306 

93636 

28652616 

17-4928557 

6-7386641 

•003267974 

307 

94249 

28934443 

17-5214155 

6-7459967 

•003257329 

308 

94864 

29218112 

17-5499288 

6-7533134 

•003246753 

309 

95481 

29503609 

17-5783958 

6-7606143 

•003236246 

310 

96100 

29791000 

17-6068169 

6-7678995 

•003225806 

311 

96721 

30080231 

17-6351921 

6*7751690 

•003215434 

312 

97344 

30371328 

17-6635217 

6-7824229 

•003205128 























118 Table of Squares, Cubes, Square and Cube Roots. 


i Number. 

Squares. 

Cubes. 

\/ Roots. 

3 / " “ 

V Roots. 

Reciprocals. 

313 

97969 

30664297 

17-6918060 

6-7896613 

•003194888 

314 

98596 

30959144 

17-7200451 

6-7968844 

•003184713 

315 

99225 

31255875 

17-7482393 

6-8040921 

•003'74603 

315 

99856 

31554496 

17-7763888 

6-8112847 

•003164557 

317 

100489 

31855013 

17-8044938 

6-8184620 

.003154574 

318 

101124 

32157432 

17-8325545 

6-8256242 

•003144654 i 

319 

101761 

32461759 

17-8605711 

6-8327714 

•003134796 

320 

102400 

32768000 

17-8885438 

6-8399037 

•003125000 

321 

103041 

33076161 

17-9164729 

6-8470213 

•003115265 

322 

103684 

33386248 

17-9443584 

6-8541240 

•003105590 

323 

104329 

33698267 

17-9722008 

6-S612120 

•003095975 

324 

104976 

34012224 

18-0000000 

6-8682855 

•003086420 

325 

105625 

34328125 

18-0277564 

6-8753433 

•003076923 

32C 

106276 

34645976 

18-0554701 

6-8823888 

•003067485 

327 

106929 

34965783 

18-0831413 

6-8894188 

•003048104 

328 

107584 

35287552 

18-1107703 

6-8964345 

•003048780 

329 

108241 

35611289 

18-1383571 

6-9034359 

•003039514 

330 

108900 

35937000 

18-1659021 

6-9104232 

•003030303 

331 

109561 

36264691 

18-1934054 

6-9173964 

•003021148 

332 

110224 

36594368 

18-2208672 

6-9243556 

•003.012048 

333 

110889 

36926037 

1S-2482876 

6-9313088 

•003003003 

334 

111556 

37259704 

18-2756689 

6-9382321 

•002994012 

335 

112225 

37595375 

18-3030052 

6-9451496 

•002985075 

336 

112896 

37933056 

18-3303028 

6-9520533 

•002976190 

337 

113569 

38272753 

18-3575598 

£-9589434 

•002967359 

338 

114244 

38614472 

18-3847763 

6-9658198 

•002958580 

339 

114921 

38958219 

18-4119526 

6-9726826 

•002949853 

340 

115600 

39304000 

18-4390889 

6-9795321 

•002941176 

341 

116281 

39651821 

18-4661853 

6-9863681 

•002932551 

342 

116964 

40001688 

18-4932420 

6-9931906 

•002923977 

343 

117649 

40353607 

18-5202592 

7-0000000 

•002915452 

344 

11S336 

40707584 

18-5472370 

7-0067962 

•002906977 

345 

119025 

41063625 

18-5741756 

7-0135791 

•002898551 

346 

119716 

41421736 

18-6010752 

7-0203490 

•002890173 

347 

120409 

41781923 

18-6279360 

7-0271058 

•002881844 

348 

121104 

42144192 

18-6547581 

7-0338497 

•002873563 

349 

121801 

42508549 

18-6815417 

7-0405860 

•002865330 

350 

122500 

42875000 

18-7082869 

7-0472987 

•002857143 

351 

123201 

43243551 

18-7349940 

7-0540041 

•002849003 

352 

123904 

43614208 

18-7616630 

7-0606967 

•002840909 

353 

124609 

43986977 

18-7882942 

7-0673767 

•002832861 

354 

125316 

44361864 

18-8148S77 

7-0740440 

•002824859 

355 

126025 

44738875 

18-8414437 

7-0806988 

•00281690i 

356 

126736 

45118016 

18-8679623 

7-0873411 

•002808989 

357 

127449 

45499293 

18-8944436 

7-0939709 

•002801120 

358 

128164 

45882712 

18-9208879 

7-1005885 

•002793296 

359 

128881 

46268279 

18-9472953 

7-1071937 

•002785515 

360 

129600 

46656000 

18-9736660 

7-1137866 

•002777778 

361 

130321 

47045831 

19-0000000 

7-1203674 

•002770083 

362 

131044 

47437928 

19-0262976 

7-1269360 

•002762431 

363 

131769 

47832147 

19-0525589 

7-1334925 

•002754821 

364 

132496 

48228544 

19-0787840 

7-1400370 

•002747253 
























Ta;le op Squares, Cubes, Square and Cube Roots. 


Number. 

Squares.' 

Cubes. 

V' Roots. 

V Roots. 

Reciprocals. 

365 

133225 

48627125 

19*1049732 

7*1465695 

*002739726 

366 

133956 

49027896 

19*1311265 

7*1530901 

*002732240 

367 

134689 

49430863 

19*1572441 

7*1595988 

*002724796 

368 

135424 

49836032 

19*1833261 

7*1660957 

*002717391 

369 

136161 

50243409 

19*2093727 

7*1725809 

*002710027 

370 

136900 

50653000 

19*2353841 

7*1790544 

*002702703 

371 

137641 

51064811 

19*2613603 

7*1855162 

*002695418 

372 

138384 

51478848 

19*2873015 

7*1919663 

*002688172 

373 

139129 

5189-5117 

19*3132079 

7*1984050 

*002680965 

374 

139876 

52313624 

19*3390796 

7*2048322 

•002673797 

375 

140625 

52734375 

19*3649167 

7*2112479 

*002666667 

376 

141376 

53157376 

19*3907194 

7*2176522 

*002659574 

377 

142129 

53582633 

19*4164S78 

7*2240450 

•002652520 

378 

142884 

'54010152 

19*4422221 

7*2304268 

•002645503 

379 

143641 

1 54439939 

19*4679223 

7*2367972 

•002638521 

380 

144400 

54872000 

19*4935887 

7*2431565 

•002631579 

381 

145161 

55306341 

19*5192213 

7*2495045 

•002624672 

382 

145924 

55742968 

19*5448203 

7*2558415 

•002617801 

383 

146689 

56181887 

19*5703858 

7*2621675 

•002610966 

3S4 

147456 

56623104 

19*5959179 

7*2684824 

•002604167 

385 

148225 

57066625 

19*6214169 

7-2747S64 

•002597403 

386 

148996 

57512456 

19*6468827 

7*2810794 

•002590674 

387 

149769 

57960603 

19*6723156 

7*2873617 

•002583979 

388 

150544!58411072 

19*6977156 

7*2936330 

•002577320 

389 

151321 

58863869 

19*7230829 

7*2998936 

•002570694 

390 

152100 

59319000 

19*7484177 

7*3061436 

•002564103 

391 

152881 

59776471 

19*7737199 

7*3123828 

•002557545 

392 

153664 

60236288 

19*7989899 

7*3186114 

•002551020 

393 

154449 

60698457 

19*8242276 

7*3248295 

•002544529 

394 

155236 

61162984 

19*8494332 

7*3310369 

•002538071 

395 

156025 

61629875 

19*8746069 

7*3372339 

•002531646 

396 

156816 

62099136 

19*8997487 

7*3434205 

•002525253 

397 

157609 

62570773 

19*9248588 

7*3495966 

•002518892 

398 

158404 

63044792 

19*9499373 

7*3557624 

•002512563 

399 

159201 

63521199 

19*9749844 

7*3619178 

•002506266 

400 

160000 

64000000 

20-0000000 

7*3680630 

•002500000 

401 

160801 

64481201 

20-0249S44 

7*3741979 

•002493760 

402 

161604 

64964S08 

20*0499377 

7*3803227 

•002487562 

403 

162409 

65450827 

20*0748599 

7*3864373 

•002481390 

404 

163216 

65939264 

20*0997512 

7*3925418 

•002475248 

405 

164025 

66430125 

20*1246118 

7*3986363 

•002469136 

406 

164836 

66923416 

20*1494417 

7*4047206 

•002463054 

407 

165649 

67419143 

20*1742410 

7*4107950 

•002457002 

408 

166464 1 

67917312 

20*1990099 

7-416S595 

•002450980 

409 

167281 ; 

68417929 

20*2237484 

7*4229142 

•0024449S8 

410 

168100 1 

68921000 

20*248454)7 

7*4289589 

•002439024 

411 

168921 

69426531 

20*2731349 

7*4319938 

•002433090 

412 

169744 

69934528 

20*2977831 

7*4410189 

•002427184 

413 

170569 

70444997 

20*3224014 

7*4470343 

•002421308 

414 

171396 

70957944 

20*3469899 

7*4530399 

•002415459 

415 

172225 

71473375 

20*3715488 

7*4590359 

•002409639 

41€ 1 

173056 

71991296 

20*3960781 

7*4650223 

• 

•002406846 j 
































Table of Squares. Cubes, Square and Cube Roots 


Number. 

Squares 

Cubes. 

y/ Roots. 

<y/ Roots. 

Reciprocals. 

417 

1738S9 

72511713 

20*4205779 

7-4709991 

•002398082 

418 

174724 

73034632 

20-4450483 

7-4769664 

•002392344 

419 

175561 

73560059 

20-4694895 

7-4829242 

•002386635 

420 

176400 

74088000 

20-4939015 

7-4888724 

•002380952 

421 

177241 

74618461 

20-5182S45 

7-4948113 

•002375297 

422 

178084 

75151448 

20-5426386 

7-5007406 

•002369668 

423 

178929 

756S6967 

20-5669638 

7-5066607 

•002364066 

424 

179776 

76225024 

20-5912603 

7-5125715 

•002358491 

425 

180625 

76765625 

20-6155281 

7-5184730 

•002352941 

426 

181476 

77308776 

20-6397674 

7-5243652 

•002347418 

427 

182329 

77854483 

20-6639783 

7-5302482 

•002341920 

428 

183184 

78402752 

20-6881609 

7-5361221 

•002336449 

429 

184041 

78953589 

20-7123152 

7-5419867 

•002331002 

430 

184900 

79507000 

20-7364414 

7-5478423 

•002325581 

431 

185761 

80062991 

20-7605395 

7-5536888 

•002320186 

432 

186624 

80621568 

20-7846097 

7-5595263 

•002314815 

433 

187489 

81182737 

20-8086520 

7-5653548 

•002309469 

434 

188356 

81746504 

20-8326667 

7-5711743 

•002304147 

435 

189225 

82312875 

20-8566536 

7-5769849 

•002298851 

436 

190096 

82S81856 

20-8806130 

7-5827865 

•002293578 

437 

190969 

83453453 

20-9045450 

7-5885793 

•002288330 

438 

191844 

84027672 

20-9284495 

7-5943633 

•002283105 

439 

192721 

84604519 

20-9523268 

7-6001385 

•002277904 

440 

193600 

85184000 

20-9761770 

7-6059049 

•002272727 

441 

194481 

85766121 

21-0000000 

7-6116626 

•002267574 

442 

195364 

863508S8 

21-0237960 

7-6174116 

•002262443 

443 

196249 

86938307 

21-0475652 

7-6231519 

•002257336 

444 

197136 

87528384 

21-0713075 

7-6288837 

•002252252 

445 

198025 

88121125 

21-0950231 

7-6346067 

•002247191. 

446 

198916 

88716536 

21-1187121 

7-6403213 

•002242152 

447 

199809 

89314623 

21-1423745 

7-6460272 

•002237136 

448 

200704 

89915392 

21-1660105 

7-6517247 

•002232143 

449 

201601 

90518849 

21-1896201 

7-6574138 

•002227171 

450 

202500 

91125000 

21-2132034 

7-6630943 

•002222222 

451 

203401 

91733851 

21-2367606 

7-6687665 

•002217295 

452 

204304 

92345408 

21-2602916 

7-6744303 

•002212389 

453 

205209 

92959677 

21-2837967 

7-6800857 

•002207506 

454 

206116 

93576664 

21-3072758 

7^857328 

*002202643 

455 

207025 

94196375 

21-3307290 

7-6913717 

•0021 7802 

456 

207936 

94818816 

21-3541565 

7-6970023 

•0021929S2 

457 

208849 

95443993 

21-3775583 

7-7026246 

•002188184 

458 

209764 

96071912 

21-4009346 

7-7082388 

•002183406 

459 

210681 

96702579 

21-4242853 

7-7188448 

•002178649 

460 

211600 

97336000 

21-4476106 

7-7194426 

•002173913 

461 

212521 

97972181 

21-4709106 

7-7250325 

•002169197 

462 

213444 

98611128 

21-4941853 

7-7306141 

•002164502 

463 

214369 

99252847 

21-5174348 

7-7361877 

•002159827 

464 

215296 

99897344 

21-5406592 

7-7417532. 

•002155172 

465 

216225 

100544625 

21-5638587 

7-7473109 

•002150538 

468 

217156 

101194696 

21-5870331 

7-752S606 

•002145923 

467 

218089 

101847563 

21-6101828 

7-7584023 

•002141328 

468 

219024 

1102503232 

21-6333077 

7-7639361 

•002136752 





















Table oe Squares. Cubes, Square and Cube Roots. 


N umber. 

Squares. 

| Cubes. 

s/ Roots. 

\/ Roots. 

469 

219961 

103161709 

21-6564078 

7-7094620 

470 

220900 

103823000 

21-6794834 

7-7749801 

471 

221841 

104487111 

21-7025344 

7-7804904 

472 

222784 

105154048 

21-7255610 

7-7859928 

473 

223729 

105828817 

21*7485632 

7-7914875 

474 

224676 

106496424 

21*7715411 

7*7969745 

475 

225625 

107171875 

21-7944947 

7-8024538 

476 

226576 

107850176 

21-8174242 

7-8079254 

477 

227529 

108531333 

21*8403297 

7-8133892 

478 

228484 

109215352 

21-8632111 

7*8188456 

479 

229441 

109902239 

21-8860686 

7-8242942 

480 

230400 

110592000 

21-9089023 

7-8297353 

481 

231361 

111284641 

21-9317122 

7-8351688 

482 

232324 

111980168 

21-9544984 

7-8405949 

483 

233289 

112678587 

21-9772610 

7-8460134 

484 

234256 

113379904 

22-0000000 

7-8514244 

4S5 

235225 

1140S4125 

22-0227155 

7-8568281 

486 

236196 

114791256 

22-0454077 

7-8622242 

487 

237169 

115501303 

22-06S0765 

7-8676130 

488 

238144 

116214272 

22-0907220 

7-8729944 

489 

239121 

116930169 

22-1133444 

7-8783684 

490 

240100 

117649000 

22-1359436 

7-8837352 

491 

241081 

118370771 

22-1585198 

7*8890946 

492 

242064 

119095488 

22-1810730 

7-8944468 

493 

243049 

119823157 

22-2036033 

7-8997917 

494 

244036 

120553784 

22-2261108 

7-9051294 

495 

245025 

121287375 

22-2485955 

7-9104599 

496 

246016 

122023936 

22-2710575 

7-9157832 

497 

247009 

122763473 

22-2934968 

7-9210994 

498 

248004 

123505992 

22-3159136 

7-9264085 

499 

249001 

124251499 

22-3383079 

7*9317104 

500 

250000 

125000000 

22-3606798 

7-9370053 

501 

251001 

125751501 

22-3830293 

7-9422931 

502 

252004 

126506008 

22-4053565 

7-9475739 

503 

253009 

127263527 

22-4276615 

7-9528477 

504 

254016 

1280240641 

22-4499443 

7-9581144 

505 

255025 

128787625 

22-4722051 

7-9633743 

506 

256036 

129554216 

22-4944438 

7-9686271 

507 

257049 

130323843 

22-5166605 

7-9738731 

508 

258064 

131096512 

22-5388553 

7-9791122 

509 

259081 

131S72229 

22-5610283 

7-9843444 

510 

260100 

132651000 

22-5831796 

7-9895697 

511 

261121 

133432831 

22-6053091 

7-9947883 

512 

262144 

134217728 

22-6274170 

8-0000000 

513 

263169 

135005697! 

22-6495033 

8-0052049 

514 

264196 

135796744 

22*6715681 

8-0104032 

515 

265225 

136590S75 

22-6936114 

8-0155946 

516 

266256 

137388096 

22-7156334 

8-0207794 

517 

267289 

138188413 

22-7376341 

8-0259574 

518 

2683241 

138991832 

22-7596134 

8-03112S7 

619 

2693611 

139798359 

22-7815715 

8-0362935 

520 1 

270400 1 

14060S900 1 

22-8035085 

S-0414515 


121 


Reciprocals. 

*002132196 

*002127660 

•002123142 

•002118644 

•002114165 

•002109705 

•002105263 

•002100S40 

•002096436 

•002092050 

•002087683 

•002083333 

•002079002 

•002074689 

•002070393 

•002066116 

•002061856 

•002057613 

•002053388 

•002049180 

•002044990 

•002040816 

•002036660 

•002032520 

•002028398 

•002024291 

•002020202 

•002016129 

•002012072 

•002008032 

•002004008 

•002000000 

•001996008 

•001992032 

•001988072 

•001984127 

•001980198 

•001976285 

•001972387 

•001968504 

•001964637 

•001960784 

•001956947 

•001953125 

•001949318 

•001945525 

•001941748 

•001937984 

•001934236 

•001930502 

•001926782 

•001923077 



































Table of Squares, Cubes, Square and Cube Hoots. 


Number. 

Squares. 

Cubes. 

\/ Roots. 

v Roots. 

Reciprocals. 

521 

271441 

14142C761 

22*8254244 

8-0466030 

•001919386 

522 

272484 

142236648 

22-8473193 

8-0517479 

•001915709 

523 

273529 

143055667 

22-8691933 

8-0568862 

•001912046 

524 

274576 

143877824 

22-8910463 

8-0620180 

•001908397 

525 

275625 

144703125 

22-9128785 

8-0671432 

•001904762 

526 

276676 

145531576 

22-9346899 

8-0722620 

•001901141 

527 

277729 

146363183 

22-9564806 

8-0773743 

•001897533 

528 

278784 

147197952 

22-9782506 

8-0824800 

•001893939 

529 

279841 

148035889 

23-0000000 

8-0875794 

*001890359 

530 

280900 

148877001 

23-0217289 

8-0926723 

•001886792 

53 L 

281961 

149721291 

23-0434372 

8-0977589 

•001S83239 

532 

283024 

150568768 

23-0651252 

8-1028390 

•001879699 

533 

284089 

151419437 

23-0867928 

8-1079128 

•001876173 

534 

285156 

152273304 

23-1084400 

8-1129803 

•001872659 

535 

286225 

153130375 

23-1300670 

8-11S0414 

•001869159 

536 

287296 

153990656 

23-1516738 

8-1230962 

•001865672 

537 

288369 

154854153 

23-1732605 

8-1281447 

•001862197 

s 538 

2S9444 

155720S72 

23-1948270 

8-1331870 

•001858736 

539 

290521 

156590819 

23-2163735 

8-1382230 

•001855288 

540 

291600 

157464000 

23-2379001 

8-1432529 

•001851852 

541 

292681 

J58340421 

23-2594067 

8-1482765 

•001848429 

542 

293764 

159220088 

23-2808935 

8-1532939 

•001S4501S 

543 

294849 

160103007 

23-3023604 

8-1583051 

•001841621 

544 

295936 

160989184 

23-3238076 

8-1633102 

•001838235 

545 

297025 

161878625 

23-3452351 

8-1683092 

•001834862 

546 

298116 

162771336 

23-3666429 

8-1733020 

•001831502 

547 

299209 

163667323 

23-3880311 

8-1782888 

•001828154 

548 

300304 

164566592 

23-4093998 

8-1832695 

•001824818 

549 

301401 

165469149 

23-4307490 

8-1882441 

•001821494 

550 

302500 

166375000 

23-4520788 

8-1932127 

•001818182 

551 

303601 

1672S4151 

23-4733892 

8-1981753 

•001814882 

552 

304704 

168196608 

23-4946802 

8-2031319 

•001811594 

553 

305809 

169112377 

23-5159520 

8-2080825 

•001S0S318 

554 

306916 

170031464 

23-5372046 

8-2130271 

•001S05054 

555 

308025 

170953875 

23-55843S0 

8-2179657 

•001801802 

556 

309136 

171879616 

23-5796522 

8-2228985 

•001798561 

557 

310249 

172808693 

23-6008474 

8-2278254 

•001795332 

558 

311364 

173741112 

23-6220236 

8-2327463 

•001792115 

559 

312481 

174676879 

23-6431808 

8-2376614 

*0017SS909 

560 

313600 

175616000 

23-6643191 

8-2425706 

•001785714 

561 

314721 

176558481 

23-6854386 

8-2474740 

•001782531 

562 

315844 

177504328 

23-7065392 

8-2523715 

•001779359 

563 

316969 

178453547 

23-7276210 

8-2572635 

•001776199 

564 

318096 

179406144 

23-7486842 

8-2621492 

•001773050 

565 

319225 

180362125 

23-7697286 

8-2670294 

•001769912 

566 

320356 

181321496 

23-7907545 

8-2719039 

•001766784 

567 

321489 

182284263 

23-8117618 

8-2767726 

•001763668 

563 

322624 

183250432 

23-8327506 

8-2816255 

•001760563 

569 

323761 

184220009 

23-8537209 

8-2864928 

•001757469 

570 

324900 

185193000 

23-8746728 

8-2913444 

•001754386 

571 

326041 

186169411 

23-8956063 

8-2961903 

•001751313 

572 

327184 

187149248 

23-9165215 

8-3010304 

•001748252 





























Table of Squares; Cubes, Square and Cube .Roots. 


Number. 

Squares. 

Cubes. 

V^Koote. 

yj Roots. 

Reciprocals. 

573 

328329 

188132517 

23-9374184 

8-3058651 

•001745201 

574 

329476 

189119224 

23-9582971 

8-3106941 

•001742160 

575 

330625 

!190109375 

23-9791576 

8-3155175 

•001739130 

576 

331776 

191102976 

24-0000000 

8-3203353 

•001736111 

577 

332927 

192100033 

24-0208243 

8-3251475 

•001733102 

578 

334084 

193100552 

24-0416306 

8-3299542 

•001730104 

579 

335241 

194104539 

24-0624188 

8-3347553 

•001727116 

580 

336400 

195112000 

24-0831891 

8-3395509 

•001724138 

581 

337561 

196122941 

24-1039416 

8-3443410 

•001721170 

582 

338724 

197137368 

24-1246762 

8-3491256 

•001718213 

583 

339889 

198155287 

24-1453929 

8-3539047 

•001745266 

584 

341056 

199176704 

24-1660919 

8-35S6784 

•001712329 

585 

342225 

200201625 

24-1867732 

8-3634466 

•001709402 

586 

343396 

201230056 

24-2074369 

8-3682095 

•001706485 

587 

344569 

202262003 

24-2280829 

8-3729668 

•001703578 

588 

345744 

203297472 

24-2487113 

8-3777188 

•001700680 

589 

346921 

204336469 

24-2693222 

8-3824653 

•001697793 

590 

348100 

205379000 

24-2899156 

S-3S72065 

•001694915 

591 

349281 

206425071 

24-3104996 

8-3919428 

•001692047 

592 

350464 

207474688 

24-3310501 

8-3966729 

•001689189 

593 

351649 

208527857 

24-3515913 

8-4013981 

•001686341 

594 

352836 

209584584 

24-3721152 

8-4061180 

•001683502 

595 

354025 

210644875 

24-3926218 

8-4108326 

•001680672 

596 

355216 

211708736 

24-4131112 

8-4155419 

*001677852 

597 

356409 

212776173 

24-4335834 

8-4202460 

•001675042 

598 

357604 

213847192 

24-4540385 

8-4249448 

•001672241 

599 

358801 

214921799 

24-4744765 

8-4296383 

•001669449 

600 

360000 

216000000 

24-4948974 

8-4343267 

•001666667 

601 

361201 

217081801 

24-5153013 

8-4390098 

•001663894 

602 

.362404 

218167208 

24*5356883 

8-4436877 

•001661130 

603 

363609 

219256227 

24-55605S3 

8-4483605 

•001658375 

604 

364816 

220348864 

24-5764115 

8-4530281 

•001655629 

605 

366025 

221445125 

24-5967478 

8-4576906 

•001652893 

606 

367236 

222545016 

24-6170673 

8-4623479 

•001650165 

607 

368449 

223648543 

24-6373700 

8-4670001 

•001647446 

608 

369664 

224755712 

24-6576560 

8-4716471 

•001644737 

609 

370881 

225866529 

24-6779254 

8-4762892 

•001642036 

* 610 

372100 

226981000 

24-6981781 

8-4809261 

•001639344 

611 

373321 

228099131 

24-7184142 

8-4855579 

•001636661 

612 

374544 

229220928 

24-7386338 

8-4901848 

•001633987 

613 

375769 

230346397 

24-7588368 

8-4948065 

•001631321 

614 

376996 

231475544 

24-7790234 

8-4994233 

•001628664 

615 

378225 

232608375 

24-7991935 

8-5040350 

•001626016 

616 

379456 

233744896 

24-8193473 

8-5086417 

•001623377 

617 

380689 

234885113 

24-8394847 

8-5132435 

•001620746 

618 

381924 

236029032 

24-8596058 

8-5178403 

•001618123 

619 

383161 

237176659 

24-8797106 

8-5224331 

•001615509 

620 

384400 

238328000 

24-8997992 

8-5270189 

•001612903 

621 

385641 

239483061 

24-9198716 

8-5316009 

•001610306 

622 

386884 

240641848 

24-9399278 

8-5361780 

•001607717 

623 

388129 

241804367 

24-9599679 

8-5407501 

•001665136 

624 

389376 242970624 

24-9799920 

8-5453173 

001602561 



























124 Table of Squares, Cubes, Square A?n> Cube Roots. 


dumber. 1 

Squares. 

Cubes. 

s/ Roots. 

4 / Roots. 

Reciprocals. 

625 

390025 

244140625 

25-0000000 

8-5498797 

•001600000 

626 

391876 1 

245134376 

25*0199920 

8-5544372 

•001597444 

627 

.393129 

246491883 

25-0399681 

8-5589899 

•001594896 

628 

394384 

247673152 

25-0599282 

8-5635377 

•001592357 

629 

395041 

248858189 

25-0798724 

8-5680807 

•001589825 

630 

390900 

250047000 

25-0998008 

8-5726189 

•001587302 

631 

39S161 

251239591 

25-1197134 

8-5771523 

•001584786 

632 

399424 

252435908 

25-1396102 

8-5816809 

•001582278 

633 

4006S9 

253636137 

25-1594913 

8-5862247 

•001579779 

634 

401956 

254840104 

25-1793566 

8-5907238 

•0015772S7 

635 

403225 

256047875 

25-1992063 

8-5952380 

•001574803 

636 

404496 

257259456 

25-2190404 

8-5997476 

•001572327 

637 

405769 

258474853 

25-2388589 

8-6042525 

•001569859 

638 

407044 

259694072 

25-2586619 

8-6087526 

•001567398 

639 

408321 

260917119 

25-2784493 

8-6132480 

•001564945 

6*0 

409600' 

262144000 

25-2982213 

8-6177388 

•001562500 

641 

410881' 

263374721 

25-3179778 

8-6222248 

•001560062 

642 

412164 

264609288 

25-3377189 

8-6267063 

•001557632 

643 

413449 

265847707 

25*3574447 

8-6311830 

•001555210 

644 

414736 

207089984 

25-3771551 

8-6356551 

•001552795 

645 

416025 

268330125 

25-3968502 

8-6401226 

•001550388 

646 

417316 

269585136 

25-4165302 

8-6445855 

•001547988 

647 

418609 

270840023 

25-4361947 

8-6490437 

•001545595 

648 

419904 

272097792 

25-4558441 

8-6534974 

•001543210 

649 

421201 

273359449 

25-4754784 

8-6579465 

•001540832 

650 

422500 

274625000 

25-4950976 

8-6623911 

•001538462 

651 

423801 

275894451 

25-5147013 

8-6668310 

•001536098 

652 

425104 

277167808 

25-5342907 

8*6712065 

•001533742 

653 

426409 

27S445077 

25*5538647 

8-6756974 

•001531394 

654 

427716 

279726264 

25*5734237 

8-6801237 

•001529052 

655 

429025 

281011375 

25-5929678 

8-6845456 

•001526718 

656 

430336 

282300416 

25-6124969 

8-6889630 

•001524390 

657 

431619 

283593393 

25-6320112 

8-6933759 

•001522070 

658 

432964 

284890312 

25-6515107 

8-6977843 

*001519757 

659 

434281 

286191179 

25-6709953 

8-7021882 

•001517451 

660 

435600 

287496000 

25-6904652 

8-7065877 

•001515152 

601 

436921 

2S8804781 

25.7099203 

8-7109827 

•001512859 

602 

438244 

290117528 

25-7293607 

8-7153734 

•001510574 

603 

439569 

291434247 

25*7487864 

8-7197596 

•001508296 

604 

440890 

292754944 

25-7681975 

8-7241414 

•001506024 

665 

442225 

294079625 

25-7875939 

8-7285187 

•001503759 

660 

443556 

295408290 

25-8069758 

8-7328918 

•001501502 

667 

444889 

296740963 

25-8263431 

8-7372604 

•001499250 

608 

446224 

298077632 

25-8456960 

8-7416246 

•001497006 

609 

447501 

299418309 

25-8650343 

8-7459S46 

•001494768 

670 

448900 

300763000 

25-8843582 

8-7503401 

•001492537 

671 

450241 

1302111711 

25-9036677 

8-7546913 

•001490313 

672 

451584 

1303404448 

25-9229628 

8-7590383 

•001488095 

673 

452929 

304821217 

25-9422435 

8-7633809 

•001485884 

674 

454276 

306182024 

25-9615100 

O . rr -1 A O 

O 1 U 1 1 J. Oju 

•001483680 

675 

455625 

307546875 

25-9807621 

8-7720532 

•001481481 

676 

456976 

308915776 

26-0000000 

8-*763S30 

•001479290 





























Table of Squares, Cubes, Square and Cube Roots. 


125 


Number. 

Squares. 

Cubes. 

Roots. 

3 / - 

V Roots. 

Reciprocals. 

677 

458329 

310288733 

26-0192237 

8-7807084 

•001477105 

678 

459684 

311665752 

26-0384331 

8-7850296 

•001474926 

679 

461041 

313046839 

26-0576284 

' 8-7893466 

•001472754 

680 

462400 

314432000 

26-0768096 

8-7936593 

•001470588 

681 

463761 

315821241 

26-0959767 

8-7979679 

•001468429 

682 

465124 

317214568 

26-1151297 

8-8022721 

•001466276 

683 

466489 

318611987 

26-1342687 

8-8065722 

•001464129 

684 

467856 

320013504 

26-1533937 

8-81086S1 

•001461988 

685 

469225 

321419125 

26-1725047 

8-8151598 

•001459854 

686 

470596 

322828856 

26-1916017 

8-8194474 

•001457726 

6S7 

471969 

324242703 

26-2106848 

8-8237307 

•001455604 

6S8 

473344 

325660672 

26-2297541 

8-8280099 

•001453488 

689 

474721 

327082769 

26-2488095 

8-8322850 

•001451379 

690 

476100 

328509000 

26-2678511 

8-8365559 

•001449275 

691 

477481 

329939371 

26-2868789 

8-8408227 

•001447178 

692 

478864 

331373888 

26-3058929 

8-8450854 

•001445087 

693 

480249 

332812557 

26-3248932 

8-8493440 

•001443001 

694 

4S1636 

334255384 

26-3438797 

8-8535985 

•001440922 

695 

483025 

335702375 

26-3628527 

8-8578489 

•001438849 

696 

4S4416 

337153536 

26-3818119 

8-8620952 

•001436782 

697 

4S5809 

338608873 

26-4007576 

8-8663375 

•001434720 

698 

487204 

340068392 

26-4196896 

8-8705757 

•001432665 

699 

4SS601 

341532099 

26-4386081 

8-8748099 

•001430615 

700 

490000 

343000000 

26-4575131 

8-8790400 

•001428571 

701 

491401 

344472101 

26-4764046 

8-8832661 

•001426534 

702 

492804 

345948408 

26-4952S26 

8-8874882 

•001424501 

703 

494209 

347428927 

26-5141472 

8-8917063 

•001422475 

704 

495616 

348913664 

26-5329983 

8-8959204 

•001420455 

705 

497025 

350402625 

26-5518361 

8-9001304 

•001418440 

706 

498436 

351895816 

26-5706605 

8-9043366 

•001416431 

707 

499849 

353393243 

26-5894716 

8-9085387 

•001414427 

708 

501264 

354S94912 

26-6082694 

8-9127369 

•001412429 

709 

5026S1 

356400829 

26-6270539 

8-9169311 

•001410437 

710 

504100 

357911000 

26-6458252 

8-9211214 

•001408451 

711 

505521 

359425431 

26-6645833 

8-9253078 

•001406470 

712 

506944 

360944128 

26-6833281 

8-9294902 

•001404494 

713 

508369 

362467097 

26-7020598 

8-9336687 

•001402525 

714 

509796 

363994344 

26-7207784 

8-9378433 

•001400560 

715 

511225 

365525875 

26-7394839 

8-9420140 

•001398601 

716 

512656 

367061696 

26-7581763 

8-9461809 

•001396648 

717 

514089 

368601813 

26-7768557 

8-9503438 

•001394700 

71S 

515524 

370146232 

26-7955220 

8-9545029 

•001392758 

719 

516961 

371694959 

26-SI 41754 

8-9586581 

•001390821 

720 

518400 

373248000 

26-8328157 

8-9628095 

•0013SS889 

721 

519841 

374805361 

26-8514432 

8-9669570 

•001386963 

722 

521284 

376367048 

26-8700577 

8-9711007 

•001385042 

723 

522729 

377933067 

26-8886593 

8-9752406 

•001383126 

724 

524176 

379503424! 

26-9072481 

8-9793766 

•0013812!5 

725 

525625 

381078125 

26-9258240 

8-9835089 

•001379310 

726 

527076 

382657176 

26-9448872 

8*9876373 

•001377410 

727 

528529 

3S4240583 

26-9629375 

8-9917620 

•001375516 

728 

629984 

38582S352 

26-9814751 

8-9958899 

•001373626 




















Table of Squares, Cubes, Square and Cube Hoots. 


Number. 

Squares. 

Cubes. 

\J Roots. 

V Hoots. 

Reciprocals. 

729 

531441 

387420489 

27-0000000 

9-0000000 

001371742 

730 

532900 

389017000 

27-0185122 

9-0041134 

•001369863 

731 

534361 

390617891 

27-0370117 

9-0082229 

•001367989 

732 

535824 

392223168 

27-0554985 

9-0123288 

•001366120 

733 

537289 

393832837 

27-0739727 

9-0164309 

•001364256 

734 

538756 

395446904 

27-0924344 

9-0205293 

•001362398 

735 

540225 

397065375 

27-1108834 

9-0246239 

•001360544 

736 

541696 

398688256 

27-1293199 

9-0287149 

•001358696 

737 

543169 

400315553 

27-1477149 

9-0328021 

•001356852 

738 

544644 

401947272 

27-1661554 

9-0368857 

•001355014 

789 

546121 

403583419 

27-1845544 

9-0409655 

•001353180 

740 

547600 

405224000 

27-2029140 

9-0450419 

•001351351 

741 

549081 

406869021 

27-2213152 

9-0491142 

•001349528 

742 

550564 

408518488 

27-2396769 

9-0531S31 

•001347709 

743 

552049 

410172407 

27-2580263 

9-0572482 

•001345895 

744 

553536 

411830784 

27-2763634 

9 0613098 

•001344086 

745 

555025 

413493625 

27-2946881 

9-0653677 

•001342282 

746 

556516 

415160936 

27-3130006 

9-0694220 

•001340483 

747 

55S009 

416832723 

27-3313007 

9-0734726 

•001338688 

748 

559504 

418508992 

27-3495S87 

9-0775197 

•001336898 

749 

561001 

420189749 

27-3678644 

9-0815631 

•001335113 

750 

562500 

421875000 

27-3S61279 

9-0856030 

•001333333 

751 

564001 

423564751 

27-4043792 

9-0896352 

•001331558 

752 

565504 

425259008 

27-4226184 

9-0936719 

•001329787 

753 

567009 

426957777 

27-4408455 

9-0977010 

•00132S021 

754 

568516 

428661064 

27-4590604 

9-1017265 

•001326260 

755 

570025 

430368875 

27-4772633 

9-1057485 

•001324503 

756 

571536 

432081216 

27-4954542 

9-1097669 

•001322751 

757 

573049 

433798093 

27-5136330 

9-1137818 

•001*321004 

758 

574564 

435519512 

27-5317998 

9-1177931 

•001319261 

759 

5760S1 

437245479 

27-5499546 

9-1218010 

•001317523 

760 

577600 

438976000 

27-5680975 

9-1258053 

•001315789 

761 

579121 

440711081 

27-5862284 

9-1298061 

•001314060 

762 

580644 

442450728 

27-6043475 

9-1338034 

•001312336 

763 

582169 

444194947 

27-6224546 

9-1377971 

•001310616 

764 

583696 

445943744 

27-6405499 

9-1417874 

•001308901 

765 

585225 

447697125 

27-6586334 

9-1457742 

•001307190 

766 

586756 

449455096 

27-6767050 

9-1497576 

•001305483 

767 

588289 

451217663 

27-6947648 

9-1537375 

•001303781 

768 

589S24 

452984832 

27-7128129 

9-1577139 

•001302083 

769 

591361 

454756609 

27-7308492 

9-1616869 

•001300390 

770 

592900 

456533000 

27-7488739 

9-1656565 

•001298701 

771 

594441 

458314011 

27-7668868 

9-1696225 

•001297017 

772 

595984 

460099648 

27-7S48880 

9-1735852 

•001295337 

773 

597529 

461889917 

27-8028775 

9-1775445 

•001293661 

774 

599076 

463684824 

27-8208555 

9-1815003 

•001291990 

775 

600625 

465484375 

27-8388218 

9-1854527 

•001290323 

776 

602176 

467288576 

27-8567766 

9-1894018 

•001288660 

777 

603729 

469097433 

27-8747197 

9-1933474 

001287001 

778 

605284 

470910952 

27-8926514 

9-1972897 

•001285347 

779 

606841 

472729139 

27-9105715 

9-2012286 

•001283697 

780 

6^8400 

474552000 

27-9284801 

9-2051641 

•0012S2051 





























Tabije op Squares, Cubes, Square and Cube Roots. 


127 


Number. 

Squares. 

Cubes. 

Roots. 

$/ Roots. 

Reciprocals. 

781 

609961 

476379541 

27*9463772 

9-2090962 

•001280410 

782 

611524 

47S21176S 

27-9642629 

9-2130250 

•001278772 

783 

613089 

4800486S7 

27-9821372 

9-2169505 

•001277139 

784 

614656 

481890304 

28-0000000 

9-2208726 

•001275510 

785 

616225 

483736625 

28-0178515 

9-2247914 

•001273885 

786 

617796 

485587656 

28-0356915 

9-2287068 

•001272265 

787 

619369 

487443403 

28-0535203 

9-2326189 

•001270648 

788 

620944 

4S9303872 

28-0713377 

9-2365277 

•001269036 

789 

622521 

491169069 

28-0891438 

9-2404333 

•001267427 

790 

624100 

493039000 

28-1069386 

9-2443355 

•001265823 

791 

625681 

1 494913671 

28-1247222 

9-2482344 

•001264223 

792 

627264 

496793088 

28-1424946 

9-2521300 

•001262626 

793 

628849 

498677257 

28-1602557 

9-2560224 

•001261034 

794 

630436 

500566184 

28-1780056 

9-2599114 

•001259446 

795 

632025 

502459875 

28-1957444 

9-2637973 

•001257862 

796 

633616 

504358336 

28-2134720 

9-2676798 

•00125G2S1 

797 

635209 

506261573 

28-2311884 

9-2715592 

•00 1 254705 

798 

636804 

508169592 

28-2488938 

9-2754352 

•001253133 

799 

638401 

510082399 

28-2665881 

9-2793081 

•001251564 

800 

640000 

512000000 

28-2842712 

9-2831777 

•001250000 

801 

641601 

513922401 

28-3019434 

9-2870444 

•001248439 

802 

643204 

515849608 

28-3196045 

9-2909072 

•001246883 

803 

644809 

517781627 

28-3372546 

9-2947671 

•001245330 

804 

646416 

519718464 

28-3548938 

9-2986239 

•0012437S1 

805 

648025 

521060125 

28-3725219 

9-3024775 

•001242236 

806 

649636 

523606616 

28-3901391 

9-3063278 

•001240695 

807 

651249 

525557943 

28-4077454 

9-3101750 

•001239157 

808 

652864 

527514112 

28-4253408 

9-3140190 

•001237624 

809 

654481 

529475129 

2S-4429253 

9-3178599 

•001236094 

810 

656100 

531441000 

28-4604989 

9-3216975 

•001234568 

811 

657721 

533411731 

28-4780617 

9-3255320 

•001233046 

812 

659344 

535387328 

28-4956137 

9-3293634 

•001231527 

813 

660969 

537367797 

28-5131549 

9-3331916 

•001230012 

814 

662596 

539353144 

28-5306852 

9-3370167 

•001228501 

815 

664225 

541343375 

28-5482048 

9-3408386 

•001226994 

816 

665856 

543338496 

28-5657137 

9-3446575 

•001225499 

817 

667489 

545338513 

28-5832119 

9-3484731 

•001223990 

818 

669124 

547343432 

28-6006993 

9-3522857 

•001222494 

819 

670761 

549353259 

28-6181760 

9-3560952 

•001221001 

820 

672400 

551368000 

28-6356421 

9-3599016 

•001219512 

821 

6740411 

553387661 

28-6530976 

9-3637049 

•001218027 

822 

675684 

555412248 

28-6705424 

9-3675051 

•001216545 

823 

677329 

557441767 

28-6879716 

9-3713022 

•001215067 

824 

678976 

559476224 

28-7054002 

9-3750963 

•001213592 

825 

680625 

561515625 

28-7228132 

9-3788S73 

•001212121 

826 

682276 

563559976 

28-7402157 

9-3826752 

•001210654 

827 

683929 

565609283 

28-7576077 

9-3864600 

•001209190 

82S 

685584 

567663552 

28-7749891 

9-3902419 

•001207729 

829 

687241 

569722789 

28-7923601 

9-3940206 

•001206273 

830 

688900 

571787000 

28-8097206 

9-3977964 

•001204819 

831 

690561 

573856191 

28-8270706 

9-4015691 

•001203369 

832 

692224 

575930368 

28-8444102 

9-4053387 

•001201923 



































128 


T/uitK OF Squares, Cubes, Square and Cube Roots. 


Number. 

Squares. 

Cubes. 

\/ Roots. | 

V Hoots. 

Reciprocals 

833 

693889 

578009537 

28-8617394 

9-4091054 

*001200480 

834 

695556 

580U93704 

28-8790582 

9-4128690 

*001199041 

835 

697225 

5S2182875 

28*8963666 

9*4166297 

*001197605 

836 

693896 

584277056 

28*9136646 

9*4203873 

•001196172 

837 

700569 

586376253 

28-9309523 

9-4241420 

*001194743 

838 

702244 

5S8480472 

28-9482297 

9-4278936 

*001193317 

839 

703921 

590589719 

28-9654967 

9*4316423 

*001191895 

840 

705600 

592704000 

28-9827535 

9-4353800 

*001190476 

841 

707281 

594823321 

29-0000000 

9-4391307 

*001189061 

842 

708964 

5969476S8 

29*0172363 

9-442S704 

*001187648 

843 

710649 

599077107 

29*0344623 

9-4466072 

*001186240 

844 

712336 

601211584 

29*0516781 

9-4503410 

*001184834 

845 

714025 

603351125 

29-0688837 

9-4540719 

*001183432 

846 

715716 

605495736 

29-0860791 

9*4577999 

*001182033 

847 

717409 

607645423 

29-1032644 

9*4615249 

*001180638 

848 

719104 

609800192 

29-1204396 

9-4652470 

•001179245 

849 

720801 

611960049 

29-1376046 

9-4689061 

•001177856 

850 

722500 

614125000 

29-1547595 

9-4726824 

• *001176471 

851 

724201 

616295051 

29*1719043 

9*4763957 

•001175088 

852 

725904 

618470208 

29-1890390 

9-4801061 

•001173709 

853 

727609 

620650477 

29-2061637 

9*4838136 

•001172333 

854 

729316 

622835864 

29-22327S4 

9-4875182 

•001170960 

855 

731025 

625026375 

29-2403830 

9-4912200 

•001169591 

856 

732736 

627222016 

29-2574777 

9-4949188 

•001168224 

857 

734449 

629422793 

29-2745623 

9*4986147 

•001166861 

858 

736164 

631628712 

29-2916370 

9-5023078 

•001165501 

859 

737881 

633839779 

29-3087018 

9-5059980 

•001164144 

860 

739600 

636056000 

29-3257566 

9-5096854 

•001162791 

861 

741321 

63S277381 

29-3428015 

9-5133699 

•001161440 

862 

743044 

640503928 

29-3598365 

9-5170515 

•001160093 

863 

744769 

642735647 

29-3768616 

9-5207303 

•001158749 

864 

746496 

644972544 

29-3938769 

9*5244063 

•001157407 

865 

748225 

647214625 

29-4108823 

9-5280794 

•001156069 

866 

749956 

649461896 

29-4278779 

9-5317497 

•001154734 

867 

751689 

651714363 

29-4448637 

9-5354172 

•001153403 

868 

753424 

653972032 

29-4618397 

9-5390818 

•001152074 

869 

755161 

656234909 

29-4788059 

9-5427437 

•001150748 

870 

756900 

658503000 

29-4957624 

9-5464027 

•001149425 

871 

758641 

660776311 

29-5127091 

9-5500589 

•001148106 

872 

760384 

663054848 

29-5296461 

9-5537123 

•001146789 

873 

762129 

665338617 

29*5465734 

9-5573630 

•001145475 

874 

763876 

667627624 

29*5634910 

9-5610108 

•001144165 

875 

765625 

669921875 

29-5803989 

9"5646559 

•001142857 

876 

767376 

672221376 

29-5972972 

9-5682782 

•001141553 

877 

769129 

674526133 

29-6141858 

9-5719377 

•001140251 

878 

7708S4 

1 676836152 

29-6310648 

9-5755745 

•001138952 

879 

772641 

679151439 

29-6479342 

9-5792085 

•001137656 

880 

774400 

1681472000 

29-6647939 

9-5828397 

•001136364 

881 

776161 

683797841 

29-6816442 

9-5864682 

•001135074 

882 

777924 

686128968 

29*6984848 

9-5900937 

•001133787 

883 

779689 

688465387 

29-7153159 

9-5937169 

•001132503 

884 

781456 

690807104 

29-7321375 

9-5973373 

*001131222 


I.. 



























Table of Squares, Cubes, Square and Cube Roots. 129 


Number. 

Squares. 

Cubes. 

>/ Roots. 

4/ Roots. 

Reciprocals. 

8S5 

783225 

693154125 

29*7489496 

9*6009548 

*001129944 

886 

784996 

695506456 

29*7657521 

9*6045696 

*001128668 

887 

786769 

697864103 

29*7825452 

9*6081817 

*001127396 

8S8 

7S8544 

700227072 

29*7993289 

9*6117911 

*001126126 

889 

790321 

702595369 

29*8161030 

9*6153977 

*001124859 

89(1 

792100 

704969000 

29*8328678 

9*6190017 

*001123596 

89 : 

793881 

707347971 

29*8496231 

9*6226030 

*001122334 

892 

795664 

707932288 

29*8663690 

9*6262016 

*001121076 

893 

797449 

712121957 

29*8831056 

9*6297975 

*001119821 

894 

799236 

714510984 

29*8998328 

9*6333907 

•001118568 

895 

801025 

716917375 

29*9165506 

9*6369812 

•001117818 

896 

802816 

719323136 

29*9332591 

9*6405690 

•001116071 

897 

804609 

721734273 

29*9499583 

9*6441542 

•001114827 

89S 

806404 

724150792 

29*9666481 

9*6477367 

•001113586 

899 

808201 

726572699 

29*9833287 

9*6513166 

•001112347 

900 

810000 

729000000 

30*0000000 

9*6548938 

•001111111 

901 

811801 

731432701 

30*0166621 

9*6584684 

•001109878 

902 

813604 

733S70S08 

30*0333148 

9*6620403 

•001108647 

903 

815409 

736314327 

30*0499584 

9*6656096 

•001107420 

904 

817216 

738763264 

30*0665928 

9*6691762 

•001106195 

905 

819025 

741217625 

30*0832179 

9*6727403 

•001104972 

906 

820S36 

743677416 

30*0998339 

9*6763017 

•001103753 

907 

822649 

746142643 

30*1164407 

9*6798604 

•001102536 

908 

824464 

748613312 

20*1330383 

9*6834166 

•001101322 

909 

826281 

751089429 

30*1496269 

9*6869701 

•001100110 

910 

828100 

753571000 

30*1662063 

9*6905211 

•001098901 

911 

829921 

75605S031 

30*1827765 

9*6940694 

•001097695 

912 

831744 

758550828 

30*1993377 

9*6976151 

•001096491/ 

913 

833569 

761048497 

30-2158S99 

9*7011583 

•001095290' 

914 

835396 

763551944 

30*2324329 

9*7046989 

•0010940921 

915 

837225 

760060875 

30*2489669 

9*7082369 

•001092896 

916 

839056 

768575296 

30*2654919 

9*7117723 

•001091703 

917 

840889 

771095213 

30*2820079 

9*7153051 

•001090513 

918 

842724 

773620632 

30*2985148 

9*7188354 

•001089325 

919 

844561 

776151559 

30*3150128 

9*7223631 

•001088139 

920 

846400 

778688000 

30*3315018 

9*7258883 

•001086957 

921 

848241 

781229961 

30*3479818 

9.7294109 

•001085776 

922 

850084 

783777448 

30*3644529 

9*7329309 

•001084599 

923 

851929 

786330467 

30*3809151 

9*7364484 

•001083423 

924 

853776 

788889024 

30*3973683 

9*7399634 

•001082251 

925 

855625 

791453125 

30*4138127 

9*7434758 

•001081081 

92-6 

857476 

794022776 

30*4302481 

9*7469857 

•001079914 

92? 

859329 

796597983 

30*4466747 

9*7504930 

•001078749 

928 

861184 

79917S752 

30*4630924 

9*7539979 

•001077586 

929 

863041 

8017650S9 

30*4795013 

9*7575002 

•001076426 

930 

864900 

8043-57000 

30*4959014 

9*7610001 

•001075269 

931 

866761 

806954491 

30*5122926 

9*7644974 

•001074114 

93? 

868624 

809557568 

30*5286750 

9*7679922 

•001072961 

933 

870489 

812166237 

30*5450487 

9*7714845 

•001071811 

934 

872356 

814780504 

30*5614136 

9*7749743 

•001070664 

935 

874225 

8174.00.375 

30*5777697 

9*7784616 

•001069519 

936 

876096 

820025856 

30*5941171 

9*7819466 

•001068376 


9 

























130 

Table op Squares, 

Cubes, Square and Cube Roe-re, 


Number. 

Squares. 

Cubes. 

vRoots. 

^ Roots. 

Reciprocals. 

937 

877969 

S22656953 

30-6104557 

9-7854288 

•001067236 

938 

879844 

825293672 

30-6267857 

9-7889087 

•001066098 

939 

881721 

827936019 

30-6431069 

9-7923861 

•001064963 

940 

883600 

830584000 

30-6594194 

9-7958611 

•001063830 

941 

885481 

833237621 

30-6757233 

9-7993336 

•001062699 

942 

887364 

835896888 

30-6920185 

9-8028036 

•001061571 

943 

889249 

838561807 

30-7083051 

9-8062711 

•001060445 

944 

891136 

8412323S4 

30-7245830 

9-8097362 

•001059322 

945 

893025 

843908625 

30-7408523 

9-8131989 

•001058201 

946 

894916 

846590536 

30-7571130 

9-8166591 

•001057082 

947 

896809 

849278123 

30-7733651 

9-8201169 

•001055966 

948 

898704 

851971392 

30-7896086 

9-8235723 

•001054852 

949 

900601 

854670349 

30-8058436 

9-8270252 

•001053741 

950 

902500 

857375000 

30-8220700 

9-8304757 

•001052632 

951 

904401 

860085351 

30-8382879 

9-8339238 

•001051525 

952 

906304 

862801408 

30-8544972 

9-8373695 

•001050420 

953 

908209 

865523177 

30-8706981 

9-8408127 

•001049318 

954 

910116 

868250664 

30-8868904 

9-8442536 

•001048218 

955 

912025 

870983875 

30-9030743 

9-8476920 

•001047120 

956 

913936 

873722816 

30-9192477 

9-8511280 

•001046025 

957 

915849 

876467493 

30-9354166 

9-8545617 

•001044932 

958 

917764 

879217912 

30-9515751 

9-8579929 

•001043841 

959 

919681 

881974079 

30-9677251 

9-8614218 

•001042753 

960 

921600 

884736000 

30-9838668 

9-8648483 

•001041667 

961 

923521 

887503681 

31-0000000 

9-8682724 

•001040583 

962 

925444 

890277128 

31-0161248 

9-8716941 

•001039501 

963 

927369 

893056347 

31-0322413 

9-8751135 

•001038422 

964 

929296 

895841344 

31-0483494 

9-8785305 

•001037344 

965 

931225 

898632125 

31-0644491 

9-8819451 

•001036269 

966 

933156 

901428696 

31-0805405 

9-8853574 

•001035197 

967 

935089 

904231063 

31-0966236 

9-8887673 

•001034126 

968 

937024 

907039232 

31-1126984 

9-8921749 

•001033058 

969 

938961 

909853209 

31-1287648 

9-8955801 

•001031992 

970 

940900 

912673000 

31-1448230 

9-8989830 

•001030928 

971 

942841 

'915498611 

31-1608729 

9-9023835 

•001029866 

972 

944784 

918330048 

31-1769145 

9-9057817 

•001028807 

973 

946729 

921167317 

31-1929479 

9-9091776 

•001027749 

974 

948676 

924010424 

31-2089731 

9-9125712 

•001026694 

975 

950625 

926859375 

31-2249900 

9-9159624 

•001025641 

976 

952576 

,929714176 

31-2409987 

9-9193513 

•001024590 

977 

954529 

932574833 

31-2569992 

9-9227379 

•001023541 

978 

956484 

935441352 

31-2729915 

9-9261222 

•001022495 

979 

958441 

938313739 

31-2889757 

9-9295042 

•001021450 

980 

960400 

941192000 

31-3049517 

9-9328839 

•001020408 

9S1 

962361 

944076141 

31-3209195 

9-9362613 

•001019168 

982 

964324 

94696616S 

31-3368792 

9-9396363 

•001018330 

983 

966289 

949862087 

31-3528308 

9-9430092 

•001017294 

984 

968256 

952763904 

31-3687743 

9-9463797 

•001016260 

985 

970225,955671625 

31-3847097 

9-9497479 

•001015228 

986 

972196 

958585256 

31-4006369 

9-9531138 

•001014199 

987 

974169 

961504803 

31-4165561 

9-9564775 

•001013171 

988 

976144 

,964430272 

31-4324673 

9-9598389 

1 U01012146 





























Tabi* of Squares, Cubes, Square and cube koots. 


131 




1 



Reciprocals. 

Number. 

Squares. 

Cubes. 

j \/ Hoots. 

S/ Roots. 

9S9 

978121 

967361669 

31-4483704 

9-9631981 

•001011122 

990 

980100 

970299000 

31-4642654 

9-9665549 

•001010101 

991 

982081 

973242271 

31-4801525 

9-9699055 

•001009082 

992 

984064 

976191488 

31-4960315 

9-9732619 

•001008065 

993 

9S6049 

979146657 

31-5119025 

9-9766120 

•001007049 

994 

988036 

982107784 

31-5277655 

9-9799599 

•001006036 

995 

990025 

985074875 

31-5436206 

9-9S33055 

•001005025 

996 

992016 

988047936 

31-5594677 

9-9866488 

•001004016 

997 

994009 

991026973 

31-5753068 

9-9899900 

•001003009 

998 

996004 

994011992 

31-5911380 

9-9933289 

•001002004 

999 

998001 

997002999 

31-6069613 

J9-9966656 

•001001001 

1000 

1000000 

1000000000 

31-6227766 

10-0000000 

•001000000 

1001 

1002001 

1003003001 

31-6385840 

10-0033222 

•0009990010 

1002 

1004004 

1006012008 

31-6543866 

10-0066622 

•0009980040 

1003 

1006009 

1009027027 

31-6701752 

10-0099899 

•0009970090 

1004 

1008016 

1012048064 

31-6859590 

10 0133155 

•0009960159 

1005 

1010025 

1015075125 

31-7017349 

10-0166389 

•0009950249 

1006 

1012036 

1018108216 

31-7175030 

10-0199601' 

•0009940358 

1007 

1014049 

1021147343 

31-7332633 

10-0232791 

•0009930487 

1008 

1016064 

1024192512 

31-7490157 

10-0265958 

•0009920635 

1009 

1018081 

1027243729 

31-7647603 

10-0299104 

•0009910803 

1010 

1020100 

1030301000 

31-7804972 

10-0332228 

•0009900990 

1011 

1022121 

1033364331 

31-7962262 

10-0365330 

•0009891197 

1012 

1024144 

1036433728 

31-8119474 

10-0398410 

•0009881423 

1013 

1026169 

1039509197 

31-8276609 

10-0431469 

-0009871668 

1014 

1028196 

1042590744 

31-8433666 

10-0464506 

•0009861933 

1015 

1030225 

1045678375 

31-8590646 

10-0497521 

•0009852217 

1016 

1032256 

1048772096 

31-8747549 

10-0530514 

•0009842520 

1017 

1034289 

1051871913 

31-8904374 

10-0563485 

•0009832842 

1018 

1036324 

1054977832 

31-9061123 

10-0596435 

•0009823183 

1019 

1038361 

1058089859 

31-9217794 

10-0629364 

•0009813543 

1020 

1040400 

1061208000 

31-9374388 

10-0662271 

•0009803922 

1021 

1042441 

1064332261 

31-9530906 

10-0695156 

•0009794319 

1022 

1044484 

1067462648 

31-9687347 

10-0728020 

•0009784736 

1023 

1046529 

1070599167 

31-9843712 

10-0760863 

•0009775171 

1024 

1048576 

1073741824 

32-0000000 

10-0793684 

•0009765625 

1025 

1050625 

1076890625 

32-0156212 

10-0826484 

•0009756098 

1026 

1052676 

1080045576 

32-0312348 

10-0.859262 

•0009746589 

1027 

1054729 

1083206683 

32-0468407 

10-0892019 

•0009737098 

1028 

1056784 

10S6373952 

32-0624391 

10-0924755 

•0009727626 

1029 

1058841 

1089547389 

32-0780298 

10-0957469 

•000971S173 

1030 

1060900 

1092727000 

32-0936131 

10-0990163 

•0009708738 

1031 

1062961 

1095912791 

32-1091887 

10-1022835 

•0009699321 

1032 

1065024 

1099104768 

32-1247568 

10-1055487 

•0009689922 

1033 

10670S9 

1102302937 

32-1403173 

10-1088117 

•0009680542 

1034 

1069156 

1105507304 

32-1558704 

10-1120726 

•0009671180 

1035 

1071225 

1108717875 

32-1714159 

10-1153314 

•0009661836 

1036 

1073296 

1111934656 

32-1869539 

10-1185882 

•0009652510 

1037 

1075369 

1115157653 

32-2024844 

10*1218428 

•0009643202 

1038 

1077444 

1118386872 

32-2180074 

10-1250953 

•0009633911 

1039 

1079521 

1121622319 

32-2335229 

10-1283457 

•0009624639 

1040 

■ .- 

1081600 

1124864000 

32-2490310 

10-1315941 

•0009615385 
























132 Table of Squares, Cubes, Square and Cube Roots. 


Number. 

Squares. 

Cubes. 

y/ ROOtS. 

%/ Roots. 

Reciprocals. 

1041 

1083681 

1128111921 

32-2645316 

10-1348403 

•0009606148 

1042 

1085764 

1131366088 

32-2800248 

10-1380845 

•0009596929 

1043 

1087849 

1134626507 

32-2955105 

10-1413266 

•0009587728 

1044 

10S9936 

1137893184 

32-3109888 

10-1445667 

•0009578544 

1045 

1092025 

1141166125 

32-3264598 

10-1478047 

•0009569378 

1046 

1094116 

1144445336 

32-3419233 

10-1510406 

•0009560229 

1047 

1096209 

1147730823 

32-3573794 

10-1542744 

•0009551098 

1048 

1098304 

1151022592 

32-3728281 

10-1575062 

*0009541985 

1049 

1100401 

1154320649 

32-3882695 

10-1607359 

*0009532888 

1050 

1102500 

1157625000 

32-4037035 

10-1639636 

•0009523810 

1051 

1104601 

1160935651 

32-4191301 

10-1671893 

*0009514748 

1052 

1106704 

1164252608 

32-4345495 

10-1704129 

•0009505703 

1053 

1108809 

1167575877 

32-4499615 

10-1736344 

•0009496676 

1054 

1110916 

1170905464 

32-4653662 

10-1768539 

*0009487666 

1055 

1113025 

1174241315 

32-4807635 

10-1800714 

*0009478673 

1056 

1115136 

1177583616 

32-4961536 

10-1S32868 

*0009469697 

1057 

1117249 

1180932193 

32-5115364 

10-1865002 

•0009460738 

1058 

1119364 

1184287112 

32-5269119 

10-1897116 

•0009451796 

1059 

1121481 

1187648379 

32-5422802 

10-1929209 

•0009442871 

1060 

1123600 

1191016000 

32-5576412 

10-1961283 

*0009433902 

1061 

1125721 

1194389981 

32-5729949 

10-1993336 

*0009425071 

1002 

1127844 

1197770328 

32-5883415 

10-2025369 

'0009416196 

1063 

1129969 

1201157047 

32-6035807 

10-2057382 

*0009407338 

1064 

1132096 

1204550144 

32-6190129 

10-2089375 

•0009398496 

1065 

1134225 

1207949625 

32-6343377 

10-2121347 

•0009389671 

1066 

1136356 

1211355496 

32-6496554 

10-2153300 

*0009380863 

1067 

1138489 

1214767763 

32-6649659 

10-2185233 

•0009372071 

1068 

•1140624 

1218186432 

32-6802693 

10-2217146 

•0009363296 

1069 

1142761 

1221611509 

32-6955654 

10-2249039 

*0009354537 

1070 

1144900 

1225043000 

32-7108544 

10-2280912 

*0009345794 

1071 

1147041 

1228480911 

32-7201363 

10-2312766 

•0009337068 

1072 

1149184 

1231925248 

32-7414111 

10-2344599 

*0009328358 

1073 

1151329 

1235376017 

32-7566787 

10-2376413 

•0009319664 

1074 

1153476 

1238833224 

32-7719392 

10-2408207 

*00093109S7 

1075 

1155625 

1242296875 

32-7871920 

10-2439981 

*0009302326 

1076 

1157776 

1245766976 

32-8024398 

10-2471735 

*0009293680 

1077 

1159929 

1249243533 

32-8176782 

10-2503470 

•0009285051 

1078 

1162084 

1252726552 

32-8329103 

10-2535186 

•0009276438 

1079 

1164241 

1256216039 

32-8481354 

10-2566881 

•0009267841 

1080 

1166400 

1259712000 

32-8633535 

10-2598557 

•0009259259 

1081 

1168561 

1263214441 

32-8785644 

10-2630213 

•0009250694 

10 S2 

1170724 

1266723368 

32-8937684 

10-2661850 

•0009242144 

1083 

1172889 

12702387S7 

32-9089653 

10-2693467 

•0009233610 

1084 

1175056 

1273760704 

32-9241553 

10-2725065 

*0009225092 

1085 

1177225 

1277289125 

32-9393382 

10-2756644 

*0009216590 

1086 

1179396 

1280824056 

32-9545141 

10-2788203 

•0009208103 

1087 

1181569 

1284365503 

32-9696830 

10-2819743 

*0009199632 

1088 

1183744 

1287913472 

32-9848450 

10-2851264 

•0009191176 

1089 

1185921 

1291467969 

33-0000000 

10-2882765 

*0009182736 

1090 

1188100 

1295029000 

33-0151480 

10-2914247 

•0009174312 

1 091 

1190281 

1298596571 

33-0302891 

10-2945709 

•0009165903 

1092 

L192404 

1302170688 

33-0454233 

i 10-2977153 

*0009157509 



























Table of Squares, Cubes, Square and Cube Roots. 133 


Number. 

Squares. 

Cubes. 

V^Roots. 

Roots. 

Reciprocals. 

1093 

1191649 

1305751357 

33-0605505 

10-3008577 

•0009149131 

1094 

1196S36 

130933S584 

33-0756708 

10-3039982 

•0009140768 

1 095 

1199025 

1312932375 

33-0907842 

10-3071368 

•0009132420 

1096 

1201216 

1316532736 

33-1058907 

10-3102735 

•0009124008 

i09r 

1203409 

1320139673 

33-1209903 

10-3134083 

•0009115770 

1098 

1205604 

1323753192 

33-1360830 

10-3165411 

•0009107468 

1099 

1207801 

1327373299 

33-1511689 

10-3196721 

•0009099181 

1100 

1210000 

1331000000 

33-1662479 

10-322S012 

•0009090909 

1101 

1212201 

1334633301 

33-1813200 

10-3259284 

•0009082652 

1102 

1214404 

1338273208 

33-1963853 

10-3290537 

•0009074410 

1103 

1216609 

1341919727 

33-2114438 

10-3321770 

•0009066183 

1104 

1218816 

1345572864 

33-2266955 

10-3352985 

•0009057971 

1105 

1221025 

1349232625 

33-2415403 

10-3384181 

•0009049774 

1106 

1223236 

1352899016 

33-2565783 

10-3415358 

•0009041591 

1107 

1225449 

1356572043 

33-2716095 

10-3446517 

•0009033424 

1108 

1227664 

1360251712 

33-2866339 

10-3477657 

•0009025271 

1109 

12298S1 

1363938029 

33-3016516 

10-3508778 

•0009017133 

1110 

1232100 

1367631000 

33-3166625 

10-3539880 

•0009009009 

1111 

1234321 

1371330631 

33-3316666 

10-3570964 

•0009000900 

1112 

1236544 

1375036928 

33-3466640 

10-3602029 

•0008992S06 

1113 

1238769 

137S749897 

33-3616546 

10-3633076 

•0008984726 

1114 

1240996 

1382469544 

33-3766385 

10-3664103 

•000S976661 

1115 

1243225 

1386195875 

33-3916157 

10-3695113 

•0008968610 

1116 

1245456 

1389928896 

33*4065862 

10-3726103 

•0008960753 

1117 

1247689 

1393668613 

33-4215499 

10-3757076 

•0008952551 

1118 

1249924 

1397415032 

33-4365070 

10-3738030 

•0008944544 

1119 

1252161 

1401168159 

33-4514573 

10-3818965 

•0008936550 

1120 

1254400 

1404928000 

33-4664011 

10-3849882 

•000S928571 

1121 

1256641 

1408694561 

33-4813381 

10-3880781 

•0008960607 

1122 

1258884 

1412467848 

33-4962684 

10-3911661 

•0008922656 

1123 

1261129 

1416247867 

33-5111921 

10-3942527 

•OO0S9O472O 

1124 

1263376 

1420034624 

33-5261092 

10-3973366 

•0008S96797 

1125 

1265625 

1423S28125 

33-5410196 

10-4004192 

•0008888889 

1126 

1267876 

1427628376 

33-5559234 

10-4034999 

•0008S80995 

1127 

1270129 

1431435383 

33-5708206 

1 0-4065787 

•0008873114 

1128 

1272381 

1435249152 

33-5857112 

10-4096557 

•0008865248 

1129 

1274641 

1439069689 

33-6005952 

10-4127310 

•0008857396 

1130 

1276900 

1442897000 

33-6154726 

10-4158044 

•0008849558 

1131 

1279161 

1446731091 

33-6303434 

10-4188760 

•0008841733 

1132 

1281424 

1450571968 

33-6452077 

10-4219458 

•0008833922 

1133 

1283689 

1454419637 

33-6600653 

10-4250138 

•0008826125 

1134 

1285956 

1458274104 

33-6749165 

10-4280800 

•0008818342 

1135 

1288225 

1462135375 

33-6897610 

10-4311443 

•000S810573 

1136 

1290496 

1466003456 

83-7045991 

10-4342069 

•0008S02817 

1137 

1292769 

1469878353 

33*7174306 

10-4372677 

•0008795075 

1138 

1295044 

1473760072 

33-7340556 

10-4403677 

•000S787346 

1139 

1297321 

1477648619 

33-7490741 

10-4433839 

•0008779631 

1140 

1299600 

1481544000 

33-7638860 

10-4464393 

•0008771930 

1141 

1301881 

1485446221 

33-7786915 

10-4494929 

•0008764242 

1142 

1304164 

1489355288 

33-7934905 

10-4525448 

•0008756567 

1143 

1306449 

1493271207 

33-8082830 

10-4555948 

•0008748906 

1144 

1308736 

1497193984 

33-8230691 

10-4586431 

•0008741259 


































134 Table op Squares, Cubes, Square axd Cube Roots. 


s- 

Number 

Squares. 

Cubes. 

V Roots. 

•s Roots. 

Reciprocals. 

1145 

1311025 

1501123625 

33-8378486 

10-4616896 

•0008733624 

1146 

1313316 

1505060136 

33-8526218 

10-4647343 

•0008726003 

1147 

1315609 

1509003523 

33-8673SS4 

10-4677773 

•000871S396 

1148 

1317904 

1512953792 

33-S821487 

10-4708158 

•0008710S01 

1149 

1320201 

1516910949 

33-8969025 

10-4738579 

•0008703220 

1150 

1322500 

1520875000 

33-9116499 

10-4768955 

•0008695652 

1151 

1324801 

1524845951 

33*9263909 

10-4799314 

•0008688097 

1152 

1327104 

1528823808 

33*9411255 

10-4829656 

•0008680556 

1153 

1329409 

1532808577 

33-9558537 

10-4859980 

-0008673027 

1154 

1331716 

1536800264 

33-9705755 

10-4890286 

•0008665511 

1155 

1334025 

1540798875 

33-9S52910 

10-4920575 

•0008658009 

1156 

1336336 

1544804416 

34-0000000 

10-4950847 

•0008650519 

1157 

1338649 

1548816893 

34-0147027 

10-4981101 

•0008643042 

1158 

134096 1 

1552836312 

34*0293990 

10-5011337 

•0008635579 

1159 

1343281 

1556862679 

34-0440890 

10-5041556 

•0008628128 

1160 

1345600 

1560896000 

34*0587727 

10-5071757 

•0008620690 

1161 

1347921 

1564936281 

34-0734501 

10-5101942 

•0008613264 

1162 

1350244 

1568983528 

34-0881211 

10-5132109 

•0008605852 

1163 

1352569 

1573037747 

34-0127858 

10-5162259 

•000S598452 

1164 

1354896 

1577098944 

34-1174442 

10-5192391 

•0008591065 

1165 

1357225 

1581167125 

34-1320963 

10-5222506 

•0008583691 

1166 

1359556 

1585242296 

34-1467422 

10-5252604 

•0008576329 

1167 

1361889 

1589324463 

34-1613817 

10-5282685 

•0008568980 

1168 

1364224 

1593413632 

34-1760150 

10-5312749 

•0008561644 

1169 

1366561 

1597509809 

34-1906420 

10-5342795 

•000S554320 

1170 

1368900 

1601613000 

34-2052627 

10-5372825 

•0008547009 

1171 

1371241 

1605723211 

34-2198773 

10-5402S37 

•0008539710 

1172 

1373584 

1609840448 

34-2344855 

10-5432832 

•0008532423 

1173 

1375929 

1613961717 

34-2490875 

10-5462810 

•0008525149 

1174 

1378276 

1618096024 

34*2636834 

10-5492771 

•0008517888 

1175 

1380625 

1622234375 

34-2782730 

10-5522715 

■0008510638 

J176 

1382976 

1626.379776 

34*2928564 

10-5552642 

•0008503401 

1177 

1385329 

1630532233 

34-3074336 

10-5582552 

•0008496177 

1178 

13S7684 

1634691752 

34-3220046 

10-5612445 

•0008488964 

1179 

1390041 

1638858339 

34-3365694 

10-5642322 

•0008481764 

1180 

1392400 

1643032000 

34-3511281 

10-5672181 

•0008471576 

1181 

1394761 

1647212741 

34-3656805 

10-5702024 

•000S467401 

1182 

1397124 

1651400568 

34-3802268 

10-5731849 

•0008460237 

1183 

1399489 

1655595487 

34-3947670 

10-5761658 

•0008453085 

1184 

1401856 

1659797504 

34-4093011 

10-5791449 

•0068445946 

1185 

1404225 

1664006625 

34-4238289 

10-5821225 

•0008438819 

1186 

1406596 

1668222856 

34-4383507 

10-5850983 

•0008431703 

1187 

1408969 

1672446203 

34-4528663 

10-5880725 

•0008424600 

1188 

1411344 

1676670672 

34-4673759 

10-5910450 

•0008417508 

11S9 

1413721 

1680914629 

34-4818793 

10-5940158 

•0008410429 

1190 

1416100 

1685159000 

34-4963766 

10-5969850 

•000S403361 

1191 

1418481 

1689410871 

34-5108678 

10-5999525 

•0008396306 

1192 

1420864 

1693669888 

34-5253530 

10-60291S4 

•0008389262 

1193 

1423249 

1697936057 

34-5398321 

10-6058826 

•0008382320 

1194 

1425636 

1702209384 

34-5543051 

10-6088451 

•0008375209 

1195 

1428025 

17064S9875 

34-5687720 

10-6118060 

•0008368201 

1196 1 

1430416 

1710777536 

34-5832329 

10-6147652 

•0008361204 


V. 






























Table of Squares, Cubes, Square and Cube Hoots. 


135 


Numter 

Squares. 

Cubes. 

i \/' Roots. 

\/ Roots. 

1197 

1432809 

1715072373 

34*5976879 

10-6177228 

1198 

1435204 

1719374392 

34-6121366 

10*6206788 

1199 

1437601 

1723683599 

34-6265794 

10-6236331 

1200 

1440000 

1728000000 

34*6410162 

10*6265857 

1201 

1442401 

1732323601 

34-6554469 

10-6295367 

1202 

1444804 

1736654408 

34-6698716 

10-6324860 

1203 

1447209 

1740992427 

34-6842904 

10-6354338 

1204 

1449616 

1745337664 

34-6987031 

10-6383799 

1205 

1452025 

1749690125 

34-7131099 

10-6413244 

1206 

1454436 

1754049816 

34*7275107 

10-6442672 

1207 

1456849 

1758410743 

34*7419055 

10-6472085 

1208 

1459264 

1762790912 

34-7562944 

10-6501480 

1209 

1461681 

1767172329 

34*7706773 

10-6530860 

1210 

1464100 

1771561000 

34*7850543 

10-6560223 

1211 

1406521 

11775956931 

34*7994253 

10-6589570 

1212 

1468944 

i1780360128 

34-8137904 

10-6618902 

1213 

1471369 

.1784770597 

34*8281495 

10-6648217 

1214 

1473796 

1789188344 

34-8425028 

10-6677516 

1215 

1476225 

1793613375 

34*8568501 

10*6706799 

1216 

1478656 

1798045696 

34*8711915 

10-6736066 

1217 

1481089 

1802485313 

34*8855271 

10*6765317 

1218 

1483524 

1806932232 

34-8998567 

10*6794552 

1219 

1485961 

1811386459 

34*9141805 

10-6823771 

1220 

1488400 

1815848000 

34*9284984 

10-6852973 

1221 

1490841 

1820316861 

34-9428104 

10-6882160 

1222 

1493284 

1824793048 

34*9571166 

10-6911331 

1223 

1495729 

1829276567 

34-9714169 

10-6940486 

1224 

1498176 

1833704247 

34*9857114 

10-6969625 

1225 

1500625 

1838265625 

35-0000000 

10-6998748 

122 6 

1503276 

1842771176 

35-0142828 

10-7027855 

1227 

1505529 

1847284083 

35-0285598 

10*7056947 

1228 

1507984 

1851804352 

35-0 128309 

10-7086023 

1229 

1510441 

1856331989 

35-0570963 

10-7115083 

1230 

1512900 

1860867000 

35-0713558 

10-7144127 

1231 

1515361 

1865409391 

35-0856096 

10-7173155 

1232 

1517824 

1869959168 

35-0998575 

10*7202168 

1233 

1520289 

1874516337 

35*1140997 

10-7231165 

1234 

1522756 

1879080904 

35-12S3361 

10-7260146 

1235 

1525225 

1883652875 

35-1425668 

10-7289112 

1236 

1527696 

1888232256 

35-1567917 

10-7318062 

1237 

1530169 

1892819053 

35-1710108 

10*7346997 

1238 

1532644 

1897413272 

35-1852242 

10-7375916 

1239 

1535121 

1902014919 

35-1994318 

10-7404819 

1240 

1537600 

1906624000 

35-2136337 

10-7433707 

1241 

1540081 

1911240521 

35-2278299 

10-7462579 

1242 

1542564 

1915864488 

35-2420204 

10-7491436 

1243 

1545049 

1920495907 

35*2562051 

10-7520277 

1244 

1547536 

1925134784 

35*2703842 

10-7549103 

1245 

1550025 

1929781125 

35-28455.75 

10-7577913 

1246 

1552516 

1934434936 

35-2987252 

10-7606708 

1247 

1555009 

1939096223 

35-3128872 

10*' 7 63548S 

1248 

1557504 

1943764992 

35-3270435 

10*7664252 


L 


Reciprocals. 

*0008354219 

•0008347245 

*0008340284 

*0008333333 

•0008326395 

•0008319468 

*0008312552 

•0008305648 

*0008298755 

•0008291874 

•00082S5004 

*0008278146 

•0008271299 

*0008264463 

*0008257638 

*0008250825 

•0008244023 

*0008237232 

*0008230453 

•0008223684 

*0008216927 

•0008210181 

*0008203445 

•0008196721 

•0008190008 

•0008183306 

*0008176615 

•0008169935 

•0008163265 

*0008156607 

•000S149959 

•0008143322 

•0008136696 

•0008130081 

*0008123477 

*0008116883 

•0008110300 

•0008103728 

•0008097166 

•0008090615 

•0008084074 

•0008077544 

•0008071025 

•0008064516 

•000S058018 

•0008051530 

•0008045052 

*0008038585 

•0008032129 

•0008025682 

•0008019246 

•0008012821 











































136 


Table of Squares, Cubes, Square and Cube Roots 


Number 

Squares. 

Cubes. 

>/ Roots. 

1 $/ Roots. 

Reciprocals. 

1249 

1560001 

1948441249 

35-3411941 

10-7693001 

•0008006405 

1250 

1562500 

1953125000 

35-3553391 

10-7721735 

•ooosoooooo 

1251 

1565001 

1957816251 

35-3694784 

10-7750453 

•0007993605 

1252 

1567504 

1962515008 

35-3836120 

10-7779156 

-0007987220 

1253 

1570009 

1967221277 

35-3977400 

10-7807843 

•0007980846 

1254 

1572516 

1971935064 

35-411S624 

10-7S36516 

•0007974482 

1255 

1575025 

1976656375 

35*4259792 

10-7865173 

•0007968127 

1256 

1577536 

1981385216 

35-4400903 

10-7893815 

•0007961783 

1257 

1580049 

1986121593 

35-4541958 

10-7922441 

•0007955449 

1258 

1582564 

1990865512 

35-4682957 

10-7951053 

•0007949126 

1259 

1585081 

1995610979 

35-4823900 

10-7979649 

0007942812 

1260 

1587600 

2000376000 

35-4964787 

10-8008230 

•0007936508 

1261 

1590121 

2005142581 

35-5105618 

10-8036797 

•0007930214 

1262 

1592644 

2009916728 

35*5246393 

10-8065348 

•0007923930 

1263 

1595169 

2014698447 

35-53S7113 

10-8093884 

•0007917656 

1264 

1597696 

2019487744 

35-5527777 

10-8122404 

•0007911392 

1265 

1600225 

2024284625 

35*5668385 

10-8150909 

•0007905138 

1266 

1602756 

2029089096 

35-5808937 

10-8179400 

•0007898894 

1267 

1605289 

2033901163 

35-5949434 

10-8207876 

•0007892660 

1268 

1607824 

2038720832 

35-6089876 

10-8236336 

•0007886435 

1269 

1610361 

2043548109 

35-6230262 

10-8264782 

•0007880221 

1270 

1612900 

2048383000 

35-6370593 

10-8293213 

•0007874016 

1271 

1615441 

2053225511 

35*6510869 

10-8321629 

•0007867821 

1272 

1617984 

2058075648 

35*6651090 

10-8350030 

•0007S61635 

1273 

1820529 

2062933417 

35*6791255 

10-8378416 

•0007855460 

1274 

1623076 

2067798824 

35-6931366 

10-8406788 

•0007849294 

1275 

1625625 

2072671875 

35-7071421 

10-8435144 

•0007843137 

1276 

1628176 

2077552576 

35*7211422 

10-8463485 

•0007836991 

1277 

1630729 

2082440933 

35*7351367 

10-8491812 

•0007830854 

1278 

1633284 

2087336952 

35-7491258 

10-8520125 

•0007824726 

1279 

1635841 

2092240639 

35-7631095 

10-8548422 

•0007S18608 

1280 

1638400 

2097152000 

35-7770S76 

10-8576704 

•0007812500 

1281 

1640961 

2102071841 

35-7910603 

10-8604972 

•0007806401 

1282 

1643524 

2106997768 

35-8050278 

10-8633225 

•0007800312 

1283 

1646089 

2111932187 

35-8189894 

10-8661454 

•0007794232 

1284 

1648656 

2116874304 

35*8329457 

10-8689687 

•0007788162 

1285 

1651225 

2121824125 

35-846S966 

10-8717897 

•0007782101 

1286 

1653796 

2126781656 

35-8608421 

10-8746091 

•0007776050 

1287 

1656369 

2131746903 

35-8747822 

10-8774271 

•0007770008 

1288 

1 658944 

2136719872 

35*8887169 

10-8802436 

•0007763975 

1289 

1661521 

2141700569 

35-9026461 

10-8830587 

•000*7757952 

1290 

1664100 

2146689000 

35-9165699 

10-8858723 

•0007751938 

1291 

1666681 

21516S5171 

35-9304884 

10-8886845 

•0007745933 

1292 

1669264 

2156689088 

35-9444015 

10-8914952 

•0007739938 

1293 

1671849 

2161700757 

35-9583092 

10-8943044 

•0007733952 

1294 

1674436 

2166720184 

35-9722115 

10-8971123 

•0007727975 

1295 

1677025 

2171747375 

35-9861084 

10-8999186 

•0007722008 

1296 

1679616 

2176782336 

36-0000000 

10-9027235 

•0007716049 

1297 

1682209 

2181825073 

36-0138862 

10-9055269 

•0007710100 

1298 

1684804 

2186S75592 

36-0277671 

10-9083290 

•0007704160 

1299 

1687401 

2191933899 

36-0416426 

10-9111296 

•0007698229 

1300 

1690000 

2197000000 

36-0555128 

10-9139287 

•0007692308 
























Table op Squares, Cubes, Square and Cube Roots, 


137 


Number. 

Squares. 

Cubes. 

\/ Roots. 

Roots. 

Reciprocals. 

1301 

1692601 

2202073901 

36-0693776 

10*9167265 

•0007686395 

1302 

1695204 

3207155608 

36-0832371 

10-9195228 

•00076S0492 

1303 

1697S09 

2212245127 

36-0970913 

10*9223177 

*0007674579 

1304 

1700416 

2217342464 

36-1109402 

10*9251111 

*0007668712 

1305 

1703025 

2222447625 

36-1247837 

10-9279031 

*0007662835 

1306 

1705636 

2227560616 

36-1386220 

10*9306937 

*0007656968 

1307 

1708249 

2232681443 

36-1524550 

JO-9334829 

*0007651109 

1308 

1710864 

2237S10112 

36-1662S26 

10-9362706 

*0007645260 

1309 

1713481 

2242946629 

36-1801050 

10*9390569 

*0007639419 

1310 

1716100 

2248091000 

36-1939221 

10*9418418 

•00076335S8 

1311 

1718721 

2253243231 

36-2077340 

10*9446253 

*0007627765 

1312 

1721344 

2258403328 

36-2215406 

10*9475074 

*0007621951 

1 1313 

1723969 

2263571297 

36-2353419 

10*9501880 

*0007616146 

1314 

1726596 

2268747144 

36-2491379 

10-9529673 

*0007610350 

1315 

1729225 

2273930875 

36-2626287 

10*9557451 

*0007604563 

3316 

1731856 

2279122496 

36-2767143 

10*9585215 

*0007598784 

1317 

17344S9 

2284322013 

36-2904246 

10*9612965 

*0007593014 

1318 

1737124 

2289529432 

36-3042697 

10-964 0701 

*0007587253 

1319 

1739761 

2294744759 

36-3180396 

10*9668423 

•0007581501 

3320 

1742400 

2299968000 

36*3318042 

10*9696131 

•0007575758 

1321 

1745041 

2305199161 

36-3455637 

10-9723825 

*0007570023 

1322 

1747684 

2310438248 

36*3593179 

10-9751505 

*0007564297 

1323 

1750329 

2315685267 

36*3730670 

10-9779171 

•0007558579 

1324 

1752976 

2320940224 

36*3868108 

10*9806823 

•0007552870 

1325 

1755625 

2326203125 

36*4005494 

10*9834462 

*0007547170 

1326 

1758276 

2331473976 

36*4142829 

10-9862086 

*0007541478 

1327 

1760929 

2336752783 

36-4280112 

10*9889696 

•0007535795 

1328 

17635S4 

2342039552 

36*4417343 

10*9917293 

•0007530120 

1329 

1766241 

2347334289 

36-4554523 

10*9944876 

*0007524454 

1330 

1768900 

2352637000 

36-4691650 

10-9972445 

*0007518797 

1331 

1771561 

2357947691 

36-482S727 

11-0000000 

•0007513148 

1332 

1774224 

2363266368 

36*4965752 

11 *0027541 

•0007507508 

1333 

1776889 

2368593037 

36*5102725 

11-0055069 

•0007501875 

3334 

1779556 

2373927704 

36-5239647 

11 *0082583 

•0007496252 

1335 

1782225 

2379270375 

36*5376518 

11-01 10082 

•0007490637 

1336 

1784S96 

2384621056 

36*5513388 

11-0137569 

•0007485030 

1337 

1787569 

23S9979753 

36*5650106 

11-0165041 

•0007479432 

1338 

1790244 

2395346472 

36-5786S23 

11-0192500 

•0007473842 

1339 

1792921 

2400721219 

36*5923489 

11-0219945 

•0007468260 

1340 

1795600 

2406104000 

36-6060104 

11-0247377 

•0007462687 

1341 

17982SI 

2411494821 

36-6196668 

11-0274795 

•0007457122 

1342 

1800964 

2416S936S8 

36-6333181 

11-0302199 

•0007451565 

1343 

1803619 

2122300607 

36-6469144 

11 -0329590 

•0*00744 6016 

1344 

1806336 

2^27715584 

36-6606056 

110356967 

*0007440476 

1345 

1S09025 

2133138625 

36-6742416 

11-0384330 

•0007434944 

1346 

1811716 

2438569736 

36-6878726 

11-04116S0 

•00074294 21 

1347 

1814409 

244400S923 

36-7014986 

11-0439017 

•0007423905 

1348 

1S17104 

2449156192 

36-7151195 

11*0466339 

*0007418398 

1349 

1S19S01 

2454911549 

36*7287353 

11*0493649 

•0007412898 

1350 

1822500 

2460375000 

36-7423461 

1 1-0520945 

•0007407407 

1351 

1S25201 

2465846551 

36*7 559519 

11*0548227 

•0007401924 

1352 

1827904 

2471326208 

36-7695526 

11*0575197 

•0007396450 




































1?,8 Table of Squares, Cubes, Square and Cube Roots 


Number 

Squares. 

1 Cubes. 

\/ Roots. 

v/ Roots. 

Reciprocals. 

1353 

1830609 

2476813977 

36*7831483 

11*0602752 

•0007390983 

1354 

1833316 

2482309864 

367967390 

11*0629994 

•0007385524 

13 5 5 

1836025 

2487813875 

36*8103246 

11*0657222 

•0007380074 

1356 

1838736 

2493326016 

36*8239053 

11*0684437 

•0007374631 

1357 

1S41449 

2498846293 

36-8374809 

11*0711639 

•0007369197 

1358 

1844164 

2504374712 

36*8510515 

11-0738S28 

•0007363770 

1359 

1846881 

2509911279 

36*8646172 

11*0766003 

•0007358352 

1360 

1849600 

2515456000 

36*8781778 

11*0793165 

•0007352941 

1361 

1852321 

2521008881 

36-8917335 

11*0820314 

•0007347539 

1362 

1S55044 

2526569928 

36*9052842 

11*0847449 

•0007342144 

1363 

1857769 

2532139147 

36*9188299 

11*0874571 

•0007336757 

1364 

1860496 

2537716544 

36*9323706 

11*0901679 

•0007331378 

1365 

1863225 

2543302125 

36*9459064 

11*0928775 

•0007326007 

1366 

1865956 

2548S95896 

36*9594372 

11-0955S57 

•0007320644 

1367 

186S689 

2554497863 

36*9729631 

11*0982926 

•0007315289 

1368 

1871424 

2560108032 

36*9864840 

11*1009982 

•0007309942 

1369 

1874161 

2565726409 

37*0000000 

11*1037025 

•0007304602 

1370 

1876900 

2571353000 

37*0135110 

11*1064054 

•0007299270 

1371 

1879641 

2576987811 

37*0270172 

11*1091070 

•0007293946 

1372 

1882384 

2582630848 

37*0405184 

11*1118073 

•0007288630 

1373 

1885129 

2588282117 

37*0540146 

11*1145064 

•0007283321 

1374 

18S7876 

2593941624 

37*0675060 

11*1172041 

•0007278020 

1375 

1890625 

2599609375 

37-0S99924 

11*1199004 

•0007272727 

1376 

1893376 

2605285376 

37*0944740 

11*1225955 

•0007267442 

1377 

1896129 

2610969633 

37*1079506 

11*1252893 

•0007262164 

1378 

1898884 

2616662152 

37*1214224 

11*1279817 

•0007256894 

1379 

1901641 

2622362939 

37*1348893 

11*1306729 

•0007251632 

1380 

1904400 

2628072000 

37*1483512 

11*1333628 

•0007246377 

1381 

1907161 

26337S9341 

37*1618084 

11*1360514 

*0007241130 

1382 

1909924 

2639514968 

37*1752606 

11*1387386 

•0007235890 

1383 

19126S9 

2645248887 

37*1887079 

11*1414246 

•0007230658 

1384 

1915456 

2650991104 

37*2021505 

11*1441093 

•0007225434 

1385 

1918225 

2656741625 

37*2155881 

11*1467926 

•0007220217 

1386 

1920996 

2662500456 

37*2290209 

11*1494747 

•0007215007 

1387 

1923769 

2668267603 

37*2424489 

11*1521555 

*0007209805 

1388 

1926544 

2674043072 

37*2558720 

11*1548350 

•0007204611 

1389 

1929321 

2679826869 

37*2692903 

11*1575133 

•0007199424 

1390 

1932100 

2685619000 

37*2827037 

11*1601903 

•0007194245 

1391 

1934881 

2691419471 

37*2961124 

11*1628659 

•0007189073 

1392 

1937664 

2697228288 

37*3095162 

11*1655403 

•0007183908 

1393 

1940449 

2703045457 

37*3229152 

11*1682134 

*0007178751 

1394 

1943236 

2708870984 

37*3363094 

11*1708852 

*0007173601 

1395 

1946025 

2714704875 

37*3496988 

11*1735558 

*0007168459 

1396 

1948816 

2720547136 

37*3630834 

11*1762250 

•0007163324 

1397 

1951609 

2726397773 

37*3764632 

11*1788930 

•000715S196 

1398 

1954404 

2732256792 

37*3898382 

11*1815598 

*0007153076 

1399 

1957201 

2738124199 

37*4032084 

11*1842252 

*0007147963 

1400 

1960000 

2744000000 

37*4165738 

11*1868894 

*0007142857 

1 1401 

1962801 

2749884201 

37*4299345 

11*1895523 

*0007137759 

1402 

1965604 

2755776808 

37*4432904 

11*1922139 

*0007132668 

1403 

1968109 

2761677827 

37*4566410 

11*1948743 

*0007127584 

1404 

....— ...... ■ .... 

1971216 

27675S7264 

37*4699880 

11*1975334 

*0007122507 
























139 


Table of Squares, Cubes, Square and Cube Roots. 


Number 

! Squares. 

| Cubes. 

Roots. 

1405 

1974025 

2773505123 

37-4833296 

1406 

1976836 

2779431416 

j 37*4966665 

1407 

1979649 

2785366143 

. 37*5099987 

1408 

1982404 

!2791309312 

! 37-5233261 

1409 

19S52S1 

1 2797260929 

37*5366487 

1410 

1988100 

!2803221000 

37*5499667 

1411 

1990921 

2809189531 

37-5632799 

1412 

1993744 

2815166528 

j 37*5765885 

1413 

1996509 

2821151997 

37-5898922 

1414 

1999396 

2827145944 

37*6031913 

1415 

2002225 

2833148375 

1 37*6164857 

1416 

2005056 

2839159296 

37*6297754 

1417 

2007889 

2845178713 

37*6430604 

1418 

2010724 

2851206632 

37*6563407 

1419 

2013561 

2857243059 

37*6696164 

1420 

2016400 

2863288000 

37*6828874 

1421 

2019241 

2S69341461 

37*6961536 

1422 

20220S4 

2875403448 

37*7094153 

1423 

2024929 

2S81473967 

; 37-7226722 

1424 

2027776 

2887553024 

! 37-7359245 

1425 

2030625 

2893640625 

! 37*7491722 

1426 

2033476 

2S99736776 

37-7624152 

1427 

2036329 

2905841483 

37*7756535 

1428 

2039184 

2911954752 

37-7888873 

1429 

2042041 

2918076589 

37-8021163 

1430 

2044900 

2924207000 

37*8153408 

1431 

2047761 

2930345991 

37*8285606 

1432 

2050624 

2936493568 

37*8417759 

1433 

20534S9 

2942649737 

37*8549864 1 

1434 

2056356 

2948814504 

37*8681924 

1435 

2059225 

2954987875 

37-8813938 1 

1436 

2062096 

2961169856 

37-8945906 

1437 

2064969 

2967360453 

37-9077828 i 

1438 

2067844 

2973559672 

37-9209704 j 

1439 

2070721 

2979767519 

37-9341535 

1440 

2073600 

2985984000 

37-9473319 

1441 

2076481 

2992209121 

37-9605058 

1442 

2079364 

2998442888 

37-9736751 

1443 

2082249 

3004685307 

37*9868398 

1444 

2085136 

3010936384 

38-0000000 

1445 

20S8025 

3017196125 

38*0131556 

1446 

2090916 3023464536 

38*0263067 

1447 

2093809 ! 3029741623 

38*0394532 

1448 

2096704| 

3036027392 

38*0525952 

1449 

2099601 

3042321849 

38*0657326 

1450 

21025001 

3048625000 

38*0788655 

1451 

2105401 

3054936851 

38*0919939 

1452 

2108304! 

3061257408 

38*1051178 

1453 

2111209 

3067586777 

38*1182371 i 

1454 , 

2114116 

3073924664 

38-1313519 

1455 

2117025 

3080271375 

38-1444622 

1456 

2119936 1 

3086626816 

38-1575681 


%/ Roots. 

11*2001913 
11*2028479 
11*2055032 
11*2081573 
11*2108101 
11*2134017 
11*2101120 
11*2187611 ! 
11-2214089 : 
11*2240054 I 
11*2207007 
11*2293448 
11*2319876 
11*2340292 
11*2372096 
11*2399087 
11*2425465 
11*2451831 
11*2476185 
11*2504527 
11*2530856 
11*2557173 
11*2583478 
1 1*2009770 
11*2036050 
11*2002318 
11*2088573 
11*2714816 
11*2741047 
11*2767206 
11*2793472 
11*2819006 
11*2845849 
11*2872019 
11-2898177 
11*2924323 
11*2950457 
11*2976579 j 
11*3002088 
11*3028786 
11*3054871 
11*3080945 
11*3107006 
11*3133056 
11*3159094 
11*3185119 
11*3211132 
11*3237134 
11*3263124 
11*3289102 
11*3315067 
11*3341022 


Reciprocals. 

•0007117438 
•0007112376 
•0007107321 
•0007102273 
*0007097232 
•0007092199 
•0007087172 
•0007082153 
•0007077141 
•0007072136 
•0007067138 
•0007002147 
*0007057163 
*0007052186 
•0007047216 
*0007042254 
•0007037298 
*0007032349 
•0007027407 
•0007022472 
•0007017544 
•0007012023 
•0007007708 
•0007002801 
•0000997901 
•0000993007 
•0000988120 
*0000983210 
•0006978307 
•0000973501 
•0006968641 
•0006903788 
•000095S942 
•0000954103 
•0000949270 
•0000941444 
•0006939625 
*0006934813 
•0006930007 
•0000925208 
•0000920415 
•0000915629 
0000910850 
0000900078 
0006901312 
0000896552 
0006891799 
0000887052 
0006882312 
0000877579 
0006872852 
0006368132 



































140 


Table ov Squares, Cubes, Square and Cube Roots. 


Number. 

Squares. | Cubes. 

\/ Roots. 

y Roots. 

Reciprocals. 

1457 

2122849 3092990993 

38-1706693 

11-3366964 

•0006863412 

1458 ' 

2125764 3099363912 

38-IS37662 

] 1 -3392894 

•000685S7I1 

1459 

212S681 3.105745579 

3S-196S585 

11-3418813 

*0006854010 

1460 

2131600 3112136000 

38-2099463 

11-3444719 

•0006849315 

1461 

2134521 3118535181 

38-2230297 

11-3470614 

•0006S44627 

1462 

2137444 3124943128 

38-2361085 

11-3496497 

'0006839945 

1463 

2140369 3131359847 

38*2491829 

11-3522368 

*0006835270 

1464 

2143296 3137785344 

33-2022529 

11-3548227 

•0006830601 

1465 

2146225 3144219625 

38-2*53184 

11-3574075 

•0006825939 

1466 

2149156 3150662696 

38-28o3794 

11-3599911 

*0006821282 

1467 

2152089 3157114563 

38-3014360 

11-3625735 

•0006816633 

1468 

2155024 3163575232 

38-3144881 

11-3651547 

•0006811989 

1469 

2157961 3170044709 

38-3275358 

11-3677347 

'0006807352 

1470 

2160900 3176523000 

38-3405790 

11-3703136 

•0006802721 

1471 

2163841 3183010111 

38-3536178 

11-3728914 

*0006798097 

1472 

2166784 3189506048 

38-3666522 

1 1-3754679 

*0006793478 

1473 

2169729 3196010817 

38-3796821 

11-3780433 

*0006788866 

1474 

2172676 3202524424 

38-3927076 

11-3806175 

•0006784261 

1475 

2175625 3209046875 

38-4057287 

11-3831906 

*0006779661 

1476 

2178576 3215578176 

38-4187454 

11-3857625 

•0006775068 

1477 

2181529 3222118333 

38-4317577 

11-3883332 

*0006770481 

1478 

2184484 3228667352 

38-4447656 

11-3909028 

•0006765900 

1479 

2187441 3235225239 

38-4577691 

11-3934712 

*0006761325 

1480 

2190400 3241792000 

38-4707681 

11-3960384 

•0006756757 

1481 

2193361 3248367641 

38-4837627 

11-3986045 

•0006752194 

1482 

2196324 3254952168 

38-4967530 

1 1-401 1695 

*0006747638 

1483 

2199289 3261545587 

38-5097390 

1 1-4037332 

*0006743088 

1484 

2202256 3268147904 

38-5227206 

1 1-4062959 

•0006738544 

1485 

2205225 3274759125 

38-5356977 

1 1-4088574 

•0006734007 

I486 

2208196 3281379256 

38-5486705 

11-4114177 

*0006729474 

1487 

2211169 3288008303 

38-5616389 

11-4139769 

*0006724950 

1488 

2214144 3294646272 

38-5746030 

1 1-4165349 

*0006720430 

1489 

2217121 3301293169 

3S-5875627 

11-4190918 

•0006715917 

1490 

2220100 3307949000 

38-6005181 

11-4206476 

•0006711409 

1491 

2223081 3314613771 

38-6134691 

11-4242022 

*0006796908 

1492 

2226064 3321287488 

38-6264158 

11-4267556 

*0006702413 

1493 

2229049 3327970157 

38-6393582 

11-4293079 

•0006697924 

1494 

2232036 3334661784 

38-6522962 

11-4318591 

•0006693440 

1495 

2235025 3341362375 

38-6652299 

11-4344092 

*0006688963 

1496 

2238016 3348071936 

38-6781593 

11-4369581 

•0006684492 

1497 

2241009 3354790473 

38-6910843 

11-4395059 

*0006680027 

1498 

2244004 3361517992 

38-7010050 

11-4420525 

*0006675567 

1499 

2247001 3308254499 

38-7169214 

11-4445980 

*0006671114 

1500 

2250000 3375000000 

38-7298335 

11-4471424 

*0006666667 

1501 

2253001 3381754501 

38-7427412 

11-4496857 

*0006662225 

1502 

2256004 3388518008 

38-7556447 

11-4522278 

*0006657790 

1503 

2259009 3395290527 

38-7685439 

11-4547688 

*0006553360 

1504 

2262016 3402072064 

38-7814389 

11-4573087 

*0006648936 

1505 

2265025 3408862625 

38-7943294 

11-4598476 

*0006644518 

1506 

226S036 3415662216 

38-8072158 

11-4623850 

*0006640106 

1507 1 

2271049 3422470843 

38-8200978 

11-4649215 

*0006635700 

1508 1 

227406413429288512 

38-8329757 

11-4674568 

*0000631300 































Table of Squares, Cubes, Square and Cube Roots. 141 


’ 

Number. 

Squares. 

Cubes. 

Roots. 

4/Ttoots. 

Reciprocals. 

1509 

22770S1 

3436115229 

38-845S491 

11-4699911 

•0006626905 

1510 

2280100 

3442951000 

38-8587184 

11-4725242 

•0006622517 

1511 

2283121 

3449795831 

38-8715834 

11-4750562 

1 -0006618134 

1512 

22S6144 

3456649728 

38-8844442 

11-4775871 

•0006613757 

1513 

2289169 

3463512697 

38-8973006 

11-4801169 

•0006609385 

1514 

2292196 

3470384744 

38-9101529 

11-4826455 

•0006605020 

1515 

2295225 

'3477265875 

38-9230009 

11-4851731 

•0006600660 

1516 

2298256,3484156096 

38-9358447 

11-4876995 

•0006596306 

1517 

230I2S9 

3491055413 

33-9486841 

11-4902249 

•0006591958 

1518 

2304324 

3597963832 

38-9615194 

11-4927491 

•0006587615 

1519 

2307361 

3504881359 

38-9743505 

11-4952722 

•0006583278 

1520 

2310400 

3511808000 

38-9871774 

11-4977942 

•0006578947 

1521 

2313441 

3518743761 

39-0000000 

11-5003151 

•0006574622 

1522 

2316484 

3525688648 

39-01281S4 

11-5028348 

•0006570302 

1523 

2319529 

3532642667 

39-0256326 

11 *5053535 

•0006565988 

1524 

2322576 

3539605824 

39-0384426 

11-5078711 

•0006561680 

1525 

2325625 

3546578125 

39-0512483 

11-5103876 

•0000557377 

1520 

2328676 

3553559576 

39-0640499 

11-5129030 

•0006553080 

1527 

2331729 

3560558183 

39-0768473 

11-5154173 

•0006548788 

1528 

2334784 

3567549552 

39-0896406 

11-5179305 

•0006544503 

1529 

2337841 

3574558889 

39-1024296 

11-5204425 

•0006540222 

1530 

2340900 

3581577000 

39-1152144 

11-5229535 

•0006535948 

1531 

2343961 

3588604291 

39-1279951 

11-5254634 

•0006531679 

1532 

2347024 

3595640768 

39-1407716 

11-5279722 

•0006527415 

1533 

2350089 

3602686437 

39-1535439 

11-5304799 

•0006523157 

1534 

2353156 

3609741304 

39-1663120 

11 -5329865 

•0006518905 

1535 

2356225 

3616805375 

39-1790760 

11-5354920 

•0006514658 

1536 

2359296 

3623878656 

39-1918359 

11-5379965 

•0006510417 

1537 

2362369 

3630961153 

39-2045915 

11-5404998 

•0006506181 

1538 

2365444 

3638052872 

39-2173431 

11-5430021 

•0006501951 

1539 

2368521 

3645153819 

39-2300905 

11-5455033 

•0006497726 

1540 

2371600 

3652264000 

39-2428337 

11-5480034 

•0006493506 

1541 

2374681 

3657983421 

39-2555728 

11-5505025 

•0006489293 

1542 

2377764 

3666512088 

39-2683078 

11-5530004 

•0006485084 

1543 

2380849 

3673650007 

39-2S10387 

11-5554972 

•0006480881 

1544 

2383936 

3680797184 

39-2937654 

11-5579931 

•0006476684 

1545 

2387025 

3687953625 

39-3064880 

11-5604878 

•0006472492 

1546 

2390116 

3695119336 

39-3192065 

11-5629815 

•0006468305 

1547 

2393209 i 

3702294323 

39-3319208 

11-5654740 

•0006464124 

1548 

2396304 

3709478592 

39-3446311 

11-5679655 

•0000459948 

1549 

2399401 

3716672149 

39-3573373 

11-5704559 

•0006455778 

1550 

2402500 

3723875000 

39-3700394 

11-5729453 

•0006451613 

1551 

2405601 

3731087151 

39-3827373 

11-5754336 

•0006447453 

1552 

2408704 

3738308608 

39-3954312 

11-5779208 

•0006443299 

1 

2411809 

3745539377 

39-4081210 

11-5804069 

•0006439150 

1554 

2414916 

3752779464 

39-4208067 

11-582S919 

•0006435006 

1555 

2418025! 

3760028875 

39-4334883 

11-5853759 

•0006430868 

1556 

2421136 

3767287616 

39-4461658 

11-5S7S588 

•0000426735 

1557 

2424249 

3774555693 

39-4588393 

1 1-5903407 

•0006422608 

1558 

2427364 3781833112 

39-4715087 

11 -5928215 

•0006418485 

1559 

2430481[ 

3789119879 

39-4841740 

11-5953013 

•0006414368 

1560 

2433600 

3796416000 

39-496S353 

11-5977799 

•0006410253 



































142 


Table op Squares, Cubes, Square and Cube Roots. 








Number 

Squares. 

Cubes. 

V Roots. 

\/ Koois. 

Reciproeals. 

1561 

2436721 

3803721481 

39-5094925 

11 0002576 

•0006406150 

1562 

2439844 

3811036328 

39-5221457 

11-0027342 

•0000402049 

1563 

2442969 

3818360547 

39-5347948 

11-6052097 

•0006397953 

1564 

2446096 

3825641444 

39-5474399 

11-0076841 

•0000393862 

1565 

2449225 

3833037125 

39-5600809 

11-0101575 

•00063S9776 

1566 

2452356 

38403S9496 

39-5727179 

11-0120299 

•0006385096 

1567 

2455489 

3847751263 

39-5853508 

11-0151012 

•0000381621 

1568 

2458624 

3855123432 

39-5979797 

11.-0175715 

•0006377551 

1569 

2461761 

3802503009 

39-6106046 

11-6200407 

•0006373486 

1570 

2464900 

3869883000 

39-0232255 

11-6225088 

•0000369427 

1571 

2468041 

3877292411 

39-0358424 

11-0249759 

•0000365372 

1572 

2471184 

3884701248 

39-6484552 

11-6274420 

•0006361323 

1573 

2474329 

3892119157 

39-6010040 

11-0299070 

•0006357279 

1574 

2477476 

3899547224 

39-6736688 

11-6323710 

•0000353240 

1575 

2480625 

3906984375 

39-6862096 

11-0348339 

•0006349206 

1576 

2483776 

3914430976 

39-6988665 

11-6372957 

•0006345178 

1577 

2486929 

3921887033 

39-7114593 

11-0397506 

•0006341154 

1578 

2490084 

3929352552 

39-7240481 

11-0422104 

•0006337136 

1579 

2493241 

3936827539 

39-7360329 

11-6440751 

•0006333122 

1580 

2496400 

3944312000 

39-7492138 

11-6471329 

•0006329114 

1581 

2499561 

3951805941 

39-7617907 

11-6495895 

•0006325111 

1582 

2502724 

395930936S 

39-7743636 

11*6520452 

•0006321113 

1583 

2505889 

3966822287 

39-7869325 

11-0544998 

•0006317119 

1584 

2509056 

3974344704 

39-7994976 

11*6569534 

•0000313131 

1585 

2512225 

3981876625 

39-8120585 

11-6594059 

•0006309148 

1586 

2515396 

3989418056 

39-8240155 

11-6618574 

•0006305170 

1587 

2518569 

3996969003 

39-S3716S6 

11-6043079 

•0006301197 

1588 

2521744 

4004529472 

39-8497177 

11-6007574 

•0000297229 

1589 

2524921 

4012099469 

39-8622628 

11-0092058 

•0006293206 

1590 

2528100 

4014679000 

39-8748040 

11-6710532 

•0006289308 

1591 

2531281 

402726S071 

39-8873413 

11-0740996 

•0000285355 

1592 

2534464 

4034866688 

39-8998747 

11-6765449 

•0006281407 

1593 

2537649 

4042474857 

39-9124041 

11-6789892 

•0006277404 

1594 

2540836 

4050092584 

39-9249295 

11-6814325 

•0006273526 

1595 

2544025 

4057719875 

39-9374511 

11-6838748 

•0000269592 

1596 

2547210 

4065350736 

39-9499687 

11-6S63161 

•0000265604 

1597 

2550409 

4073003173 

39-9024824 

11-0887563 

•0006261741 

1598 

2553604 

4080059192 

39-9749922 

11-6911955 

•0000257822 

1599 

2556S01 

4088324799 

39-9874980 

11-6930337 

•0000253909 

1600 

2560000 

4096000000 

40-0000000 

11-6960709 

•0000250000 

N. 

2V*. 

N*. 

\/Y. 

iV n. 

1 

N 

1 

i/tv 

N. 

i /n*- 


Vn. 

- — 

-[/ iV 

fx. 

■j/w. 

N. 

Vxl 

fiV. 

1 

f N 

X*. 

N*. 

iV°. 

N. 

v'Y. 

1 < 




_ _ _ 


iV2 

2V3. 

N'\ 

N». 

V N\ 

N. 

1 

1 

1 

1 

J 1 

*1 1 


N’ 

IV s " 

N* ' 

\ N m 

ViV 

X. 


























Evolution. 


143 


When the number contains Integer and Decimals. 

Example 5. Required the Square Root of 7845*45? In the column of Sguares 
you will hnd, 

+7849*96 = 88*62, +7849*96 = 88*62, 

—7845*4 5 = 88*62-, —7832*25 = 88*52, 

• 451 divided by 1771 = 00*0256. 

^7845*45 = 88*5256 nearly. 

&tfWhen the number of ciphers in the integer is even, the number of 
figures taken in the Square column must also be even ; but when the number 
of figures in the integer is odd, the number taken in the Square column must 
also be odd. 

To find the Cube Root of Numbers exceeding 1600. 

Example 6. Required the Cube Root of 5694958? In the Cube column you will 

find, 

+5735339 = 1793 +5735339 = 1793. 

—56 94958 = 1783- —5639752 = 1783. 

40381 divided by 95587 = 000*4225, 

$ 6694958 = 178*4225 nearly. 

When the number contains Integer and Decimals. 

Example 7. Required the Cube Root of 4186*586? In the column of Cubes you 
will find, 

+4251*528 = 16*23 4251*528 = 16*2 3 

—4186*585 = 16*13- 4173*281 = 16*1 3 

64942 ~ 7824?" = 00*083 

•$' 4186*586 = 16*183 nearly. 

4®=*The following notice must be particularly attended ro, when extracting 
Cube Root of numbers with decimals. 

2 ciphers in the integer must be 5, 8, or 11 ciphers in the Cube column. 


3 

66 

66 

66 

3, 6, or 

9 

66 

66 

4 

66 

66 

66 

4, or 

7 

66 

66 

5 

a 

66 

66 

5, or 

8 

66 

66 

6 

a 

66 

66 

6, or 

9 

66 

66 

7 

a 

66 

6* 

7, or 10 

66 

66 


Example 8. Required the Cube Root of 61358*75? In the Cube column and 8 
figures you will find, 

+61629*875 = 395 3 +61629875 = 39*5 3 

— 61358*750 = 3943- —611 62984 = 39*4 3 

271*125 divided by 466891 = 00*05807 

^61358*75 =- 39*45807. 

To find the Fourth Root. 

Rule,. Extract the Square Root of the number as before described, and of 
that root extract the Square Root again, then the last is the Eourth root of 
the number. 

Example 9. Required the fourth root of 2469781 ? 

2469781 = V j/2469781 — yl571*4463 = 39*6467, the answer. 

To find the Sixth Root. 

Rule. Find the Cube Root of the number as before described, and of that 
root extract the Square Root, and then the last is the Sixth root of tlio 
number. 

























144 


Powers and Roots. 


To Find Powers and Roots not Noted in the Table. 

At. the foot of the last page of square and cube roots is an algebraic table I 
which indicates how higher power and root can be found in the above tables. , 
Example 1. {Suppose the 6th root is required of the number 3914430976. 

On the 4th line in the algebraic table will be found \/N', on the same line 
find the letter N, which is in the column of cubes. Find the given number 
3914430976 in the column of cubes, which answers to 39.6988 in the column 

of square roots; therefore l/'3914430976 = 39.6988. 

Example 2. Find the q/ / 5735339 2 . On the third line in the algebraic table 

find \/N i , which is in the columns of squares; and N is in the column of 
cubes. Find the number 5735339 in the column of cubes; and in the column 

of squares is found the required number, namely ]> 5735339 2 = 32041. 

Thus, a variety of powers and roots can be found by reference to the alge¬ 
braic tabic. 

Table of Hie First Nine Powers of Numbers. 


1st. 

2d. 

3d. 

4th. 

5th. 

6th. 

7th. 

8th. 

9th. 

1 

1 

1 

1 

1 

1 

1 

1 

1 

2 

4 

8 

16 

32 

64 

128 

256 

512 

3 

9 

27 

SI 

243 

729 

2187 

6561 

19683 

4 

16 

64 

256 

1024 

4096 

16384 

65536 

262144 

5 

25 

125 

625 

3125 

15625 

78125 

390625 

1953125 

6 

36 

216 

1296 

7776 

46656 

279936 

1679616 

10077696 

7 

49. 

343 

2401 

16807 

117649 

823543 

5764801 

40353607 

8 

64 

512 

4096 

32768 

262144 

2097152 

16777216 

134217728 

9 

81 

729 

6561 

59049 

531441 

4782969 

43046721 

387420489 


Table of Permutation. Seepage 29. 

2 How many different numbers can be written by the 
6 nine Arabic digits? From the table we have the per- 
mutation 9 = 362880 different numbers. 

720 How many different words can be written by the 
5040 seven letters algebra f 

362880 The permutation of 7 is 5040, but there are two 
3628800 a’s; and the permutation of 2 is 2. Therefore, 
39916800 5040 : 2 = 2520 different words. 

479001600 

6227020800 

87178291200 

1307674368000 

20922789888000 

355687428096000 

6402373705728000 

121645100408832000 

2432902008176640000 

51090942171709440000 

1124000727777607680000 

25852016738884976640000 

620448401733239439360000 


1 

2 

3 

4 

5 

6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 








































































































Table of the Fourth and Fifth Powers of Numbers. 145 


No. 

4th 

Power. 

5th 

Power. 

No. 

4th 

Power. 

5th 

Power. 

No. 

4th 

Power. 

5th 

Power. 

1 

1 

1 

51 

6765201 

345025251 

101 

104060401 

10510100501 

2 

16 

32 

52 

7311616 

380204032 

102 

108243216 

11040808032 

3 

81 

243 

53 

7890181 

418195493 

103 

1125508S1 

11592740743 

4 

256 

1024 

54 

8503056 

459165024 

104 116985856 

12166529024 

5 

625 

3125 

55 

9150625 

503284375 

105 121550625 

12762815625 

6 

1296 

7776 

56 

9834496 

550731776 

106 126247696 

13382255776 

7 

2401 

16807 

57 

10556001 

601692057 

107 

131079601 

14025517307 

8 

4096 

32768 

58 

11316496 

656356768 

108 

136048896 

14693280768 

9 

6561 

59049 

59 

12117361 

714924299 

109 

141158161 

15386239549 

10 

10000 

100000 

60 

12960000 

777600000 

110 

146410000 

16105100000 

11 

14641 

161051 

61 

13845841 

844596301 

111 

151807041 

16850581551 

12 

20736 

248832 

62 

14776336 

916132832 

112 

157351936 

17623416832 

13 

28561 

371293 

63 

15752961 

992436543 

113 

163047361 

18424351793 

14 

38416 

537824 

64 

16777216 

1073741824 

114 

168896016 

19254145824 

15 

50625 

759375 

65 

17850625 

1160290625 

115 

174900625 

20113571875 

16 

65536 

1048576 

66 

.18974736 

1252332576 

116 

181063936 

21003416576 

17 

83521 

1419857 

67 

20151121 

1350125107 

117 

187388721 

21924480357 

18 

104976 

1889568 

68 

21381376 

1453933568 

118 

193877776 

22877577568 

19 

130321 

2476099 

69 

22667121 

1564031349 

119 

200533921 

23863536599 

20 

160000 

3200000 

70 

24010000 

1680700000 

120 

207360000 

2488:1200000 

21 

194481 

4084101 

71 

25411681 

1804229351 

121 214358881 

25937424601 

22 

234256 

5153632 

72 

26873856 

1934917632 

122 

221533456 

27027081632 

23 

279841 

6436343 

73 

28398241 

2073071593 

123 

228886641 

2815305(1843 

24 

331776 

7962624 

74 

29986576 

2219006624 

124 

236421376 

29316250624 

25 

390625 

9765625 

75 

31640625 

2373046875 

125 

244140625 

30517578125 

26 

456976 

11881376 

76 

33362176 

2535525376 

126 

252047376 

31757969376 

27 

531441 

14348907 

77 

35153041 

2706784157 

127 

260144641 

33038369407 

28 

614656 

17210368 

78 

37015056 

2887174368 

128 

268435456 

34359738368 

29 

707281 

20511149 

79 

38950081 

3077056399 

129 

276922881 

35723051649 

30 

810000 

24300000 

80 

40960000 

3276800000 

130 

285610000 

37129300000 

31 

923521 

28629151 

81 

43046721 

3486784401 

131 

294499921 

38579489651 

32 

1048576 

33554432 

82 

45212176 

3707398432 

132 

303595776 

40074642432 

S3 

1185921 

39135393 

83 

47458321 

3939040643 

133 

312900721 

41615795893 

34 

1336336 

45435424 

84 

49787136 

4182119424 

134 

322417936 

43204003424 

35 

1500625 

52521875 

85 

52200625 

4437053125 

135 

332150625 

44840334375 

36 

1679616 

60466176 

86 

54708016 

4704270176 

136 

342102016 

46525874176 

37 

1874161 

69343957 

87 

57289761 

4984209207 

137 

352275361 

48261724457 

38 

2085136 

79235168 

88 

59969536 

5277319168 

138 

362673936 

50049003168 

39 

2313441 

90224199 

89 

62742241 

5584059449 

139 

373301041 

51888844699 

40 

2560000 

102400000 

90 

65610000 

5904900000 

140 

384160000 

53782400000 

41 

2825761 

115856201 

91 

68574961 

6240321451 

141 

395254161 

55730836701 

42 

3111696 

130691232 

92 

71639296 

6590815232 

142 

406586896 

57735339232 

43 

3418801 

147008443 

93 

74805201 

6596883693 

143 

418161601 

59797108943 

44 

3748096 

164916224 

94 

78074896 

7339040224 

144 

429981696 

61917364224 

45 

4100625 

184528125 

95 

81450625 

7737809375 

145 

442050625 

64097340625 

46 

4477456 

205962976 

96 

84934656 

8153726976 

146 

454371856 

66338290976 

47 

4879681 

229345007 

97 

88529281 

8587340257 

147 

466948881 

68641485507 

48 

5308416 

254803968 

98 

92236816 

9039207968 

148 

479785216 

71008211968 

49 

5764801 

282475249 

99 

96059601 

9509900499 

149 492884401 

73439775749 

50 

5250000 

312500000 1 

100 

100000000 

10000000000 

150|506250000l 

75937500000 


10 




















































146 


Irregular Figures. 


To find tlie Area and Solidity of Irregular Figures. 

Chapman's rule in the construction of ships, Stockholm, 1775. 



Divide the base A B into any even number of equal parts. 6 = distance between 
the ordinates ; Q = area of the projecting figure. 

Q — —(a -f- 46 -}■ 2c -f- 4 d -f* 2e -j- 4f+(/)• ... 1. 

3 

Suppose this area to revolve around the axis A B and form a solid figure like a 
handle, an urn or a gun ; then the solidity C of the figure will be— 

C= + 46* + 2c* + 4<P + 2e* + 4/* + g t). . . . 2 . 

3 


The practical calculation of these formulas is set up 
as in table for Formula 1. Suppose a = 1.25. b = 1.15, 
c— 1.52, d = 1.86. e — 2,f= 1.77, and g= 1.20. 

The distance between the ordiuates being 4 = 2 ? 

then the area will be, Q = Ax 28.51 = 19 square of 

3 

whatever measure used. 

The convex surface 5 of the figure will be, S= 2n Q 
= 2 X 3.14 X19 = 119.3 square. 


Ordinates. 

Mult. 

Product. 

a 

1.25 

1 

1.25 

b 

1.15 

4 

4.50 

c 

1.52 

2 

3.04 

d 

1.86 

4 

7.44 

e 

2. 

2 

4.00 

f 

1.77 

4 

7.08 

9 

1.20 

1 

1.20 

Q 

9.506 

% 

28.51 


This rule can also be employed in calculating the 
cubic contents of earth-work in excavations and 
embankments, in which the ordinates are expressed 
in areas of the sections. 

Suppose a — 36 square feet, yards, metres, or 
whatever unit of measure, 6=30, c = 42, d = 56, 
e = 84, f=72, and <7 = 50, the distance between 
the sections being, say, 50 feet. The calculation is 
set up as in the preceding table, namely: 

Volume C= 50 X 323.3 = 1616.5 cubics of what¬ 
ever unit of measure used. 

This rule is universally employed for calculating the areas of water-lines, cross- 
sections and cubic contents of displacement in ships (Jcnoum as Simpsoti's rule). 

When the cubic content is required between each section, calculate it as ex¬ 
plained in Excavation and Embankment. 

Surface of Revolution. 

The surface generated by a line revolving around an axis, is equal to the length 
of the line multiplied by tho circumference of its centre of gravity. 

N. B. The line, whether straight or curved, must bo in the same plane as the 
axis. 

Solidity of Revolution. 

The solidity generated by a plane revolving around an axis, is equal t;o the area 
of tire plane multiplied by the circumference of its centre of gravity. 

N. B. The revolving plane must be in the same plane as the axis. 


Ordinates. 

Mult. 

Product. 

a 

36 

1 

36 

b 

30 

4 

120 

c 

42 

2 

84 

d 

56 

4 

224 

e 

84 

2 

168 

f 

72 

4 

288 

9 

50 

1 

50 

C 

323.3 

34 

970 


































147 


Table of 8tli OrdinateSf for Railroad Curves, 


Angle. 

W 

1. 7 . 

Ordinates. 

2 . Q. | 3. 5. 

4, li. 

Angle. 

W 

H 

Ordx 

2. 6 

nates. 

3. 5. 

4. h. 

l c 

•00084 

•00164 

•00193 

•00218 

5 3° 

•05313 

•08932 

•11063 

•11773 

2 

•00191 

•00327 

•00409 

•00436 

54 

*05422 

•09130 

•11318 

•12003 

3 

•00299 

•00522 

•00561 

•00659 

55 

•05531 

•09308 

•11510 

•12235 

4 

•00382 

•00654 

•00818 

•00872 

56 

•05646 

•09487 

•11731 

•12466 

5 

•00437 

•00818 

•01023 

•01091 

57 

•05760 

•09673 

•11950 

•12698 

6 

•00573 

•00928 

•01228 

•01309 

58 

•05875 

•09853 

•12170 

•12932 

7 

•00675 

•01173 

•01432 

•01527 

59 

•05989 

•10037 

•12393 

•13162 

8 

•00764 

•01309 

•01639 

•01746 

GO 

•06094 

•10220 

•12612 

•13397 

9 

•0()845 

•01474 

•01842 

•01964 

61 

•06261 

•10427 

•12840 

*13631 

10 

•00955 

•01637 

•02047 

•02183 

62 

•06331 

•10593 

•13054 

•13866 

1 1 

•01053 

•01801 

•02250 

•02402 

63 

•06451 

•10781 

*13281 

•14101 

1 2 

•01146 

•01965 

•02456 

•02620 

64 

•06570 

•10964 

•13505 

•14337 

13 

•01245 

•02129 

*02662 

•02839 

65 

•06681 

•11101 

•13765 

•14573 

14 

•01284 

•02271 

•02861 

•03058 

66 

•06805 

•11342 

•13956 

•14810 

15 

•01438 

•02461 

•03081 

•03282 

67 

•06914 

•11532 

•14181 

•15048 

16 

•01535 

•02625 

•03277 

•03496 

68 

•07040 

•11721 

*14409 

•15286 

17 

•01630 

•02789 

*03484 

•03715 

69 

•07168 

•11912 

•14637 

•15526 

18 

•01730 

•02956 

•03693 

•03935 

70 

•07284 

•12103 

•14864 

•15765 

1 9 

•0L858 

•03125 

•03996 

•04154 

7 1 

•07407 

•12294 

*15087 

•16005 

20 

•01922 

•03286 

•04103 

•04374 

72 

•07535 

•12485 

•15323 

•16245 

21 

•02022 

•03453 

•04309 

•04594 

73 

•07656 

•12685 

•15555 

•16487 

22 

•02119 

•03619 

•04522 

•04814 

74 

•07784 

•12S77 

•15785 

•16729 

23 

•02215 

•03787 

•04720 

•05034 

75 

•07912 

•13078 

*16016 

•16972 

24 

•02311 

•03934 

•04930 

•05255 

76 

•08040 

•13292 

•16247 

•17216 

25 

•02413 

•04117 

•05138 

•05475 

77 

•08168 

•13472 

•16482 

•17460 

2 6 

•02508 

•04283 

•05346 

•05696 

78 

•08297 

•13670 

•16716 

•17706 

2 7 

•02610 

•04457 

•05552 

•05917 

79 

•08426 

•13868 

•16951 

•17951 

28 

•02708 

•04621 

•05761 

•06139 

80 

•08560 

•14070 

•17187 

•18198 

29 

•02813 

•04793 

•05970 

•06361 

81 

•08695 

•14274 

•17423 

•18445 

30 

•02911 

•04970 

•06188 

•06582 

82 

•08829 

•14477 

•17660 

•18694 

31 

•03005 

•05125 

•06386 

•06804 

83 

•08944 

•14681 

•17901 

•18943 

32 

•03107 

•05298 

•06596 

•07027 

84 

•09105 

•14888 

•18140 

•19193 

33 

•03191 

*05464 

•06806 

•07250 

85 

•09235 

•15120 

•18379 

•19444 

34 

•03310 

•05637 

•07016 

•07477 

86 

•09377 

•15304 

•18622 

•19695 

35 

•03412 

•05804 

•07424 

•07695 

87 

•09518 

•15509 

•18865 

•19946 

36 

•03515 

•05992 

•07452 

•07919 

88 

•09660 

•15756 

•19108 

•20201 

37 

•03616 

•06147 

•07646 

•08143 

89 

•09780 

•15931 

•19350 

•20555 

38 

•03718 

•06327 

•07858 

•08367 

90 

•09944 

•16144 

•19597 

•20710 

39 

•03821 

•06492 

•08069 

•08591 

91 

•10098 

•16359 

•19842 

•20966 

40 

•03905 

•06631 

•08243 

•08816 

92 

•10240 

•16575 

•20092 

•21223 

41 

•04030 

•06836 

•08494 

•09041 

93 

•10384 

•16787 

•20338 

•21481 

42 

•04133 

•07012 

•08707 

•09266 

94 

•10537 

•17005 

•20589 

•21740 

43 

•04241 

•07182 

•08920 

•09492 

95 

•10692 

•17224 

•20837 

•22000 

44 

•04363 

•07353 

•09130 

•09719 

96 

•10851 

•17444 

•21091 

•22262 

4 5 

•0i522 

•07531 

•09346 

•09945 

97 

•10997 

•17666 

•21342 

•22523 

46 

•04556 

•07706 

•09562 

•10172 

98 

•11150 

•17888 

•21596 

•22786 

47 

•04682 

•07894 

•09790 

•10400 

9 9 

•11310 

•18111 

•22800 

•23050 

4 8 

•04833 

•08059 

•09991 

•10627 

100 

•11468 

•18354 

•22107 

•23315 

4 9 

•04879 

•08236 

•00207 

•10856 

101 

•11626 

•18500 

•22364 

•23596 

5 0 

•04982 

•08413 

•00422 

•11085 

102 

•11791 

•18793 

•22623 

•23848 

5 1 

•05096 

•08593 

•10639 

•11314 

103 

•11959 

•19021 

•22876 

•24107 

5 2 

•05204 

•08768 

•10855 

•11543 

1 04 

•12116, 

•19256 

•23147 

•24386 















































148 


Rail Road Curves 


RAIL ROAD CURVES. 

When Railroads are to be connected by curves, we commonly have given the 
distance (Fig. 142,) between the two ends o o of the tracks, and the tangential 
angle t Dy these the curve is to be constructed. 

Example 1. Fig. 142. The chord C = 168 feet, and the tangential angle 
tj = 19° 30'. Required the centre angle w =,and the radius R = ? 

w = 2(19° 309 = 39 °- R = 39/ ^ = 1-4979X168 = 25T647 feet. 

k = See Table for Segments, &c., of a circle. 

By Tangential Angles* 

The curve to be laid out by the three tangential angles ror, ron, and noo, 
each angle = je = 6° 30'. Required the chord r = ? 

Tire centre angle for the chord r is 

2X(6° 30') = 13°, and r = R = 0-2264X 251-647 = 56-974 feet. 

By Angles uf Deflexion* 

Divide the centre angle w into an even number of parts = z. Set off at o the 
angle z — r o n, and bisect it into ror and ron .—find the chord r, and sub-chord 
a, and continue as shown by Figure. 

Example 2. Fig. 142. The tangential angle v = 78°, and the chord (7= 638 
feet. Required the centre-angle w = ? Radius R — ? Chord r = ? and the 6ub' 
chord a = ? 

w = 2X78° = 156°. R = “6t c = 0-51117 X 638 = 326-126 feet. 

Let the curve be laid out by 6 angles of deflexion, and z — 4x156° = 26°, and 
r =» R = 0-4199X326-126 = 146-73 feet. 
a ==■ * 6 k r = 0-4495X146-73 = 66-012 feet. 


Example 3. Fig. 143. 
height h — l 


By Ordinates* 

The chord C= 368 feet, and v = 36°. Required the 
h — &C( cosec.r — cot.i>). 


From ....... cosec.36° = 1-70130 

Subtract ....... cot.36° = 1-37638 

The height h = 0-32492X184 =* 59785 feet. 


At * = 92 feet from h. Required the ordinate y ? 


sin .2 = 


2X92 sin.36° 
368 


= 0-2938926 = sin.l7° 6'. 


y = iX368 


/cos.l7° 6' 
\ sin.36° 


— cot.36° 


) 


= 45-9448 feet. 


By Sub-Chords* 


Example 4. Fig. 144. The ends o and o of the tracks form different angles to 
and IF to the chord C. and therefore must be connected by two curves of differ¬ 
ent radii‘A’ a d r. The chord C — 869 feet, iu = 38°, and W= 86°. Required 


the distance 
radii R and r 


from 
= ? 


o to the height h, n = ? sub-chord b = ? sub-chord a = ? 


v = *X3S° = 19°, and F= |X86° 43°. 


n 


869 tan.19° 
tan.l9°4-tan.43° 


= 234-35 feet. 


b = 234-35 sec.43° = 320-42 feet. I R = v*ka = 1-5358X671-21 = 1030-2 ft. 
a = sec.l9°(809 — 234-35) = 671-21 ft. | r »*kb == 0-73314X320-42 = 234 91 ft. 


By Eight Ordinates* 

Exanple. 5. Fig. 146. Required 8 ordinates for a curve of chord C = 710 feet 
and the centre angle w = 69°? (See Table on the preceding page.) 

1st and 7th Ordinates 0 07168x710 = 50-8928 feet. 

2nd “ 6th “ 0T1912X710 = 84-5752 “ 

3rd “ 5th “ 0-14637X710 — 103-9227 “ 

4th or height h 015526X710 = 110 2346 “ 

















Railroad Curves. 


149 



144 


n = 


145 


143 

By Ordinates. 

h— £C( cosec.t;— cot.v). 

i/'-Zcos .2 , v 

!/ = iOl — -cot.y ), 

Vsin.i; / 

sin.: 


By Sub-chords. 

C tan.-i; , . Tr 

,- Tr , ii — n tan. V, 

tan. v+ tan. V 

j i; w = 2v 

b ~n sec.F, W = 2K> 

a sec.u(C — •><), 


Parallel tracks by a reverse curve. 

Formulas same as above. 

The length o o = 2c, length of 
a circle arc l = 0-035u R. 


142 

By angles of deflexion. 
w = 2v, R = w k C -= iC cosec. v. 1 
r = R, a = *k r = 2 r sin. \z. 



146 

The greatest radius in a reverse 
curve. 

w = £( F+3r), W = w+ V — v, 

a= w k R, b = w kR, 

R = C sec .w (sin. V —|/sin.*F — cos.*u>). 



147 

Ourve by 8 Ordinates. 

The ordinates are calculated in the 
accompanying Table, the chord C =-= 1 or 
the unit. 

If the angle w is large, or there be some 
obstacle on the chord C, find the height h 
and lay out the curve by two or more sets 
of 8 ordinates. 



















































150 


By Ordinates and Subciiouds. 


By Ordinates and Subchords. 

Example 6. Fig. 148. The tangents t being prolonged to where they 
meet at a, divide that angle into two equal parts, say W= 75°. Required 
the tangents t — 1 external secant S=i chords C'=1 and the angle tc=1 
Radius of the curve ft=1500 feet. 

t = R cot,75°=1500X0-26794=401-91 feet. 

Centre angle w=90-75°=15° for half the curve. 

S = ft (sec.l5 } — 1) =1500 (1-0352-1) =52-8 feet. 

The chords C=k ft = 0-26104X1500=391-56 feet. 

Measure off from a the tangents and the external secant. 

Draw the chords C C, and divide them each into eight equal parts. 
In the table of ordinates under w>=15° will be found the 

1st. 7th. 0-01438X391-56^6-631, 3rd. 6th. 0-03081X391-56=12-063, 

2nd. 6th. 0 02461X391-56 -9 636, 4th. 0-03282X39-56=12-851, 

Thus by only four multiplications, 16 ordinates in the curve is obtained. 

Should there be any obstacles for the chords C. C. as is often the case 
in excavations and on embankments, a line can be drawn further in on 
the track parallel to the chord and the ordinates obtained by subtraction, 
readily understood by the Engineer. 

Ellipse by Ordinates. 

By this arrangement ellipses can be constructed of any proportions. 
One of the two axes is divided into 16 equal parts. The ordinates 
drawn and calculated as shown by the figure 102. 


Parallel Tracks by a Semi-Ellipse, 

Example 7. Fig. 150. The instrument placed at b and 6', divide the 
angles W and w each into two equal parts, prolong the chords which 
will meet at a , a point in the curve. Divide the chords each into eight 
equal parts, and draw the ordinates parallel to the tracks as shown in the 
figure. The grand chord C is the unit for calculating the ordinates, 
which latter are alike on both the chords c', c'\ 

1st 2nd. 3rd. 4th. 6th. 6th. 7th. 

0-1795C 0-2058C 0“2029C 0-1830 C 0T477C 0i09lC 0-0586C. 

Suppose the grand chord to be C=2050 feet. 

Required the length of the 6th ordinate ! 0-1091X2050=223 655 feet. 


Tracks not Parallel by Elliptic, arc, 

Example 8. Fig. 151. Divide the angles W and w each into two equal 
parts, prolong the subchords until they intersect one another at a, which 
is a point in the curve. Divide the chord C into eight equal parts, join 
a with the 4th division and draw the other ordinates parallel thereto. 

Suppose the angles are 1F=18 :I and «> = 12 J , the centre angle will be 30° 
for which the ordinates are to be calculated from the table. The chord 
C=125 feet. Required the 3rd and 5th ordinates! 0-06188X125=7-335 feet. 

Springing of Bails. 


Example 9. Fig. 152. A rail of L=21 feet is to be curved to a radius 
ol ft=1250 feet. Required the spring S=l in sixteenths of an inch. 

24X21* 

S = - ——— = 8-47 sixteenths. 

12o0 


Super Elevation of the External Rail. 

Example 10. Fig. 153. A train running Af=30 miles per hour on a 
curve of ft=1550 feet radii, the gauge of the track is G -6 feet. Required 
the angle of inclination v—1 and the super elevation of the external 
rail /i=1 


tan.v =-- = 0-0387=tan. 2° 13'. 

15X1550 

h=G sin.l 0 21'=5X0"02356 =0-1178 feet, or nearly H inches. 

It is practically impossible to lay the super elevation to suit the dif¬ 
ferent speeds of trains. If a mean speed is taken, the faster passenger 
trains will wear the outer rail, and the slow or freight train will wear 
the inner rail. 












Railroad Curves. 


151 




Ellipse by ordinates. 


1 

2 : 

3: 

4: 


0-4840(7 

0-6616(7 

0-7803(7 

0-8660(7 


5; 
6 : 
7 ; 

8 : 


: 0-9204(7 
0-9682(7 
; 0-9922(7 
C the unit. 


Parallel tracks by elliptic curves 


c' — 


w — 2 v. 
(7 sin. W 


c"= 


W= 2 V, 
C sin.w 


2 sin.v ' ” 2 sin. V 

See example for ordinates. 


- - — 


152 


Spring of Rails. 

_ 1*5 L* .... 

S =—^—- — spring in inches. 


5 = 


R 

24 L 


-=16ths of an inch. 


Inclination of tracks in curves ; 


M* 

tan.u= —h = G sin.n. 
15 K 


Meaning of letters, see example. 


148 

By ordinates and subchords. 
t= R cot. W=R tan.ta, 1F=90— w, 

S = R (sec.tv —1 ) = R ('cosec. W —1^ 1 
C=k R. For k, see table of segments. 


151 

Tracks not parallel by elliptic arc. 

Angle of the arc —W-\-w. 

Ordinates to be calculated from the table, 
page 147._ 



























































152 Laying Out Railway Curves. 


Explanation of tile Figures on the Following Page. 

The most correct and accurate ways of laying out railway curves are by external 
secant or by ver sines, either to be employed, as the ground permits. The operation 
is well understood from figures 154 and 155. 

The natural secant and versin. are found in the trigonometrical tables, bub- 
tract 1 from the natural secant, and the remainder will be the external secant. 
Multiply the external secant by the assumed radius, and the product is the external 
secant s in the same unit of measure as the radius. 

The centre angle is divided by 2 and 2 as many times as may be required for 
setting out the curve. 

Fig. 154 is used when there are obstacles inside the curve, and Fig. 155 when 
the outside is inaccessible. The sinus-versus in the tables, multiplied by the 
assumed radius, will be the height of the curve above the chord. 

When the inside of the curve is obstructed, and the point T of intersection is 
also inaccessible, then the curve can be laid out as illustrated by Fig. 156. 

Fig. 157 illustrates how to lay out a curve by chords of 100 feet. 


Tangential angles for a chord of c = 100 feet, and different radii R from 500 feet to 

3 miles (fig. 157). 


R. 

tan. angle. 

R. 

tan. angle. 

R. 

tan. an 

gle. 

Feet. 

O 

§ 

// 

Feet. 

O 

t 

tt 

Miles. 

O 

i 

" 

500 

5 

43 

46 

3000 

0 

57 

18 

0.125 

4 

20 

26 

COO 

4 

46 

29 

3500 

0 

49 

6 

0.25 

2 

10 

13 

700 

4 

5 

33 

4000 

0 

42 

58 

0.5 

1 

5 

G 

800 

3 

34 

52 

4500 

0 

38 

12 

0.75 

0 

43 

25 

900 

3 

10 

59 

5000 

0 

34 

23 

1 mile. 

0 

32 

33 

1000 

2 

51 

53 

5500 

0 

31 

15 

1.25 

0 

26 

2 

1100 

2 

36 

16 

6000 

0 

28 

39 

1.5 

0 

21 

42 

1200 

2 

23 

15 

7000 

0 

24 

34 

1.75 

0 

18 

42 

1500 

1 

54 

35 

8000 

0 

21 

30 

2 

0 

16 

17 

2000 

1 

25 

56 

9000 

0 

19 

6 

2 yz 

0 

13 

1 

2500 

1 

8 

46 

10000 

0 

17 

12 

3 

0 

10 

51 


Fig. 158 illustrates a section of a cut or embankment through sloping ground. 
The meauing of letters is the same as that on the following pages on excavation 
and embankment. 

Fig. 159. Sidings for parallel tracks. — D = distance over tangent points; W= 
width between centres of tracks, aud R = radius of curvature; v = angle of frog- 
plates. 

The different operations of laying out the curves are so well 
understood by railroad engineers that it is considered unnecessary 
to enter into detailed description. The formulas and figures are 
intended only as a memorandum. 


L 
















Railroad Curves. 


153 



154 By external secants . 

External secant s = _Z?(sec .w — 1). 

W = 90 — w; w = 90 — W, w = 2v. 
tangent t = B cot.W — B tan. w. 


155 



By sinus-versus. 

w = 180— W. c — 2Bsin.w. 

C 

B = — ; -. c = 2B 6in.v. 

2 sin.w 

ver sin.A = B ver.sin.w. 


x = 90 — 


w 

" 2 * 



156 When the point T is inaccessible. 

W = 90 — v. b — 2 cl cot.w 

a -f- d — B (sec.w— 1). d = 2 &tan.r. 
a — B (sec.w—1) — \b tan.v. 



157 Tangential angle for a chord of c = 
100 feet, and different radii B from 
500 feet to 3 miles. 

c 


V) 


2v. 


sin.fw 


B = 


2 sin.^w 


2 B 

c = 2 B sin.|w. 



158 Bailway cut or embankment through 
side slopes. 

. r . b sin.(90-|-2z—s) 

b = - -ft/tan.s. c = ———-r . 

2 sin. (90 — z — s) 

x = 90 -f- z — s. v = 90 — z — s. 

b cos .s 

e — 


sin.(90 f-z — s ) 
a. — d(d tan.s -f r). 


h\— r-”— H 



Sidings for parallel tracks. 
D = 2VW{B — \W). 


— 4 w~^~ 


sin.!; = 


B 
2 K 

































154 


Excavation and Embankment, 


EXCAVATION AND EMBANKMENT. 


Example 1. The "Road-way of an excavated channel is?* = 15 feet, the depth 
D — 9 feet, and the breadth at the top b 46£ feet. Require the slope *S = ? 



Formula 6. 


„ 46-5 — 15 A „ 

S — - ^ „ =* 1*<5 or 1$ to 1* 


2X9 


Example 2. The Road way is to be r = 15, D = 18, and the slope S = If, 
Require the breadth b = ? and the cross-section A = ? 


Formula 4. 


b = 2 X 18 X 1-25 + 15 = 60 feet. 


18 / \ 

Formula 7. A = — ^ 60 + 15 j = 675 square feet. 






Example 3. The Road-way is to be r =■ 16 feet, the slope S = If, and the depth 

1 


D — 11 feet. Required the area of Cross-section A = ? 
Formula 9. A = 11 (11 X 1| + »*) = 357'5 square feet. 


Example 4. The Road-way r = 18 feet, slope S= If, d = 14 feet 6 inches, and 
the length from o is l = 55 feet. Required the cubic contents c = ? 


Formula 11. c = 55 X 
by 27 = 444.28 cubic yards. 


/14'5 V 1*25 1R\ 

14*5(—~ + -a ) = 11995*676 cubic feet,divided 


Example 5. The Road-way is r = 16 feet, slope S = If feet, D — 17*5, d = 7*4 
and the length L = 100 feet. Required the cubic content C = ? 


Formula 12. C =10o[h( — * + ' ' 4 * + 1 ‘ ‘ 5 x '1 ^(H‘5 + 7*4^] 

= 444-15 cubic feet, or 1645*4 cubic yards. 

The computation is executed thus. 


17-5 

7*4 


17*5 

7.4 


700 

1225 


24*9 

8 


129*50 

17*5’ = 306*25 ) From table 
7*4* = 54*76 J i 


199*2 


of Squares. 


3) 490-51 (163-5. J slope _ ^ , 


199*2 


X 100 = 44445. cubic feet. 





















Excavation and Embankment. 


155 



Letters In the Formulas correspond with the Figure. 


<S = cot. v , 
a = D S, • 
a = D cot. v , - 
b = 2 D S + r, 


s= *zr 


5 2 2 )’ 


A=§(J + r), - 7. 

a -j(i + r), - 8. 

A = D(D S + r), 9. 

•I = d[d S + r), - 10. 

-'*£+*). «• 


c = ^[ s ( 


+ \ (D + if)]. 


3 ' 2 

2> a + d a ,+2)^ 


Letters Denote, 

A and a = Cross-Sections in square feet, of the excavated channel or 
embankment. 

2) and d — depth in feet, of the Sections. 

r — width in feet of the Road-Way. 

b = Base in feet of the embankment, or top breadth of the channel. 

L = length in feet, between the two Sections A and a. 

I = length in feet, from the Section a to the point o where the ground is 
level with the road. 

C = cubic contents in feet, between A and a. 

c = cubic contents in feet, between a and o. 

S = slope of the sides. The slope is commonly given in proportions, thus: 
“ Slope = 1£ to 1,” which means, that the side slopes H feet horizontally for 1 
fcot vertical. 

v = angle of the slope. 


















































156 


Railroads. 


TRACTION ON ROADS. 

Letters denote. 

F= tractive force in pound avoir., necessary to overcome the rolling 
friction, and ascending inclined planes. 

Af = miles per hour of the train or force F. 

T = weight of the load in tons, including the weight of the carriages. 

On rail-roads T includes the weight of the locomotive and tender. 
t = weight of the locomotive resting on the driving wheels in tons. 
h = vertical rise in feet per 100 of inclined roads. 
b = base in feet per 100 of the inclined road or plane. 
k — tractive coefficient in pound per ton of the load T, as noted in the ac¬ 
companying Table, under the different conditions of the road. (P. 158.) 
A = area of one of the two cylinder pistons in a locomotive, in sq. in. 

P = mean pressure of steam in lbs. per sq. in. on cylinder pistons. 

S= stroke of pistons in feet. 

D = diameter of driving wheel in fefct. 

IP= actual horse power of a locomotive or the power necessary for the 
load. About 25 per cent, is allowed for friction and working pumps. 
/= adherence coefficient of the driving wheels to the rails, in pounds 
per ton of the weight t. 
n = revolutions per minute of driving wheels. 
d = continued working hours of a horse. 

v = velocity in feet per second, t' = weight of a horse in pounds. 

Example 11. Fig. 161. The area of one of the two cylinder pistons in 
a locomotive is A =314 square inches, stroke of piston P=2 feet, mean- 
pressure P=80 lbs. per square inch. Driving wheels D =4 feet diameter. 
Required the tractive force F=1 of a locomotive. 

F = 314X2X80 = 1256Q lbg> the angwer# 

4 

The adhesive force of the driving wheels to the rails, / t, must always 
be greater than the tractive force of the locomotive, otherwise the wheels 
will slip on the track. 

Example 12. Fig. 162. A locomotive of t= 15 tons on an inclined plane 
rising h = 10 feet, and the base b = 99-5 feet per 100. f= 560. other dimen¬ 
sions being the same as in the preceding example. Required the tractive, 
retroactive and adhesive forces? 

Tractive, F = 22-4X15X10=9200 lbs. 

4 

Retroactive, F= 22-4X15X10=3360 lbs. 

Adhesive, F = — ^^X 99 ' 6 ^ 835S lba> 

100 

Consequently the locomotive can ascend the inclined plane with a pulling 
force of 8358 — 3360 = 4998 lbs., without slip in the driving wheels. 

Example 13. Fig. 163. A train of T — 200 tons is to be drawn M= 20 miles 
per hour on a horizontal track in good condition, A — 4. Required retroactive 
force F— ? 

F = 200 (4-)-]/20) = 1694-4 lbs. the answer. 

Example 14. Fig. 164. A train of T=150 tons is to be drawn up an in¬ 
clined plane of h ~9 feet in 100, with a speed of Af=16 miles per hour 
k=4. Required the necessary horse power of the locomotive IP=? 

(22-4X9+4+ i/l6) = 1342-144 horses. 

370 

Example 15. Fig. 165. Required the tractive ability F=1 of a horse 
running Af=7 miles per hour, in d =4 continued hours. ’ 

375 

F = —— = 26-8 lbs. the answer. 

7 \ 4 

















Railways and Common Roads. 


157 



165 

F= T (k+^/M). »=1 -466 M. 

„ 550 375 ,. 1U . . 

F— - ■— =——;• ability of a horse. 

v y/d My/d 


162 


ASP D n 

f =~d — 22 ' 4 "‘- M =W 


ftb. 


Adhesive, *—^>22*4 th. retractive. 


F= T (k-\- y/M). </* t— Adhesive. 


MT„ . ^ | 




F=T(22-4h+k+ s /Mj. <~= Ad. 


T22-4 h+k+yil). 


F=T(22.4 h+k+z'M). 


F= 


550 


- M= 0-6821 r. 


F= 


A SP 

~ D~~' 

A SPn 


28 M 
n =-rT' 

ASPM 


11000 376 D 

Adhesive force =ft. 




















































158 


Railroads. 


Example 16. fig. 166. Required, the tractive force F= ? of a load 7 — 5.25 tons 
to be drawn M— 2 miles per hour up a turnpike of/t = 9 feet in 100, the road be¬ 
ing newly laid with coarse gravel A* = 50 ? 

F= 5.25 (22.4 X 8 + 50 + j/2) = 1328.30 lba. 


Suppose a horse to weigh t' = 1000 lbs., working continually in d = 1 hour up 
the turnpike. Required, the tractive ability F= 1 per horse? 


F= 


375 

V 1 


1000 X _9 _ 97>5 lbg> 
100 


1328 25 

Number of horses =--— = 13.6, say 14 horses, which will be necessary for 

97.5 

the load under the mentioned circumstances. In these examples it is necessary to 
take 1 . and d > 1 . 



Traction Coefficient at very Slow Speed. k 

On railroads in good condition, carriage axles well lubricated, . 4 

On railroads under ordinary, not very good condition, ... 8 

On very smooth stoue pavement, .... ... 12 

On ordinary street pavemeuts in good condition, . . . .20 

On street pavements and turnpikes,.30 

On turnpikes new laid with coarse gravel and broken stones, . . 50 

On common roads in bad condition,.150 

On natural loose ground or sand.560 


Adhesion Coefficient. 


On rails of maximum dryness,.672 

“ very dry,.560 

“ under ordinary circumstances,.450 

“ in wet weather,.315 

“ with snow or frost,.224 


In railway curves the retractive force is augmented so many per cent, as the 
whole train occupies degrees in the curve. 

Railway Gauges. Gauge 

feet. in. 

The most general gauge in coal mines,. 2 6 

Denver and Rio Grande railway,.3 

Rio Grande and Texas,.3 6 

The most general gauge in the United States, England, France, Prus¬ 
sia, Sweden, Mexico, Chili and Peru,.4 8 ^ 

The compromised gauge,.49 

Camden and Amboy,. 49 // 

In the Southern States and in Russia,.5 

Irish railways,. 5 3 

Louisiana and Texas, also in Canada and India, .... 5 6 

Great Western in England, . 

Rniu-fall in Inches at Different Seasons of the Year. 


Locations. 

Year. 

Spring. 

Summer. 

Fall. 

Winter. 

Nishny, Taguilsk, Russia, . . 

18.26 

3.35 

9.28 

3.70 

1.93 

Tobolsk, Siberia,. 

17.76 

2.29 

9.05 

4.02 

2.40 

Nertchinsk, Asia,. 

Yakoutsk, East Siberia, . . . 

18.13 

2.32 

10.5 

4.96 

0.35 

10.25 

1.40 

3.35 

3.59 

1.85 

Peking, China,. 

Macao, Quang-toug,. 

23.88 

2.17 

17.7 

3.50 

0.51 

67.81 

18.8 

28.0 

17.7 

3.31 

Saigon, India, . 

62.80 

5.86 

28.9 

28.0 

0.04 

Yokohama, Japan,. 

35.02 

7.52 

12.0 

15.2 

0.295 

Manilla, Philip. Islands, . . 

71.31 

4.77 

34.1 

25.6 

4.84 


For rain-fall, see page 491. 
































Navigation. 


159 


TRAVERSE SAILING AND SURVEYING. 


To navigate a vessel upon the supposition that the earth is a level plane, on 
which the meridians are drawn north and south, parallel with each other; and 
the parallels east and west, at right-angles to the former. 

The line NS represents a meridian north and 
south; tliQ. line WE represents a parallel east 
and west. 

A ship in l sailing in the direction of l V, and 
having reached V, it is required to know her 
position to the point l, which is measured by the 
line ll', and the angle Nil'; and located by 
the lines l a and a V 

While the vessel is running from l to V, the 
distance is measured by the log and time; and 
the course Nil' is measured by the compass 
commonly expressed in points. 

These four quantities bear the following names. 

d — 1l', distance from l to l' in miles. 

C = Nil', course, or points from the meridian, 
tl = l a, departure or difference in longitudes, in miles. 
u = a V, difference in latitudes, in miles. 

I — latitude in degrees. 

L — difference in longitude, in degrees or time. 

Traverse Formulas. 



tr — d sin.C, - 

- 

1, 

ft = u tan.C, - 

- 

2, 

13 = 60 cos./ L , 

- 

3, 

= V (P—u 1 , 

- 

4, 

u = d cos. C, - 

- 

5, 

u — h cot. C, - 

m 

6, 

_ 60Z, cos./ 

U tan. C * 

m 

7, 

u = s/ d* — 

m 

8, 

d — •—/-,» * 
sin.C 

- 

9, 

, u 

a = - • 

m 

10, 

COS. C, 



, 60 L cos./ 

d = -—.— - r% — > 
sin.C 

- 

11, 

d = V 

m 

12, 

cos./ = WL , - 

- 

13, 

, d sin.C 

"“•'“-eo l' 

- 

14, 


_? u tan.C 

cos.l =* , - 

15 

_ TJ 

60cos./ * 

16 

L^d sin. C 

60cos./ * 

17 

j 7/. tan.C 

60cos./ * 

18 

cos. C = -y , - 
d 

19, 

. r* 

sin.C =— - 

a 

20 

tan.C 

u 

21, 

. „ 60 L cos./ 

sin.C — , 

d 

22, 

, r, 60 L cos.l 

tan. C — 

23, 


U 



























Land Surveying. 


160 


Example, 1. A vessel sails east-north-east (6 points) 236 miles. Required her 
departure b, and difference in latitude u. (See page 162.) 

Formula 1. b = d sin. C= 236 X sin. 6 points = 218 miles departure, and u = d 
cos. c. = 236 X cos. 6 points = 90.3 miles difference in latitude. 

Example 2. A ship sails in north latitude in a course C= i?>S7?fi7 = 6J points; 
at a distance of 132 miles she made a difference in longitude of L = 3° 34'. VI hat 
latitude is she in? . 


Formula 14. 


cos. I = 


d sin .C 
60 L 


132 X sin. 6 J 

co x m°) 


= 0.59832; 


or 1 = 53° 15' the latitude. 

In high latitudes and very long distances, the preceding formulas will not give 
oorrect results. (See Spherical Trignometry.) 


LAND SURVEYING. 

Application of formulas on the preceding page. 



The operation is readily understood by the illustration. When only an azimuth 
compass is used, the course C at each station is measured from the magnetic needle 
or meridian to the direction of the survey. When a theodolite is employed, the 
course C is read as carefully as possible from the compass at the first station, but 
at the second station the angle v between the distances is measured, from which 
subtract the first course, and the remainder will be the second course. At the 
third station subtract the second course from the angle between the distances, 
and the remainder will be the third course, and so on. The calculated course is 
compared with that shown by the compass at each station; if a difference is ob¬ 
served, there may be some errors in the subtraction or angle measurement, or 
some local attraction of the magnetic needle, which is sometimes the case near 
great deposits of iron ores. The angles and courses are measured by the theodo- 
lito because they cannot be read so delicately on the compass. 

At the 5th station, where the 4th and 6 th stations are on the same side of the 
meridian and both north of 5, add the 4th course to the angle 4, 5, 6 , and the 
sum is the new course. On return to the 1st station, where the 7th and 2d sta¬ 
tions are both on the same side of the meridian, and one north and the other 
south, add the angle 2 , 1,7. to the 7th course, subtract the sum from 180°, and 
the remainder should be the 1 st course, which shows the accuracy of the survey. 

At station 1 when a transit is used the vernier is clamped at zero, and then the 
telescope turned until the compass needle is at zero, and the instrument clamped 
below. The vernier is then loosened and the angle read both from needle and 
vernier. At the 2d station the telescope is transited for a back sight, fixed below, 
the vernier loosened again, and the course read from vernier and needle as before. 
This is called traversing. 


















Traverse Table. 


161 


Traverse Table for the Survey. 


Sta- 

Course 

Sin. or cos. 

Dist. 

Latitude. 

Departure. 

tion. 

c. 


C. 


d. 

N. 

s. 

E. 

w. 

1 

N. 35°42 / E., 

\ 

cos. 8121 ' 
sin. 5835 


200 

162.42 

• • • 

116.70 

• • • 

2 

S. 63 43 E., 

\ 

' cos. 4415 
sin. 8972 


185 

• • • 

81.68 

165.98 

• • • 

3 

N. 68 38 E., 

\ 

cos. 3643 
sin. 9312 


263 

95.81 

• • • 

244.90 

• • • 

4 

S. 42 25 E., 


‘ cos. 7382 
sin. 6747 


228 

• • • 

168.31 

153.78 

• • • 

5 

N. 85 51 W., 

\ 

cos. 0723 
sin. 9974 


223 

16.12 

• • • 

• • • 

222.42 

6 

S. 72 18 W., 


cos. 3040 
sin. 9526 


321 

• • • 

97.58 

• • • 

305.78 

7 

N. 64 27 W., 

J 

f cos. 4313 
sin. 9022 


170 

73.32 

• • • 

• • • 

153.37 

Sum of N. S. E. and W., 

• 

• 

347.67 

347.57 

681.36 

681.57 

Subtract the smallest, 


• 

347.57 



681.36 

Errors in the measurement, 

• 

• 

0.10 



0.21 


Find the natural sines and cosines in the trigonometrical tables. 

The distance, d , multiplied by the cosine for the course C, will be the difference 
in latitude formula 5. 

The distance, d, multiplied by the sine for the course C y will be the departure 
formula 1. 

The formulas and traverse table will answer for any unit of measure, but if the 
above traverse bad been made in miles, whether on land or sea, each departure 
should be divided by cosine for the mean latitude between each two stations, formula 
16, in order to obtain the true difference in longitude. To divide by cosine is the 
same as to multiply by the secant for the same angle. 


Length of a Degree in Parallel of Latitude. 

Multiply the length of a degree at the equator (60 sea-miles = 69.03 statute 
miles = 110.83 kilometres) by cosine for the latitude, and the product will be the 
length of a degree in parallel of latitude. 

The length of a minute or second at the equator, multiplied by the cosine for 
the latitude, will be the corresponding length in the parallel of that latitude. 


Measurement over Sloping Ground. 


d = Sloping dis¬ 

b = Base, or hori¬ 

h = Difference in 

v = Angle of the 

tance. 

zontal distance. 

height. 

slopes. 

d = h cosec. v. 

b — d cos. v. 

h = d sin. v. 

h 

sin. v —-. 

d 

d = b sec. v. 

b = h cot. v. 

h = b tan. v. 

tan. v= 

b 


The horizontal distance b is equal to the inclined distance d, multiplied by 
cosine for the sloping angle v. 

The vertical height h is equal to the inclined distance d, multiplied by 
sine of the sloping angle v. 

The natural sine and cosine for any slope will be found in the tables. 


11 























162 Mariners’ Compass. 













































































Navigation. 


163 


170 



Distance and Dip of Horizon, 

from different heights above the surface of the ocean. 


Height. 

Distance. 

Dip. 

Height. 

Distance. 

Dip. 

Height. 

Distance. 

Dip. 

Feet. 

Miles. 

f 

n 

Feet. 

M iles. 

! 

tf 

Feet. 

Miles. 

0 • " 

0.582 

1 mile. 

0 

59 

16 

6.29 

3 

56 

150 

16.22 

0 14 07 

1* 

1.31 

0 

59 

17 

5.45 

4 

03 

200 

18.72 

0 16 18 

2 

1.87 

1 

24 

18 

6.61 

4 

11 

300 

22.91 

0 19 56 

3 

2.29 

1 

42 

19 

6.77 

4 

17 

400 

26.46 

0 23 03 

4 

2.63 

1 

68 

20 

6.92 

4 

24 

500 

29.58 

0 25 46 

5 

2.96 

2 

12 

25 

6.61 

4 

55 

1000 

32.41 

0 28 18 

6 

3.24 

2 

25 

30 

7.25 

6 

23 

2000 

59 20 

0 51 42 

7 

3.49 

2 

36 

35 

7.83 

5 

49 

3000 

72.50 

1 3 24 

8 

3.73 

2 

47 

40 

8.37 

6 

14 

4000 

83.70 

1 14 15 

9 

3.96 

2 

57 

45 

8.67 

6 

36 

5000 

93.50 

1 21 54 

10 

4.18 

3 

07 

50 

9.35 

6 

58 

1 mile. 

96.10 

1 24 01 

11 

4.39 

3 

16 

60 

10.25 

7 

37 

H “ 

108.96 

1 35 40 

12 

4.58 

3 

25 

70 

11.07 

8 

14 

2 “ 

123.23 

1 48 20 

13 

4.77 

3 

33 

80 

11.83 

8 

48 

2£ “ 

140.64 

2 3 50 

14 

495 

3 

41 

90 

12.55 

9 

20 

3 “ 

154.10 

2 15 50 

15 

5.12 

3 

49 

100 

13.23 

9 

51 

5 “ 

199.15 

2 57 15 


* For smaller heights, see Curvature of the Earth. 

The refraction is included in the dip of horizon. 

The distance being the tangent a b in statute miles, at the elevation a c, in feet. 

Example 1. The lighthouse at a is 100 feet above the level of the sea. Required 
the distance a b. 

Height 100 feet = 13.23 miles. 

Example 2. The flag of a ship is seen from a in d. Required the distance a d, 
when the flag is known to be 50 feet above the level d’ of the sea? 

Height of the light 100 = 13.23 miles a b, 

Height of the flag 50 = 9.35 “ b d. 

Distance to the ship = 22.58 miles a d. 

Example 3. A steamer is seen at e; the horizon b seen in the masts is assumed 
to be 16 feet above the level e'. Required the distance to the ship ? 

Height of the light 100 = 13.23 miles a b , 

The assumed height 16 = 5.29 “ e 5, 

Distance to the ship = 7.94 miles a e. 


Particular attention is called to page 159, to find the distance d, course C, 
departure fr, difference in latitude u, and difference in longitude L. When the 
course C is given by the compass, use the compass table on page 162 for sin.C, 
cos.C, and lan.C, which is handier than the full trignometrical tables. 
































f64 


Curvature of the Earth, 


CORRECTION FOR CURVATURE OF THE EARTH 

IN LEVELING. 

Notation. 

D s distance in miles from the level to the stave or other object, and 
d = the same distance in feet. 

C = correction for curvature in feet at the stave ; always negative, 
c = the same correction in inches. 


c= 2 ^ 

3 

d 7 


3486643 


z> = 1 . 22474 /c. 

d = 1867.3 ]/£ 


The accompanying table gives the curvature for distances from 100 feet to 20 
miles. For greater distances see table of Distances and Dip of Horizon. 

Difference of Apparent and True Level or Cmvature of the 
Earth, with and without Refraction. 


Distance. 

Curvature. 

Curv. and ref. 

Distance. 

Curvature. 

Curv. and ref. 

Feet. 

Inches. 

Feet. 

Miles. 

Feet. 

Feet. 

100 

.0028 

.0002 

1 

0.606 

0.575 

200 

.0115 

.0008 

2 

2.666 

2.283 

300 

.0258 

.0018 

3 

6.000 

5.141 

400 

.04S9 

.0033 

4 

10.675 

9.150 

600 

.0717 

.0051 

6 

16.675 

14.291 

600 

.1032 

.0073 

6 

24.083 

20.583 

700 

.1405 

.0100 

7 

32.683 

28.167 

800 

.1835 

.0130 

8 

42.691 

36.591 

900 

.2223 

.0158 

9 

54.025 

46.031 

1000 

.2868 

.0204 

10 

66.700 

57.175 

1500 

.6453 

.0459 

11 

80.708 

69.175 

2000 

1.147 

.0817 

12 

96.050 

82.325 

2500 

1.792 

.1276 

13 

112.716 

96.616 

3000 

2.581 

.1836 

14 

130.732 

112.058 

3500 

3.513 

.2500 

15 

150.075 

126.633 

4000 

4.589 

.372 

16 

170.750 

147.191 

4500 

6.557 

.396 

17 

192.766 

165.225 

5000 

7.170 

.5110 

18 

216.108 

185.233 

5500 

8.676 

.6185 

19 

240.783 

206.391 

6000 

10.324 

.7300 

20 

266.800 

228.683 


























Divergency of the Parallel. 


1G5 


TO FIND THE DIVERGENCY OF THE PARALLEL 

FROM THE PRIME VERTICAL. 

Notation. 

I = latitude of the parallel in degrees. 

v = distance on the prime vertical, expressed in angle of the great circle from 
the base-meridian. r 

c = divergency in feet of the parallel at the angle v. 

c = 729000 sin. 2 X h 

The divergency is calculated in the accompanying table for distances from one 
second to one degree, also expressed in feet and miles on the prime vertical. The 
coefficient c = 729000 sin 2 ?v, which, multiplied by the latitude of the parallel in 
degrees, gives the divergency in feet. 

Example 1. Suppose the distance on the prime vertical to be v = & — 6 miles 
and 4770 feet, the latitude of the parallel being 48°. Required the divergency. 

Prom the table, 0.5551 X 48° = 26.6448 feet, the divergency required. 


Divergency of the Parallel from the Prime Vertical. 


Distance on prime vertical. 

Coefficient. 

Distance on 

prime vertical. 

Coefficient 

Seconds v. 

Feet. 

C. 

Minutes v. 

Miles. 

Feet. 

C. 

1 

101.25 

0.00000434 

1 

1 

795 . 

0.0154213 

2 

202.5 

0.00001735 

H 

1 

3832.5 

0.0346979 

3 

303.75 

0.00003855 

2 

2 

1590 

0.061685 

4 

405 

0.00006916 

2 h 

2 

4627.5 

0.0964 

5 

506.25 

0.0001071 

3 

3 

2585 

0.1387917 

6 

607.5 

0.0001542 

4 

4 

3180 

0.24674 

7 

708.75 

0.0002099 

5 

5 

3975 

0.3855 

8 ' 

810 

0.00027665 

6 

6 

4770 

0.55516 

9 

911.25 

0.0003470 

7 

8 

285 

0.75564 

10 

1012.5 

0.0004284 

8 

9 

1080 

0.98696 

11 

1113.75 

0.00051833 

9 

10 

1875 

1.2491253 

12 

1215 

0.0006168 

10 

11 

2670 

1.5420 

13 

1316.25 

0.00072394 

11 

12 

3465 

1.865820 

14 

1417.50 

0.0008396 

12 

13 

4260 

2.220604 

15 

1518.75 

0.0009638 

13 

14 

5055 

2.6062 

16 

1620 

0.0010966 

14 

16 

570 

3.02256 

17 

1721.25 

0.0012380 

15 

17 

1365 

3.4696 

18 

1822.5 

0.0013879 

16 

18 

2160 

3.94783 

19 

1923.75 

0.0015464 

18 

20 

3750 

4.9965012 

20 

2025 

0.0017135 

20 

23 

GO 

6.1680 

25 

2531.25 

0.002677 

25 

28 

4035 

9.637500 

30 

3037.5 

0.0038553 

30 

34 

2730 

13.8785 

35 

3543.75 

0.0052475 

35 

40 

1425 

18.8895 

40 

4050 

0.0068539 

40 

45 

120 

24.6720 

45 

4556.25 

0.0086742 

45 

51 

4095 

31.22815 

50 

5062.5 

0.010709 

50 

57 

2790 

38.5500 

55 

5568.75 

0.012958 

55 

63 

1485 

46.6455 

60 

0075 

0.154213 

60 

69 

180 

55.5151 


The length of minutes and seconds on the parallel is equal to that in the table, 
multiplied by cosine for the latitude. 

These calculations are necessary in running a parallel of latitude by fore and 
back sighting, and also for laying out the parallels and meridians on a map. 

Empirical Formulas. 

In making use of empirical formulas, whether derived by the method of 
interpolation following or otherwise, it must always be remembered that they 
are valueless beyond the limits of the observation on which they are based. 





















166 


Interpolation. 


Interpolation is to insert numerical values between given data, for 
constructing tables or empirical formulas expressing the probable rela¬ 


tive variation of quantities. Let x and y be two variable quantities de- j 
pending on one another and measured in simultaneous stages of their 


progress, as 


x x x 2 x 3 Xi and x 0 

2 /i 2 h 2/3 2/4 and 2 /* 


0.1 ini = I 


We have y = Ayi+ByiYCyArDyfaEy-l&cc,. 


- - 1 


A= 


2 3 

V V 

fa — a;.,) fa — as 3 ) fa — *.,) (x 


4 

V 


■x b ) 


6 given 
N/data. 


B= 


fa\ X. 2 ) fa— X a ) fa £C 4 ) fal—X b ) 

(i X - Xi) ( X — cc 3 ) ( X — x A ) (x Xj) 


ci 


2 > - - 


far—Xi) fa — x,) fa,— x t ) fa *— X 6 ) 

_ J I I 


G 

d> 


0 = 


fa — Xi) fa — x. 2 ) fa — ic 4 ) (a; — x b ) 


to 


3> 


D = 


fa 3 — X x ) fa 3 — x,) fa 2 —X A ) fa.— X,) 

_ _I I 

fa — Xi) fa — X 2 ) fa —SC 3 ) fa — x b ) 


U» 

0 > 

e 4> 

S E = 


fai—xj fai—x.,) fai— x a ) t fa,— * 6 ) 

__i 

fa — X x ) fa — x a ) fa — x a ) fa —» 4 ) 


6 >- 


(as*—asi) fa-—x_) fa a —x 3 ) fa b —x^ 


J 


The values of the coefficients A, B, C, D, and E, with their given data, 
inserted in formula 1 gives an empirical formula for the variation of x 
and y. The number of observations or given data of x and y should be 
one more than the order of progression. In arithmetical progression 
two observations are sufficient for a correct formula. For all curves in 
the conic sections, or others which are of the second order, there should 
be at least three observations. Pressure of steam progresses with the 
temperature in the 6th order, for which requires seven observations to 
make a correct formula. When the order of progression is not known, 
the more observations gives the most correct result. 


Example. Let y represent the boiling-point of salt water and x the j 
percentage of salt in solution. It is found in three experiments, 


that xi=3, 
when i/i=213-2 


0-2 = 18, 
1 / 2 = 219 °, 


#3=36 per cent. salt. 
2/3=226° boiling-point. 


Find a formula that will give any intermediate value of x and yl 


(#—18)(:r—36) 


(3—18/(3—36) 




(x —3) (a;—36) 


(18—3)(18—36)’ 


c= 3 )(®— 18 ) 

(36—3) (36—18)’ 


2/=213-2 A +219 £+226 C. y=0‘4a:+212 























Frogs, 


167 
































































































































































































































































































































































































































































































































































































































































































































































































































































































168 


Trigonometry. 


TRIGONOMETRY. 

Trigonometry is that part of Geometry which treats of triangles. It is divided 
into two parts— viz., plane and spherical. 

Plane Trigonometry treats of triangles which are drawn (or imagined to be) on 
a plane. Spherical Trigonometry treats of the triangles which are drawn (or 
imagined to be) on a sphere. 

A triangle contains seven quantities—namely, three sides, three angles and the 
surface. When any three of these quantities are given, the four remaining ones 
can by them be ascertained (one side or the area must be one of the given quanti¬ 
ties), and the operation is called solving the triangle , which is only an application 
of arithmetic to geometrical objects. 

Example 1. Fig. 173. An inclined plane a = 150 feet long, and c = 27 feet, the 
height over its base. What is the angle of inclination C = ? 

Formula 14. sin.C — — = = 0.18000. 

a 150 

Find 9.1S000* in the table of sines, which will be found at 10°30', which is the 
angle C nearly. 

Example 2. Fig. 174. An oblique-angled triangle has the sides c = 27.6 feet, the 
angle C = 34° 10', and the angle A = 47° 40'. How long is the side a = ? 

Formula 1. a = c “M- = 27 -6 X sin.47°4 0 ' = 36 _ 3a feet> the answer . 
sin.C sin. 34° 10' 


By Logarithms. 

log .a = log.c + log.sin.A — log.sin.C'. 
c + log. 27.G = 1.44090 

A + log. sin. 47° 40' = 9.86878 

1.30968* 

C — log. sin. 34° 10' = 9.74942 

log. 36.4 = 1.56026, or a = 36.4 feet 

Example 3. Two ships A and B are 800 yards apart; the angles between each 
other and a castle C are A — 63° 45', B = 75° 50'. What are the two distances from 
the castle ? 

C --= ISO — 63° 4o'— 75° 50' = 40° 25'. 


To A the distance will be, 
c sin.JB 


5 = 


sin.C 


To B the distance will be, 

csin.A 
a — - 

sin.C 


Form. 4. 

800 X sin . 7 5° 50' 
sin. 40° 25' 

Form 1. 

800 X sin. 63° 45' 


1195.75 yards. 


sin. 40° 25' 


/ = 1106.6 yards. 


* The index of a logarithm for a fraction is negative; but in the logarithms for 
the trigonometrical functions, 10 is added to the index, lor which it appears so 
much less than 10 as the real negative index. Therefore, when trigonometrical 
logarithms are added, 10’s must be rejected from the sum of ihe index, which will 
be understood by the examples. 


cot.i? = "-- — cot.^4. 

b cos.4 


tan.A = 


a sin.C 


b + a cos.C" 


4 

















Trigonometry, 


169 



172 




3 Versed sine 

4 Coversed sine 

5 Tangent 

6 Cotangent 

7 Secant 

8 Cosecant 


u 

a. 

« 

« 

a 

tt 


ver.sin. C. 

co.ver.sin.C. 

tan.C. 

cot.C. 

sec.C. 

cosec. C. 


r = Radius of the circle, which is the unit by which the functions are mea» 

sored. 


tan.C 


tan.C 


r 2 = sin. 2 C+cos. 2 C. 
sin.C 


cos. C* 

1 

cot. C’ 


~ cos. C. 

COv* v —* • 7^| t 

sin. C 

1 


cot.C = 


tan.C’ 


sec.C 

me 

cosec. C = —: 


cos. C ’ 

1 


sin.C * 

sinv.C =1 — cos .C, 

cosv.C = 1 — sin. C, 

sin.2C = 2 sin.C cos.C, 

sin.|C = i/7sin. 2 C+sinv. 2 C), 

sin.(C+I?) = sin.C cos. B+ 
sin.2?cos.C. 

Positive and, Negative Signs. 


Angles. 

sin. 

cos. 

ver.sin. 

co.ver. 

sin. 

tan. 

cot. 

sec. 

cosec. 

+0° 

+0 

+1 

+0 

+i 

+0 

+ GO 

+1 

+ 00 

t-90 0 

+ 1 

+0 

+1 

+o 

+QO 

±° 

+00 

+ 1 

+180° 

±0 

-1 

+2 

+i 

To 

Too 

—1 

+00 

+270° 

—1 

To 

+1 

+2 

+00 

+ 

+oo 

— 1 

+360° 

T° 

+i 

+0 

+1 

To 

-00 

+1 

—QO 


YVllBlittminuuij -- — - 

it cannot he increased or diminished. 

Example. What is the length of the secant for an angle of 74° 18'? 


Secant C = 


cos. 74° IS' 


= 3-695. 









































170 


Right-Angled Triangle. 


FORMULAS FOR RIGHT-ANGLED TRIANGLES. 

173 


B 


O', 


a = v/6*+c*, 

_ c 
° sin. C’ 

b 

a ~ cos .C’ 


a = 2. /_%_, 

V sm.2C 

b = a cos. C, 
b = c cot. C, 
b = a sin.2?, 
b—c tan .B, 


tan.C 



1 , 

2 , 

3, 

4, 

5, 

6 , 

7, 

8 , 

9, 



n a* sin.2C 

g- ? ’ 

Q = £ J^tan. C, 

Q = £ c’ cot. C, 

Q = i c \f (a+c)(a c) 

sin. C = — , 

a 


cos. C = —, 
a 


tan. C = , 

4Q 

sin.2C= — , 

or * 

2 Q 

tan.C = 


15, 

10 , 

17, 

18, 


Say the angle to be C = G0°. In the first column of the table of sines, 60° 
corresponds with 0 , SCG02 in the next column, which is the length of sin. G0°, 
when the radius of the circle is one, or the unit, and the expression sin. 60°X36 
means 0-86602X36 = 31-17672, and likewise with all the other Trigonometrical 
expressions. 

In a triangle the functions of an angle have a certain relation to the oppo¬ 
site side; it is this relationship which enables us to 6 olvo the triangle by the ap¬ 
plication of Simple Arithmetic. 

In triangles the sides are denoted by the letters a, b, and c ; their respective 
opposite angles are denoted by A, B, and C, and the area by Q. 

Example 1. Fig. 173 The side c in a right angled Trianglo being 366 feet, and 
the angle C = 39° 20'. IIow long is the side a =1 


Formula 2. 


a = 


365 


365 


sin.C 


sin. 39°20* = 0-63383 = 575 ‘ 86 feet ’ the an,wer * 




























Oblique-angled Triangle. 


171 



FORMULAS FOR OBLIQUE-ANGLED TRIANGLEa 


a : b = sin .A : sin. B, and b : c = sin .B : sin.C. 
a: c — sin. A : sin.C, and Q : ab — sin.C: 2. 


c sin. A 
a = sin. C * 

c sin.A 


a = 

b = 

b = 


sin.(A+i?)’ 
2 Q 

b sin.C’ 
c sin.J5 


sin.C ’ 
2 Q 

c sin.A’ 


. ~ c sin .B 

sin.C =-■-— , 

b 


sin.C = 


c sin.A 


2 Q 


sin.A = Tc , 

. a sin.C 
sin.A = .-, 


1, 

2 , 

3, 

4, 

5, 

6 , 

7, 

8 , 


S = i(a+b+c) 
sm.t^ = 


12 , 


on. * A = a/ 

sin .±B ■=</ ( s ~ a )( s - c ) , 14 , 

V a C 

u -v /3 ¥ 3 ’ ,5 - 

AEr^), 16, 

V ac 

17, 

ab sin.C 


cos 


cos 


n _bc sin.A 

V 2 » 


Q 


18, 


^ c’ sin.A sin .B -.q 

Q = ~TR(a+B) 1 
Q=y/S^a){S-b){S-c)S 20, 


b'+c 1 —26 c cos. A, 10, 
/^2Q sin.A. 1 1 

" - V 3nJ?lmTZ+5) ’ 


». i = . /2Ql|P+C), 21, 

Y sin.A sin.C 


I 2Q sin.C 22 
\/ sin.A sin.(A+C) 













































172 


Spherical Trigonometry. 


SPHERICAL TRIGONOMETRY. 

Spherical Trigonometry treats of triangles which are drawn (or im¬ 
agined to be) on the surface of a sphere. Their sides are arcs of the great circle 
of the sphere, and measure by the angle of the arc. Therefore the trigonometrical 
functions bear quite a different relation to the sides. 

Every section of a sphere cut by a plane is a circle. A line drawn through the 

centre and at right angles to the sectional circle is 
called an axis , and the two points where the axis 
meets the surface of the sphere are called thepofes 
of the sectional circle. 

When the cutting plane goes through the centre 
of the sphere, it will pass through the great circle, 
and is then called the Equator for the poles. 
Axis = N.S. Equator — G.E.T.W. 

Three great circle-planes, a a'a"a'", b b'b", and 
c c'c", cutting a sphere, JVESW, will form a solid 
angle at the centre O, and a triangle ABC on the 
surface of the sphere, in which the arcs a, b, c, are 
the sides. The angles formed by each two planes 
are congruent to each of the appertinent angles 
A, B and C. 

Spherical Distances. 



For the spherical distances, letters will denote, 
l = lower latitude, in degrees from the equator. 

V — highest latitude, “ “ “ 

C — course, from the latitude l to l'. 

C" = course, from “ l' to L 

d = shortest distance between l and V in degrees of the great circle. 
L — difference in longitude between l and l', in degrees, or time, 
tan. m = cot. V cos .L. 
n — 90 =f= l — m. 

— I, when l and l' are on one side of the equator. 

+ l, when l is on one side, and V on the other. Then 

7 sin. V cos. n 

COS.rf - y 9 9 9 

cos .m 


sin.C = 


sin. O' = 


sin .L cos.V 
sin .d 

sin .L cos .1 
sin .d ’ 



1 . 

2 . 

3. 


Example. 

pool. 


Required the shortest distance and course from New York to Liver- 

l = 40° 42' N. latitude, 1 w v , 

74°42' W. longitude,} New lork * 

V = 53° 22 ' N. latitude, 1 T 

2°52' W. longitude,} Lner P° o1 - 


L = 71° 8 ' difference in longitude, 
tan. m = cot. 53° 22 ' X cos. 71° 8 ' = 13° 31'. 
n = 90° — 13° 31'— 40° 42' = 35° 47'. 

Formula 1. cos. d = f ill - 53° 22' X cos. 35° 47' = 47 o 53 /^ 

cos. 13° 31" 

Shortest distance = 47° X 60 + 58 = 2878 geographical miles. 

sin.C' — f . in -^ 0 8'X c os.53°22' = 49023 ' _ 4 a points, 
sin. 47° 58' 

course from New York NE\E. 













Right-Angled Spherical Triangle 


173 



RIGHT-ANGLED SPHERICAL TRIANGLE. 


sin./> = sin.a sin.B, 

1, 

tan.c = tan.a cos .B, 

2, 

cot.C ~ cos.a tan.B, 

3, 

tan.c = sin./> tan.C, 

4, 

cos.a = cos./; cos.c, 

5, 

cos .B = cos. b sin.C, 

0, 

tan.& 

tan.a = - 

cos.C 

7, 

. tan. 6 

sin.c j J 3 j 

tan .B 

8, 

. sin. 5 

sin. a = — , 

sin.Lf 

9, 

. ~ cos .B 

sin.G = —- r , 
cos.a ’ 

10, 

cos. a 

cos.c = ,, 

cos. o’ 

11, 


sin.B = 

sin./; 
sin. a’ 

12 

cos.C = 

tan. 6 

tan.a 

13, 

tan.C = 

tan.c 

sin.Z>’ 

14, 

tan.2? = 

tan.i 

sin.c’ 

15, 

cos.c = 

cos. C 
sin.B’ 

16, 

cos.b = 

cos .B 
sin. C’ 

17, 

cos.a = 

cot. C 
tan.B’ 

18. 


The sum of the three angles in a spherical triangle is greater than two right 
angles, and less than six right angles. 

By Spherical Trigonometry we ascertain distances and courses on the surface 
of the earth ; positions and motions of the heavenly bodies, &c,, &c. Examples 
will be furnished in Geography and Astronomy. 

Example 1. Fig. 177 In a right-angled spherical triangle the side or hypothe- 
nuse a = 36° 20', the angle B = 68° 50*. How long is the side b = ? 

Formula 1. sin.6 = sin.a.sin.B — sin.36°20'Xsin.68°60'. 
a log.sin. 36° 20' = 9:77267 

B log.sin. 68° 50' = 9:96966 

The answer, log.sin. 33° 32'= 9:74233 or b = 33° 32'. 



















































Oblique angled Spherical Triangle. 


175 


OBLIQUE-ANGLED SPHERICAL TRIANGLE. 



tan.£(m-f-n)tan^(/n— n = tan.i(a + c)tani(a— c) 


tan.m = tan.c cos. A, 

- 

27, 

tan.C = 6in ' m tan 4 
sin.(£— m) 7 

m m 

- 28, 

_ cos.ccos.(£- 

COS.Cf =* ' 

-m) 

• • • 29, 

cos.wi 

» 

___ cos .a cos .m 

cos.n = 

cos.c 

m m 

• • 30, 

b = m+n. 



, cos.c tan.A 

cot.m =» —7 -» 

tan. a 

m m 

- • • 31, 


a+b + c ri A + B+C 


S 2 A . 2 ’ 


sin. i A = 
sin.^a =- 


y. 

s/ 


sin . (5 — c) sin. (5 — b) 

sin .b sin.c 


cos.S 00 s .(S — A) 
sin.B sin.C 


32, 

33, 


To Find the Area of a Spherical Triangle* 

Let Q he the area of the triangle in square degrees; if R — radius of the 
sphere, the length of one degree will be, 

“ W ’ OT ° ne SqUare degfee * 3^58* 


cot.^Q 

sin.£Q 


cot Ac cot.ia+cos.i? 
smJ? 1 

sin.^c sin.^a sin.B 
cos.££ * 


- 1 , 

- 2 , 






























176 


Analytical Geometry. 


ANALYTICAL GEOMETRY AND CONIC SECTIONS. 

An equation of a line is generally re¬ 
ferred to rectangular lines, A B = axis 
of ordinates and C D = axis of abscissas. 
The position of any point P in the curved 
line P I Q is defined by the rectangular 
distances, y the ordinate and x the abs¬ 
cissa; x and y are variables, depending 
on one another. Any change in either of 
them will produce a change in the other, 
in accordance with the formulae for the 
line. The position of a number of points 
can be determined, located and joined 
into the required line of the equation. 

The ordinate y generally constitutes the first member of the equation, and its 
value is determined by assumed values of the abscissa x. 

The junction of the two axes is called origin , and denoted by o. The line will 
not pass through the origin when the equation lias a constant term. 

Properties of Lillies Referred, to Rectangular Co-ordinates. 


A 

L Fig. 180 


I> - T 

C 

t \ - D 

H o 

Cr fi 

1 

j 


The tangent of any curve. 


. 


dx 


The subtangent of any curve, . . . H G = y -. • 

dy 


The normal of any curve, 


The subnormal of any curve, 


' ' PE - y ^l 1 + %' 

. GE=y&. . 

dx 


The point of inflection, I, where convex and 
concave curves tangent, or where a curve re¬ 
verses, is when. I . 


= o, or oo. 


Let z denote the length of any curve, then 
The radius of curvature of any curve is . 

The ordinate y is a maximum or minimum when 


dx 1 

dz = |/ dx 1 -f- dy 2 . 
R — d * 


dx d 2 y 
dx 


0 . . 


. • 1 . 
2 . 

. 3. 
4. 

. 5. 

6 . 

. 7. 

8 . 


(See Maxima and Minima.) 

A curve is convex to the axis of abscissas when the ordinate and second differen¬ 
tial coefficient have the same sign, but concave when either of them is positive and 
the other negative I Q is convex, and P I concave, to the abscissa C D. 

A Conic Section is the section obtained when a plane cuts a cone. 

The conic sections are of five different kinds, namely: 

1st. Triangle. When the plane cuts the cone through its axis. 

2d. Circle. When the plane cuts the cone at right angles to its axis. 

3d. Ellipse. When the plane cuts the cone obliquely, passing through the two sides. 
4th. Parabola. When the plane cuts the cone parallel to one side. 

5th. Hyperbola. When the plane cuts the cone at an angle to the axis less than 
the angle of the axis and the side of the cone. 













Conic Sections. 


177 





x sin z 
sin. 2 te 


J^a; sin. (z + r) + a sin.vj, 


. 5. 


This is the general formula for all conic sections. 


Any point P in any curve of the conic sections can he calcu¬ 
lated by the formula 5. The formula will, however, be greatly 
simplified for the different sections. 

For a parabola z + t> = 180°, and sin. z = sin. v, for which 


, ax sin. 2 v 

y = ——o—• 

sin. w 

The parameter of the ellipse, parabola, and hyperbola is the 
ordinate y passing through the focus of the curve, and is generally 
denoted by the letter p. 

The origin is the point where the rectangular co-ordinates meet, 
and from which the ordinates and abscissas are measured. The 
origin is generally denoted bv the letter o. 

The radius of curvature at the vertex of any conic section is 
equal to half the parameter. 


12 
















178 


Formulas for Conic Sections. 


182. 





Circle. Origin in Circumference and Diameter . 
y = V2rx — x*, r= - 

x = r + 1 / r- — y 2 , r = radius of the circle, 
x = abscissa and y — ordinate for the circle. 


183. 

Circle. Origin in the Centre. 

y = V r 2 — x 2 , x = VV 2 — y 2 , r = j/x 2 + 1/ 2 . 

The circle can be plotted by these form¬ 
ulas. 

__ - 

Circle Arc. Origin in the Arc. 

y — V a- + c x — x 2 — a, 
c 2_4^2 

a =-—-r— , the distance of the chord from 

8 h 

the centre. 


185. 





186. 



Cycloid. 

y = 0.6371^ x( 7 r d — x), e = 1.211 d, 

p =• 0.632 d. p — parameter. 

f— focus of the cycloid. 



187. Circle and Ellipse. 

If a circle be described on the minor axis of 
an ellipse, any ordinate drawn from the minor 
axis, such as a b and ac, will be 
ab : a c — n: m. 

When a circle is described on the major axis, 
we have 

de : df — n : m. 

An ellipse can be considered a circle seen in 
perspective. An ellipse seen in perspective along 
the major axis can appear like a circle. 

-J 























































Formulas for Ellipses. 


179 




188. 



Ellipse. 

. __ 

y = — V / 2 mx — x 2 . 
m 

m = major semi-ax is. 
n — minor semi-axis. 



189. 


Ellipse. 



m = major semi-axis. 
n = minor semi-axis. 


190. 

Focus and Parameter of an Ellipse. 


e = V'm 2 — n 2 , 


P 


2ft 2 

m 


e = distance from centre for focus f. 
p = parameter of the ellipse. 


191. 


Radius Vector of an Ellipse. 

R = 2m — r, r = 2m—R, 2 m=?R+r. 

The major axis is the sum of R + r. 


192. 


Radius of Curvature of an Ellipse. 

,2 


n 2 


r = 


R 


m 


m n 

m = major semi-axis. 
n = minor semi-axis. 


r = ip- 


193. 

The normal PN at any point of an el¬ 
lipse bisects the angle formed by the radii 
vectors R and r, and a line P T drawn 
through P at right angles to the normal N 
will be tangent to the ellipse in P. 

































180 


Formulas for Parabolas. 


194. 



Parabola. 


y = V'p x, 
b 2 


p = 4m, 

b* 

4 A' 


m = 


m — distance from focus f to vertex o. 




O'— 2G 


199. 

Shield’s Anti-Friction Carve. 

The line OB is the centre line of the 
B shaft. 

R = radius of the shaft. 





















































Formulas for Hyperbolas. 


181 




200 . 

In an hyperbola the transverse axis 
rn + m is equal to the difference between 
the two radii (R — r), 

2 m = R — r. r = R—2m. 


The hyperbola can thus be plotted. 


201 . 


Formula for the Hyperbola. 


y 


= -X / 2mxi-x 2 . n — 


m y 


m 


l // 2 m x+x 2 


m — major semi-axis. 
n = minor semi-axis. 


Parameter p ■ 


2 n 2 


m 



“d— Equilateral Hyperbola. 

y — V / 2mx + x 2 . n = m. e — mV' 2. 

A A and B B are at right angles and 
cal led asymptotes. 

Steam-indicator diagrams are approxi¬ 
mately equilateral hyperbolas on the ex¬ 
pansion and compression lines. 



203. 

Equilateral Hyperbola referred to * its 
Asymptotes. 

m 2 /— - 

y — m—X 2 xu. 

s 2x 3 

In this formula—£ ?/ = constant 
x = part of the stroke of piston. 
y = steam-pressure. 



204. 

Diameter of an Hyperbola. 

Every diameter P P' of an hyperbola 
is bisected in the centre by the minor 
axis n n. 

A A is called the transverse axis. 



205. 


A Tangent T T' to an Hyperbola bisects the 
Angle formed by the Radii R and r. 


a 


R(a + b) 
R+r * 


ff'*=a+b. 





































182 


Logarithms. 


LOGARITHMS, 

A Common or Briggs Logarithm is the exponent of the power to 

which 10 must be raised to give a certain number. Thus, 


* 

c 

g 

P 

ere 

p 


s- 

o 

H 

<-♦* 

: s 

. '■» 

j-i 

S3 - 

: o 


3 

log. 100 = 2 
log. 10000 = 4 
log. 5012 = 3.7 


because 10 2 =100. 

“ lot =10000.’ 

“ 103-7 = 5012. 


The unit of the logarithm is called the characteristic or index , and the decimal 
part is called the mantissa , the sum of the characteristic and mantissa is the Loga¬ 
rithm. 10 is the base of the system of Logarithms commonly used. 

It is not necessary that the base should be 10, it can be any number, but all 
the tables of Logarithms now in common use are calculated with 10 as the 
base. 

The nature of logarithms in connection with their numbers is such, that the 
index of the logarithm is always one less than the number of figures in the 
number, (when the base of the logarithm is 10,) as, 

index 5012 = 3 
mantissa 5012 = 0'7 

logarithm 5012 = 3‘7 

Let 10 be raised to any power x, and 

10* — a or log. a = x, 

■■ b or log. b — z. 


10 * 

Let the product of ab 


c and the quotient — = d. 

b 


ab = c 



or log. c = X -f 2 . 
or log. d — x — 2 . 

or log. m — 2 X log. a. 
or log. n — log. a : 3. 

Any number represented by the letters a, b, c, or rf, can be a power of 10, whose 
exponent is the logarithm of the number. Logarithms are given for numbers 
with three figures in the accompanying Table, by which any operation in Multi¬ 
plication, Division, Involution and Evolution can be performed by simple Addition 
or Subtraction of Logarithms. Tables of Logarithms are commonly more exten¬ 
sive, and calculated for any number of four or five figures, which would occupy 
too much room in this book; but by the proportional parts, the logarithm can be 
found by this Table, to four or five figures. The index of the logarithms does not 
appear in the Table, only the mantissa. It is easily remembered that the index, is 
one less than the number of figures in the number; then when the number is only 
one figure, the index is 0 ; and when the number is a fraction, the index is 
negative. 

When the logarithm is to be found fora fraction, we commonly have the frac¬ 
tion expressed in a decimal; and then the negative index is equal to one more 
than the number of ciphers before the first figure, and is sometimes placed after 
the mantissa; thus, 

log. 305 = 2-56229 log. 0*365 = -50229—1. 

log. 40-7 = 1-06931 log. 0-0407 = *66931—2. 

log. 7-59 = 0-88024 log 0-00759 = *88024—3. 

In the accompanying Table of Logarithms, for the trigonometrical lines the 
negative index is shown thus, 10 being added to the characteristic: 

log. sin. 35° 40' = log. 0-58306 = 9:76572. 









Logarithms. 


183 


To find the Logarithm of Numbers. 

Example 1. Find the logarithm of 45. 

To 45 in tho first column of the Table, answers 65321 in the next eolumn, 
which is the mantissa; index = 1 because 45 is two figures. 

Then, log. 45 = 1*65321, the answer. 

Example 2. Find the logarithm of 768? 

Opposite 76 in the first column, answers 88536 in the column marked 8 on the 
top or bottom. Index = 2 because 768 is three figures. 

Then, log. 768 2-8S536. 

Example 3. Find the Logarithm of 6846 ? 

. log. 6840 = 3-83505 

Proportional part, 64X0-6 == 384 

log. 6846 = 3"835434 the answer. 

To find the number for a given Logarithm. 

Example 1. What number answers to the logarithm 3*87157 ? 

In the Table you will find in the column of logarithms, that 

log. 7440 = 3-S7157. 

Example 2. What number answers to the logarithm 3-801884? 

Given logarithm 3-801884, 

Subt. nearest table log. 3 801400 = log. 6330, 

Divided by proportional part, 69|4S4| 7, 

6337 the req. numb. 

Multiplication by Logarithms. 

Rule. Add together the logarithms of the factors, and the sum is the loga- 
rithm of the product. 

Example 1. Multiply 425 by 48. 

To log. 425 = 2-62839, 

Add log. 48 = 1-68124, 

The product, log. 20400 = 4-30963. 

Example 2. Multiply 79600 by 0-435. 

To log. 79600 = 4-90091, 

Add, log. 0-435 - -63848— 1, 

The product log. 34690 = 4-53939. 

Division by Logarithms. 

Rule. From the logarithm of the dividend subtract the logarithm of the di¬ 
visor, and the difference is the logarithm of the quotient. 

Example 1. Divide 43800 by 368. 

From log. 43800 = 4-64147, 

Subtract log. 368 = 2-56584, 

The quotient log, 119 = 2 07563. 

Example 2. Divide 36 by 0.625. 

From log. 36 = 1-55636, 

Subtract, log. 0*625 = *79588-1. 

The quotient, log. 57-6 = 1*76048. 

A negative index has the reverse operation of its mantissa, as if the mantissa is 
subtracted, add the negative index, and vice versd. 

Involution by Logarithms. 

Rule. Multiply the logarithm of the number by its exponent, and the pro¬ 
duct is the logarithm of the power of the number. 

Evolution by Logarithms. 

Rule. Divide the logarithm of the number by the index of the root, and the 
quotient is the logarithm of the root of the number. 
















184 


Logarithms or Numbers. 


]?SO. 100 to 1600. Logarithms. 00000 to 20412^ 


No. 

0 

1 • 

2 

3 

4 

5 

6 

7 

8 

9 


43 

100 

00000 

00043 

00087 

00130 

00173 

00217 

00260 

00303 

00346 

00389 

l 

4 

101 

0432 

0475 

0518 

0561 

0604 

0047 

0680 

0732 

0775 

0817 

2 

9 

102 

0860 

0903 

0945 

0088 

1030 

1072 

1115 

1157 

1190 

1242 

3 

13 

103 

1284 

1326 

1368 

1410 

1452 

1494 

1536 

1578 

1620 

1662 

4 

17 

104 

1703 

1745 

1787 

1828 

1870 

1912 

1953 

1005 

2036 

2078 

5 

22 

105 

02110 

02160 

02202 

02243 

02284 

02325 

02306 

02407 

02449 

02400 

6 

26 

10G 

2531 

2572 

2612 

2653 

2604 

2735 

2776 

2816 

2857 

2808 

7 

30 

107 

2938 

2970 

3019 

3060 

3100 

3141 

3181 

3222 

3262 

8302 

8 

34 

108 

3342 

3383 

3423 

3403 

3503 

3543 

3583 

3623 

3663 

3703 

9 

39 

100 

3743 

6182 

3822 

3862 

3002 

3041 

3081 

4021 

4060 

4100 


4 | 

110 

04139 

04179 

04218 

04258 

04207 

04336 

04376 

04415 

04454 

04403 


Til 

111 

4532 

4571 

4610 

4650 

4689 

4727 

4766 

4805 

4844 

4883 

1 

4 

Q 

112 

4922 

4961 

4909 

5038 

5077 

5115 

5154 

5192 

5231 

5200 


O 

113 

5303 

5346 

5385 

5423 

5461 

5500 

5538 

5576 

5614 

5652 

O 

1Z 

114 

5690 

5729 

5767 

5805 

5843 

5881 

5018 

5956 

5904 

6032 


lb 

115 

06070 

06108 

06145 

06183 

06221 

06258 

06296 

06333 

06371 

06408 

O 

Ik 

116 

G44G 

6483 

6521 

6558 

6505 

66113 

6670 

6707 

6744 

6781 

b 

lb 

117 

6819 

6856 

6893 

6930 

6967 

7004 

7041 

7078 

7115 

7151 

7 

Z\f 

118 

7188 

7225 

7262 

7298 

7335 

7372 

7408 

7445 

7482 

7518 

o 

OO 

110 

7555 

7501 

7628 

7664 

7700 

7737 

7773 

7800 

7846 

7882 

9 

61 

120 

07018 

07054 

07900 

0S027 

08063 

08090 

08135 

08171 

08207 

08243 


39 

121 

8279 

8314 

£350 

8386 

8422 

8458 

8493 

8520 

8565 

8600 

i 

4 

122 

8636 

8672 

8707 

8743 

8778 

8814 

8849 

8884 

8920 

8955 

2 

8 

123 

8901 

9026 

9061 

9096 

9132 

0167 

9202 

9237 

9272 

9307 

3 

12 

124 

o:i42 

0377 

9412 

0447 

0482 

9517 

9552 

0587 

9621 

9656 

4 

16 

125 

09691 

00726 

09760 

09795 

09830 

09864 

09809 

00934 

09068 

10003 

5 

20 

12G 

10037 

10072 

10106 

10140 

10175 

10209 

10243 

10278 

10812 

0346 

6 

23 

127 

0380 

0415 

0449 

0483 

0517 

0551 

0585 

0619 

0653 

0687 

7 

27 

128 

0721 

0755 

0789 

0823 

0S57 

0890 

0924 

0058 

0002 

1025 

8 

31 

120 

1059 

1093 

1126 

1160 

1193 

1227 

1261 

1294 

1327 

1361 

9 

35 

130 

11394 

11428 

11461 

11404 

11528 

11561 

11504 

11628 

11661 

11604 


37 

131 

1727 

1760 

1793 

1826 

1S60 

1803 

1026 

1050 

1902 

2024 


132 

2057 

2090 

2123 

2156 

2189 

2222 

2254 

2287 

2320 

2352 

1 

4 

133 

2385 

2418 

2450 

2183 

2516 

2548 

2581 

2613 

2646 

2678 

2 

7 

134 

2710 

2743 

2775 

2808 

2840 

2872 

2905 

2037 

2969 

3001 

6 

11 

135 

13033 

13066 

13098 

13130 

13162 

13194 

13226 

13258 

13200 

13322 

4 

15 

13G 

3354 

3386 

3418 

3450 

3481 

3513 

3545 

3577 

3600 

8640 

5 

19 

137 

3672 

3704 

3735 

3767 

3790 

3830 

3862 

3803 

3925 

3056 

6 

22 

138 

3988 

4010 

4051 

4082 

4114 

4145 

4176 

4208 

4230 

4270 

7 

26 

130 

4301 

43.i3 

4364 

4395 

4120 

4457 

4489 

4520 

4551 

4582 

8 

80 

140 

14613 

14644 

14675 

14706 

14737 

14768 

14799 

14829 

14860 

11891 

9 

3o 

141 

4022 

4953 

4983 

5014 

5045 

5076 

5106 

5137 

5168 

5198 


35 

142 

5229 

6259 

5290 

5320 

5351 

5381 

5412 

5442 

5473 

5503 

1 

4 

143 

5534 

5564 

5591 

5625 

5655 

5685 

5715 

5746 

577G 

5806 

2 

7 

144 

5836 

5866 

5897 

5927 

5957 

5987 

6017 

6047 

6077 

6107 

3 

11 

145 

16137 

16167 

16197 

16227 

16256 

16286 

16316 

16346 

16376 

16406 

4 

14 

14G 

6435 

6465 

6405 

6524 

6554 

6584 

6613 

6643 

'6673 

6702 

5 

18 

147 

G732 

6761 

6701 

6820 

6850 

6879 

6000 

6038 

6967 

6907 

6 

21 

148 

7026 

7056 

7085 

7114 

7143 

7173 

7202 

7231 

7260 

7289 

7 

25 

140 

7319 

7:348 

7377 

7406 

7435 

7464 

7403 

7522 

7551 

7580 

8 

28 

150 

17609 

17638 

17667 

17696 

17725 

17754 

17782 

17811 

17840 

17860 

9 

32 

151 

7898 

7926 

7955 

7984 

8013 

8041 

8070 

8009 

8127 

8156 



152 

8184 

8213 

8241 

8270 

4298 

8327 

8355 

8384 

8412 

8441 


33 

153 

8469 

8498 

8526 

8554 

8583 

8611 

8630 

8667 

8606 

8724 

L 

3 

154 

8752 

8780 

8808 

8837 

8865 

8893 

8921 

8049 

8977 

0005 

2 

7 

155 

19033 

19061 

10089 

19117 

10145 

10173 

19201 

19229 

10257 

10285 

6 

10 

156 

9312 

0340 

9368 

9396 

0424 

9451 

9470 

9507 

0535 

0562 

4 

13 

157 

9500 

9618 

9645 

0673 

0700 

0728 

9756 

9783 

0811 

9838 

5 

17 

158 

9866 

0893 

0921 

9948 

0076 

20003 

20030 

20058 

20085 

20112 

6 

20 

159 

20140 

20167 

20194 

20222 

20249 

0276 

0303 

0330 

0358 

0385 

7 

23 

No. 

0 

1 

2 

3 

4 

5 

6 

f-r 

/ 

8 

9 

8 

9 

26 

80 


















































Logarithms op Numbers.- 


185 


. . - - - -- • - 


No. 1600 to 2200. Logarithms. 20412 to 34242. 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


31 

160 

20412 

20439 

20466 

20493 

20520 

20548 

20575 

20602 

20629 

20656 

1 

3 

161 

0683 

0710 

0737 

0763 

0796 

0817 

0844 

0871 

0898 

0925 

2 

6 

162 

0952 

0978 

1005 

1032 

1059 

1085 

1112 

1139 

1165 

1192 

3 

9 

163 

1219 

1245 

1272 

1299 

1325 

1352 

1378 

1405 

1431 

1458 

4 

12 

164 

1484 

1511 

1537 

1564 

1590 

1617 

1643 

1669 

1G96 

1722 

5 

16 

165 

21748 

21775 

21801 

21827 

21854 

21880 

21906 

21932 

21958 

21985 

6 

19 

166 

2011 

2037 

2063 

2089 

2115 

2141 

2167 

2194 

2220 

2246 

7 

22 

167 

2272 

2298 

2324 

2350 

2376 

2401 

2427 

2453 

2479 

2505 

8 

25 

168 

169 

2531 

2789 

2557 

2814 

2583 

2840 

260S 

2866 

2634 

2891 

2660 

2917 

2686 

2943 

2712 

2968 

2737 

2994 

2763 

3019 

9 

28 

170 

23045 

23070 

23096 

23121 

23147 

23172 

23198 

23223 

23249 

23274 


&9 

171 

3300 

3325 

3350 

3376 

3401 

3426 

3452 

3477 

3502 

3528 

1 

o 

172 

3553 

3578 

3603 

3629 

3654 

3679 

3704 

3729 

3754 

3779 

2 

G 

173 

3805 

3830 

3855 

3880 

3905 

3930 

3955 

3980 

4005 

4030 

3 

9 

174 

4055 

4080 

4105 

4130 

4155 

4180 

4204 

4229 

4254 

4279 

4 

12 

175 

24304 

24329 

24353 

24378 

24403 

24428 

24452 

24477 

24502 

24527 

5 

15 

176 

4551 

4576 

4601 

4625 

4650 

4674 

4699 

4724 

4748 

4773 

6 

7 

17 

177 

4797 

4822 

4846 

4871 

4895 

4920 

4914 

4969 

4993 

5018 

20 

178 

5042 

5066 

5091 

5115 

5139 

5164 

5188 

5212 

5237 

5261 

8 

23 

179 

5285 

5310 

5334 

5358 

5382 

5406 

5431 

5455 

5479 

5503 

9 

26 

180 

25527 

25551 

25575 

25600 

25624 

25648 

25672 

25696 

25720 

25741 


27 

181 

5768 

5792 

5816 

5840 

5864 

5888 

5912 

5935 

5959 

5983 

i 

3 

182 

6007 

6031 

6055 

6079 

6102 

6126 

6150 

6174 

6198 

6221 

2 

5 

183 

6245 

6269 

6293 

6316 

6340 

6364 

6387 

6411 

6435 

6458 

3 

8 

184 

6482 

6505 

6529 

6553 

6576 

6600 

6623 

6647 

6670 

6694 

4 

11 

185 

26717 

26741 

26764 

26788 

26811 

26834 

26858 

26881 

26905 

26928 

5 

14 

186 

6951 

6975 

6998 

7021 

7045 

7068 

7091 

7114 

7138 

7161 

6 

16 

187 

7184 

7207 

7231 

7254 

7277 

7300 

7323 

7346 

7370 

7393 

7 

19 

188 

7416 

7439 

7462 

7485 

7508 

7531 

7554 

7577 

7600 

7623 

8 

22 

189 

190 

7646 

27875 

7669 

27898 

7 692 
27921 

7715 

27944 

7738 

27967 

7761 

27989 

7784 

28012 

7807 

28035 

7830 

28058 

7852 

28081 

9 

24 

25 

3 

191 

192 

8103 

8330 

8126 

8353 

8149 

8375 

8171 

8398 

8194 

8421 

8217 

8443 

8240 

8466 

8262 

8488 

8285 

8511 

8307 

8533 

1 

193 

194 

8556 

8780 

8578 

8803 

8601 

8825 

8623 

8847 

8646 

8870 

8668 

8892 

8691 

8914 

8713 

8937 

8735 

8959 

8758 

8981 

A 

3 

0 

8 

10 

13 

15 

18 

20 

23 

195 

29003 

29026 

29048 

29070 

29092 

29115 

29137 

29159 

29181 

29203 

4 

5 

196 

9226 

9248 

9270 

9292 

9314 

9336 

9358 

9380 

9403 

9425 

197 

9447 

9469 

9491 

9513 

9535 

9557 

9579 

9601 

9623 

9645 

o 

198 

9667 

9688 

9710 

9732 

9754 

9776 

9798 

9820 

9842 

9863 

7 

199 

9885 

9907 

9929 

9951 

9973 

9994 

30016 

30038 

30060 

30081 

o 

200 

30103 

0320 

30125 

30146 

30168 

30190 

30211 

30233 

30255 

30276 

30298 

y 

201 

0341 

0363 

03S4 

0406 

0428 

0449 

0471 

0492 

0514 


23 

202 

0535 

0557 

0578 

0600 

0621 

0643 

0664 

0685 

0707 

0728 

i 

2 

203 

0750 

0771 

0792 

0814 

0835 

0856 

0878 

0899 

0920 

0942 

2 

5 

204 

0963 

0984 

1006 

1027 

1048 

1069 

1091 

1112 

1133 

1154 

3 

7 

205 

31175 

31197 

31218 

31239 

31260 

31281 

31302 

31323 

31315 

31366 

4 

9 

206 

1387 

1408 

1429 

1450 

1471 

1492 

1513 

1534 

1555 

1576 

5 

12 

207 

1597 

1618 

1639 

1660 

1681 

1702 

1723 

1744 

1765 

1785 

6 

14 

208 

1806 

1827 

1848 

1869 

1890 

1911 

1931 

1952 

1973 

1994 

7 

16 

209 

2015 

2035 

2056 

2077 

2098 

2118 

2139 

2160 

2181 

2201 

8 

18 

210 

211 

32222 

2428 

32243 

2449 

32263 

2469 

32284 

2490 

32305 

2510 

32325 
253 L 

32346 

2552 

32366 

2572 

32387 

2593 

32408 

2613 

9 

21 

21 

2 

A 

212 

2634 

2654 

2675 

2695 

2715 

2736 

2756 

2777 

2797 

2818 

1 

213 

2838 

2858 

2879 

2899 

2919 

2940 

2960 

2980 

3001 

3021 

2 

o 

214 

3041 

3062 

3082 

3102 

3122 

3143 

3163 

3183 

3203 

3224 

6 

8 

11 

13 

T5 

215 

33244 

33264 

33284 

33304 

33325 

33345 

33365 

33385 

33405 

33425 

(\ 

216 

3445 

3465 

3486 

3506 

3526 

3546 

3566 

3586 

3606 

3626 


217 

3646 

3666 

3686 

3700 

3726 

3746 

3766 

3786 

3806 

3826 

G 

7 

218 

3846 

3866 

3885 

3905 

3925 

3945 

3965 

3985 

4005 

4025 

219 

4044 

4064 

4084 

4104 

4124 

4143 

4163 

4183 

4203 

4223 

8 

17 

No. 

0 

1 

2 

3 

4 

5 

G 

7 

8 

9 

9 

19 




































186 


Logarithms of Numbers, 


No. 2200 to 2800. Logarithms. 34242 to 44716. 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


20 

220 

34242 

34202 

342S2 

34301 

34321 

34341 

34361 

34380 

34400 

34420 

l 

2 

221 

4439 

4459 

4479 

4498 

4518 

4537 

4557 

4577 

4596 

4016 

2 

4 

222 

4035 

4055 

4074 

4694 

4713 

4733 

4753 

4772 

4792 

4811 

3 

6 

228 

4830 

4850 

4809 

4889 

4908 

4928 

4947 

4967 

4986 

5005 

4 

8 

224 

5025 

5044 

5004 

5083 

5102 

5122 

5141 

5160 

5180 

5199 

5 

1() 

225 

35218 

35238 

35257 

35276 

35295 

35315 

35334 

35353 

35372 

35392 

6 

12 

226 

5411 

6130 

6449 

5408 

5488 

5507 

5526 

6545 

5564 

65.83 

7 

14 

2-7 

5003 

5622 

6041 

5660 

5679 

5698 

6717 

6736 

6755 

6774 

8 

16 

228 

5793 

5813 

5832 

5851 

5870 

5889 

690S 

6927 

6946 

5905 

9 

IS 

220 

6084 

6003 

6021 

6040 

6059 

6078 

6097 

6116 

6135 

0154 



280 

36173 

36192 

30211 

36229 

36248 

36267 

36286 

36305 

36324 

30342 



231 

6361 

6380 

0399 

6418 

6436 

6455 

6474 

6493 

6511 

6530 

1 


232 

0549 

6508 

6586 

6605 

6624 

0642 

6661 

6680 

6698 

6717 


4 

233 

6730 

0754 

6773 

6791 

6810 

6829 

6847 

6866 

6S84 

6903 

O 

G 

284 

6922 

0940 

6959 

6977 

6996 

7014 

7033 

7051 

7070 

7088 

4 

b 

235 

37107 

37125 

37144 

37162 

37181 

37199 

37218 

3723G 

37254 

37 273 

6 

10 

230 

7291 

7310 

7328 

7346 

7365 

7383 

7401 

7420 

7438 

7467 

0 

11 

237 

7475 

7493 

7511 

7530 

7548 

7566 

75S5 

7603 

7621 

7G39 

7 

13 

23S 

7658 

7676 

7094 

7712 

7731 

7749 

7767 

7785 

7803 

7822 

8 

15 

239 

7840 

7858 

7870 

7894 

7912 

7931 

7949 

7967 

7985 

8003 

9 

17 

240 

38021 

38039 

38057 

38075 

38093 

38112 

38130 

38148 

38106 

3S184 


18 

241 

8202 

8220 

8238 

8256 

8274 

8292 

8310 

8328 

8346 

8364 

1 

2 

242 

8382 

8399 

8417 

8435 

8453 

8471 

8489 

8507 

8525 

8543 

2 

4 

243 

8501 

8578 

8596 

8014 

8632 

8650 

8668 

8686 

8703 

8721 

3 

6 

244 

S739 

8757 

8775 

8792 

8810 

882S 

8846 

8863 

8881 

8899 

4 

7 

245 

38917 

38934 

38952 

38970 

38987 

39005 

39023 

39041 

39058 

39076 

5 

9 

246 

9094 

9111 

9129 

9146 

9164 

9182 

9199 

9217 

9235 

9252 

6 

11 

247 

9270 

9287 

9305 

9322 

9340 

9358 

9375 

9393 

9410 

9428 

7 

13 

248 

9445 

9403 

9480 

9498 

9515 

9533 

9550 

95G8 

9585 

9602 

8 

14 

249 

9020 

9037 

96oo 

9672 

9690 

9707 

9724 

9742 

9759 

9777 

9 

10 

250 

39794 

39811 

39S29 

39846 

39863 

39881 

39S98 

39915 

39933 

39950 



251 

9967 

9985 

40002 

40019 

40037 

40054 

40071 

40088 

40106 

40123 


17 

252 

40140 

40157 

0175 

0192 

02o9 

0226 

0243 

0261 

0278 

0295 

1 

2 

253 

0312 

0329 

0346 

0364 

0381 

0398 

0415 

0432 

0449 

0466 

2 

3 

254 

0483 

0500 

0518 

0535 

0552 

0569 

0586 

0003 

0620 

0637 

O 

5 

255 

40654 

40671 

40688 

40705 

40722 

40739 

40756 

40773 

40790 

40807 

4 

< 

250 

0824 

0841 

0858 

0875 

0892 

0909 

0926 

0943 

0960 

0976 

5 

9 

257 

0993 

1010 

1027 

1044 

1061 

1078 

1095 

1111 

1128 

1145 

6 

10 

25S 

1162 

1179 

1196 

1212 

1229 

1246 

1263 

1280 

1296 

1313 

t 

12 

259 

1330 

1347 

1363 

1380 

1397 

1414 

1430 

1447 

1464 

1481 

8 

14 

200 

41497 

41514 

41531 

41547 

41564 

41581 

41597 

41014 

41G31 

41647 

9 

15 

201 

1004 

1081 

1697 

1714 

1731 

1747 

1764 

1780 

1797 

1814 


16 

202 

18S0 

1847 

1863 

1880 

1896 

1913 

1929 

1940 

1963 

1979 

1 

2 

203 

1990 

2012 

2029 

2045 

2062 

2078 

2095 

2111 

2127 

2144 

2 

3 

204 

2100 

2177 

2193 

2210 

2226 

2243 

2259 

2275 

2292 

2308 

3 

5 

205 

42325 

42341 

42357 

42374 

42390 

42406 

42423 

42439 

42455 

42472 

4 

6 

200 

2488 

2504 

2521 

2537 

2553 

2570 

2586 

2002 

2619 

2635 

5 

8 

207 

2051 

2607 

2084 

2700 

2716 

2732 

2749 

2765 

2781 

2797 

6 

10 

2GS 

2813 

2830 

2846 

2862 

2878 

2894 

2911 

2927 

2943 

2959 


11 

209 

2975 

2991 

3008 

3024 

3040 

3056 

3072 

3088 

3104 

3120 

8 

13 

270 

43136 

43152 

43169 

43185 

43201 

43217 

43233. 

43249 

43205 

43281 

9 

14 

271 

3297 

3313 

3329 

3345 

3361 

3377 

3393 

3409 

3425 

3441 



272 

3457 

3473 

3489 

3505 

3521 

3537 

3553 

3569 

3584 

3600 


15 

273 

3616 

3632 

3648 

3664 

3680 

3696 

3712 

3727 

3743 

3759 

1 

2 

274 

3775 

3791 

3807 

3823 

3838 

3854 

3870 

3886 

3902 

3917 

2 

3 

275 

48933 

43949 

43965 

43981 

43996 

44012 

44028 

44044 

44059 

44075 

3 

5 

276 

4091 

4107 

4122 

4138 

4154 

4170 

4185 

4201 

4217 

4232 

4 

6 

277 

4248 

4204 

4279 

4295 

4311 

4326 

4342 

4358 

4373 

4389 

0 

8 

278 

4404 

4420 

4436 

4461 

4467 

4483 

4498 

4514 

4529 

4545 

0 

9 

279 

4500 

4576 

4592 

4607 

4623 

4638 

4654 

4669 

4085 

4700 

7 

11 

No. 

0 

1 

2 

O 

O 

4 

5 

6 

7 

8 

9 

O 

9 

12 

14 







































Logarithms of Numbers, 


187 


I 


No. 2800 to 3400. Logarithms. 44716 to 53148. 


No. 

0 

1 

2 

3 

4 

5 

6 

pr 

1 

8 

9 


10 

280 

44716 

44731 

44747 

44762 

44778 

44793 

44809 

44824 

44810 

44855 

1 

2 

281 

4871 

4886 

4902 

4917 

4932 

4948 

4963 

4979 

4994 

5010 

2 

3 

282 

5025 

5040 

5056 

5071 

6086 

5102 

5117 

5133 

5148 

5163 

3 

5 

283 

5179 

5194 

5209 

5225 

5240 

5255 

5271 

5286 

5301 

5317 

4 

6 

281 

5332 

5347 

5362 

5378 

5393 

5408 

5423 

5439 

5454 

5469 

5 

8 

285 

45484 

45500 

45515 

45530 

45545 

45561 

45576 

45591 

45606 

45621 

6 

10 

286 

5637 

5652 

5667 

5682 

5097 

5712 

5728 

5743 

5758 

5773 

7 

11 

287 

5788 

5S03 

5818 

5834 

5849 

5864 

5879 

5894 

5909 

5924 

8 

13 

288 

5939 

5954 

5969 

5'.‘84 

6000 

6015 

6030 

6045 

6060 

6075 

9 

14 

280 

6090 

6105 

6120 

6135 

6150 

6165 

6180 

6195 

6210 

6225 



2 JO 

46240 

46255 

46270 

46285 

46300 

46315 

46330 

46345 

46359 

46374 



291 

6389 

6404 

6419 

6434 

6449 

6461 

6479 

6494 

6509 

6523 



292 

6538 

6553 

6568 

6583 

6598 

6613 

6627 

6642 

6657 

6672 


15 

293 

6687 

6702 

6716 

6731 

6746 

6761 

6776 

6790 

6805 

6820 

1 

2 

294 

6835 

6850 

6864 

6879 

6891 

6909 

6923 

6938 

6953 

6967 

2 

3 

295 

46982 

46997 

47012 

47026 

47041 

47056 

47070 

47085 

47100 

47114 

3 

5 

296 

7129 

7144 

7159 

7173 

7188 

7202 

7217 

7232 

7246 

7261 

4 

6 

2j7 

7276 

7290 

7305 

7319 

7334 

7349 

7363 

7378 

7392 

7407 

5 

8 

298 

7422 

7436 

7451 

7465 

7180 

7494 

7509 

7524 

7538 

7553 

6 

9 

299 

7567 

7582 

7596 

7611 

7625 

7640 

7654 

7669 

7683 

7698 

7 

11 

300 

47712 

47727 

47741 

47756 

47770 

47784 

47799 

47813 

47828 

47842 

8 

12 

301 

7857 

7871 

7885 

7900 

7914 

7929 

7943 

7958 

7972 

7986 

9 

14 

302 

8001 

8015 

8029 

8044 

8058 

8073 

8087 

8101 

8116 

8130 



303 

8144 

8159 

8173 

8187 

8202 

8216 

8230 

8244 

8259 

8273 



304 

8287 

8302 

8316 

8330 

8344 

8359 

8373 

8387 

8401 

8416 



305 

48430 

48444 

48458 

48473 

48487 

48501 

48515 

48530 

48541 

48558 


14 

306 

8572 

8586 

8601 

8615 

8629 

8643 

8657 

8671 

8686 

8700 

1 

1 

307 

8714 

8728 

8742 

8756 

8770 

8785 

8799 

8813 

8827 

8841 

2 

3 

308 

8855 

8869 

8883 

8897 

8911 

8926 

8940 

8954 

8968 

8982 

3 

4 

309 

8996 

9010 

9024 

9038 

9052 

9066 

9080 

9094 

9108 

9122 

4 

6 

310 

49136 

49150 

49164 

49178 

49192 

49206 

49220 

49234 

49218 

49262 

5 

7 

311 

9276 

9290 

9304 

9318 

9332 

9346 

9360 

9374 

9388 

9402 

6 

8 

312 

9415 

9429 

9443 

9457 

9471 

9485 

9499 

9513 

9527 

9541 

7 

10 

313 

9554 

9568 

9582 

9596 

9610 

9624 

9638 

9651 

9665 

9679 

8 

11 

314 

9693 

9707 

9721 

9734 

9748 

9762 

9776 

9790 

9803 

9817 

9 

13 

315 

49831 

49845 

49859 

49872 

49886 

49900 

49914 

49927 

49941 

49955 



316 

9969 

9982 

9996 

50010 

50024 

50037 

50051 

50065 

50079 

50092 



317 

50106 

50120 

50133 

0147 

0161 

0174 

0188 

0202 

0215 

0229 



318 

0243 

0256 

0270 

0284 

0297 

0311 

0325 

0338 

0352 

0365 


13 

319 

0379 

0393 

0406 

0420 

0433 

0447 

0461 

0474 

0488 

0501 

1 

1 

320 

50515 

50529 

50542 

50556 

50569 

50583 

50596 

50610 

50623 

50637 

2 

3 

321 

0651 

0664 

0678 

0691 

0705 

0718 

0732 

0745 

0759 

0772 

O 

4 

322 

0786 

0799 

0813 

0826 

0840 

0853 

0866 

0880 

0893 

0907 

4 

5 

323 

0920 

0934 

0947 

0961 

0974 

0987 

1001 

1014 

1028 

1041 

f) 

7 

324 

1055 

1068 

1081 

1095 

1108 

1121 

1135 

1148 

1162 

1175 

6 

8 

325 

51188 

51202 

51215 

51228 

51242 

51255 

51268 

51282 

51295 

51308 

7 

9 

326 

1322 

1335 

1348 

1362 

1375 

1388 

1402 

1415 

1428 

1441 

8 

10 

327 

1455 

1468 

1481 

1495 

1508 

1521 

1534 

1548 

1561 

1574 

9 

12 

328 

1587 

1601 

1614 

1627 

1640 

1654 

1667 

1680 

1693 

1706 



3.9 

1720 

1733 

1746 

1759 

1772 

1786 

1799 

1812 

1825 

1S38 



330 

51851 

51865 

51878 

51S91 

51901 

51917 

51930 

51943 

51957 

51970 


12 

331 

1983 

U96 

2009 

2022 

2035 

2048 

2061 

2075 

2088 

2101 

1 

1 

332 

2114 

2127 

2140 

2153 

2166 

2179 

2192 

2205 

2218 

2231 

2 

2 

333 

2244 

2257 

2270 

2284 

2297 

2310 

2323 

2336 

2349 

2362 

3 

4 

334 

2375 

2388 

2401 

2414 

2427 

2440 

2453 

2466 

2479 

2492 

4 

5 

335 

52504 

52517 

52530 

52543 

52556 

52569 

52582 

52595 

52608 

52621 

5 

6 

336 

2634 

2647 

2660 

2673 

2686 

2699 

2711 

2724 

2737 

2750' 

6 

7 

337 

2763 

2776 

2789 

2802 

2815 

2827 

2840 

2853 

2866 

2879 

7 

8 

338 

2892 

2905 

2917 

2930 

2943 

2956 

2969 

2982 

2994 

3007 

8 

10 

339 

3020 

3033 

3046 

3058 

3071 

3084 

3097 

3110 

3122 

3135 

9 

11 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 







































188 


Logarithms of Numbers. 


No. 3400 to 4000. Logarithms. I-iOg. 53148 to 60206. 


No. 

0 

1 

2 

3 

4 

5 

6 

/ 

8 

9 


13 

340 

53148 

53101 

53173 

53186 

53199 

53212 

53224 

53237 

53250 

53263 

1 

1 

341 

8276 

3288 

3301 

3314 

3326 

3,339 

3352 

3364 

3377 

3390 

2 

3 

342 

3403 

3415 

3428 

3441 

3453 

oil'6 

3479 

3191 

3504 

3517 

3 

4 

313 

3529 

3542 

3555 

3567 

3580 

3593 

3605 

3618 

3631 

3643 

4 

5 

344 

3656 

3608 

3681 

3694 

3706 

3719 

3732 

3744 

3757 

3 i 69 

5 

7 

345 

53782 

63794 

53807 

63820 

53832 

53845 

53857 

53870 

53882 

53895 

6 

8 

346 

8908 

3920 

3933 

3945 

3958 

3970 

3983 

3995 

4008 

4020 

7 

9 

347 

4033 

4045 

4068 

4070 

4083 

4095 

4108 

4120 

4133 

4145 

8 

10 

348 

349 

4158 

4283 

4170 

4295 

4183 

4307 

4195 

4320 

4208 

4332 

4220 

4345 

4233 

4357 

4245 

4370 

4258 

4382 

4270 

4394 

9 

12 



350 

54407 

54419 

64432 

54444 

54456 

54469 

54481 

54494 

54506 

54518 



351 

4531 

4543 

4555 

4568 

4580 

4593 

4605 

4617 

4630 

4642 



352 

4654 

4667 

4679 

4691 

4704 

4716 

4728 

4741 

4753 

4765 



353 

4777 

4790 

4802 

4814 

4827 

4839 

4851 

4864 

4876 

4888 



854 

4900 

4913 

4925 

4937 

4949 

4962 

4974 

4986 

4998 

5011 



355 

55023 

55035 

55047 

65060 

55072 

65084 

55096 

55108 

55121 

65133 



356 

5145 

5157 

5169 

5182 

5194 

5206 

5218 

6230 

5242 

5255 


12 

357 

5267 

5279 

5291 

5303 

5315 

5328 

6340 

6352 

6864 

5376 

1 

1 

368 

5388 

5400 

5413 

5425 

5437 

5449 

5461 

5473 

5485 

5497 

2 

2 

359 

5509 

5522 

55:14 

5546 

5558 

5570 

5582 

5594 

5606 

5618 

3 

4 

360 

55630 

55042 

55654 

55666 

55678 

55691 

55703 

55715 

56727 

55739 

4 

5 

361 

5751 

5763 

5775 

5787 

5799 

5811 

5823 

5835 

5847 

5859 

5 

6 

362 

5871 

5883 

5895 

5907 

5919 

5931 

5943 

5955 

5967 

5979 

6 

7 

363 

5991 

6003 

G015 

6027 

6638 

6050 

6062 

6074 

6086 

6098 

7 

8 

364 

0110 

0122 

613 4 

6146 

6158 

6170 

6182 

6194 

6205 

6217 

8 

10 

365 

366 

56229 

6348 

56241 

6300 

56253 

6372 

5G265 

6384 

56277 

6396 

56289 

6407 

5630 L 
0419 

56312 

6131 

56324 

0443 

56336 

6455 

9 

11 



367 

6467 

6478 

6490 

0502 

6514 

6526 

6538 

6549 

6561 

6573 



368 

6585 

6597 

6608 

0620 

6632 

0644 

6656 

6667 

6679 

6691 



369 

6703 

6714 

6726 

6738 

6750 

6761 

6773 

6785 

6797 

C808 



370 

5G820 

56832 

56^44 

50855 

56867 

56879 

56891 

56002 

56914 

56926 



371 

6937 

6949 

6961 

6972 

6984 

G996 

7008 

7019 

7031 

7043 



372 

7054 

7066 

7078 

7689 

7101 

7113 

7124 

7136 

7148 

7159 



373 

7171 

7183 

7194 

7206 

7217 

7229 

7241 

7252 

7264 

7276 


11 

374 

7287 

7299 

7310 

7322 

7334 

7345 

7357 

7368 

7380 

7392 

1 

1 

375 

57403 

57415 

5742C 

57438 

57449 

57461 

57473 

57484 

57496 

57607 

2 

2 

376 

7519 

7530 

7542 

7553 

7565 

7576 

7588 

7600 

7611 

7623 

3 

3 

377 

7 034 

7646 

7057 

7669 

7080 

7692 

7703 

7715 

7720 

7738 

4 

4 

378 

7749 

7761 

7772 

7784 

7795 

7807 

7818 

7850 

7841 

7852 

5 

6 

379 

7864 

7*75 

7887 

7898 

7910 

7921 

7933 

7944 

7955 

7967 

c 

7 

38 :> 

57978 

57990 

58001 

68013 

58024 

58035 

58047 

68058 

58070 

58081 

7 

8 

381 

8092 

8104 

8115 

8127 

8138 

8149 

8161 

8172 

8184 

8195 

8 

9 

382 

383 

8206 

8320 

8218 

8331 

8229 

8343 

8240 

8364 

8252 

8365 

8263 

8377 

8274 

8388 

8286 

8399 

8297 

8410 

8309 

8422 

9 

10 



384 

8433 

8444 

8156 

8467 

8478 

8490 

8501 

8512 

8524 

8535 



385 

58546 

58557 

58569 

58580 

58591 

58602 

58614 

58625 

58636 

58647 



386 

8659 

8670 

8681 

8692 

8704 

8715 

8726 

8737 

8749 

8760 



387 

8771 

8782 

8794 

8805 

8816 

8827 

8838 

8860 

8861 

8872 



388 

8883 

8894 

8906 

8917 

8928 

8939 

8950 

8961 

8973 

8984 



389 

8995 

9000 

9017 

9028 

9040 

9051 

9062 

9073 

9084 

9095 



390 

59100 

59118 

69129 

59140 

59151 

■ 59162 

59173 

69184 

59195 

59207 


10 

391 

9218 

9229 

9240 

9251 

9262 

9273 

9284 

9295 

9806 

9318 

1 

1 

392 

9329 

9340 

9351 

9362 

9373 

9384 

9395 

9106 

9417 

9428 

2 

2 

893 

9439 

9450 

9461 

9472 

9483 

9494 

9506 

9517 

9528 

9539 

O 

3 

394 

9550 

9561 

9572 

9683 

9594 

9605 

9616 

9627 

9638 

9619 

4 

4 

395 

59060 

69671 

59682 

69693 

59704 

59715 

59726 

59737 

59748 

59759 

5 

5 

396 

9770 

9780 

9791 

9802 

9813 

9824 

9835 

9846 

9857 

9868 

6 

6 

397 

9879 

9890 

9901 

9912 

9923 

9934 

9945 

9956 

9966 

9977 

7 

7 

398 

9988 

9999 

60010 

60021 

60632 

66643 

60654 

60065 

60076 

60086 

8 

8 

399 

60097 

60108 

0119 

0130 

0141 

0152 

0163 

0173 

0184 

0195 

9 

9 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


























Logarithms of Numbers. 


189 


No. 4000 to 4600. Logarithms. I-iOg. 60206 to 66276. 


No. 

0 

1 

2 

3 

4 

5 

6 

r* 

i 

8 

9 


11 

400 

60200 

60217 

60228 

60239 

60249 

60260 

60271 

60282 

60293 

60304 

l 

1 

401 

0314 

0325 

0336 

0347 

0358 

0869 

0379 

0390 

0401 

0412 

2 

2 

402 

0423 

0433 

0444 

0155 

0466 

0477 

0487 

0498 

0509 

0520 

3 

3 

400 

0531 

0541 

0552 

0563 

0674 

0584 

0595 

0606 

0617 

0627 

4 

4 

404 

0638 

0649 

0660 

0670 

0681 

0092 

0703 

0713 

0724 

0735 

5 

6 

405 

60746 

60756 

60767 

60778 

60788 

60799 

60810 

60821 

60831 

60842 

6 

7 

406 

0853 

0863 

0874 

0885 

0895 

0906 

0917 

0927 

0938 

0949 

7 

8 

407 

0959 

0970 

0981 

0991 

1002 

1013 

1023 

1034 

1045 

1055 

8 

9 

408 

1066 

1077 

1087 

1098 

1109 

1119 

1130 

1140 

1151 

1162 

9 

10 

409 

1172 

1183 

1194 

1204 

1215 

1225 

1236 

1247 

1257 

1268 



410 

61278 

61289 

61300 

61310 

61321 

61331 

61312 

61352 

61363 

61374 



411 

1384 

1395 

1405 

1416 

1426 

1437 

1448 

1458 

1469 

1479 



412 

1490 

1500 

1511 

1521 

1532 

1542 

1553 

1563 

1574 

1584 



413 

1595 

1606 

1616 

1627 

1637 

1618 

1658 

1069 

1679 

1G90 



414 

1700 

1711 

1721 

1731 

1742 

1752 

1763 

1773 

1784 

1794 



415 

61805 

61815 

61826 

61836 

61847 

61857 

61868 

61878 

61888 

61899 



416 

1909 

1920 

1930 

1941 

1951 

1962 

1972 

1982 

1993 

2003 



417 

2014 

2024 

2034 

2045 

2055 

2066 

2076 

2086 

2097 

2107 



418 

2118 

2128 

2138 

2149 

2159 

2170 

2180 

2190 

2201 

2211 



419 

2221 

2232 

2242 

2252 

2263 

2273 

2284 

2294 

2304 

2315 



420 

62325 

62335 

62346 

62356 

62366 

62377 

62387 

62397 

62408 

62418 



421 

2428 

2439 

2449 

2459 

2469 

2480 

2490 

2500 

2511 

2521 



422 

253J 

2542 

2552 

2562 

2572 

2583 

2593 

2003 

2613 

2624 



423 

2634 

2644 

2655 

2665 

2675 

2685 

2696 

2706 

2716 

2726 



424 

2737 

2747 

2757 

2767 

2778 

2788 

2798 

2808 

2818 

2829 


t A 

425 

62839 

62849 

62859 

62870 

628S0 

62890 

62900 

62910 

62921 

62931 

1 

1U 

1 

42G 

2941 

2951 

2961 

297 

2982 

2992 

3002 

3012 

3022 

3033 

9 

o 

427 

3043 

3053 

3063 

3073 

3083 

3094 

3104 

3114 

3124 

3134 

Q 

Q 

428 

3144 

3155 

3165 

3176 

3185 

3195 

3205 

3215 

3225 

3236 

A 

A 

429 

3246 

3256 

3266 

3276 

3286 

3296 

3306 

3317 

3327 

3337 


£ 

430 

63347 

63357 

63367 

63377 

63387 

63397 

63407 

63417 

63428 

63438 

ft 

ft 

431 

3448 

3458 

3468 

3478 

8488 

3498 

3508 

3518 

3528 

3538 

7 


432 

3548 

3558 

3568 

3579 

3589 

3599 

3609 

3619 

3629 

3639 

o 

Q 

433 

3649 

3659 

3669 

3679 

3689 

3699 

8709 

3719 

3729 

3739 

o 

Q 

434 

3749 

3759 

3769 

3779 

3789 

3799 

8809 

3819 

3829 

3839 



435 

63849 

63859 

63869 

63879 

63889 

63899 

68909 

C3919 

63929 

63939 



436 

3949 

3959 

3969 

3979 

3988 

3998 

4008 

4018 

4028 

4038 



437 

4048 

406S 

4068 

4078 

4088 

4098 

4108 

4118 

4128 

4137 



438 

4147 

4157 

4167 

4177 

4187 

4197 

4207 

4217 

4227 

4237 



439 

4246 

4256 

4266 

4276 

4286 

4296 

4306 

4316 

4326 

4385 > 



440 

64345 

64355 

64365 

64375 

64385 

64395 

64404 

64414 

64424 

64434 



441 

4444 

4454 

4464 

4473 

4483 

4493 

4503 

4513 

4523 

4532 



442 

4542 

4552 

4562 

4572 

4582 

4591 

4601 

4611 

4621 

4631 



443 

4640 

4650 

4660 

4670 

4680 

4689 

4699 

4709 

4719 

4729 



444 

4738 

4748 

4758 

4768 

4777 

4787 

4797 

4807 

4816 

4826 



445 

64836 

64846 

64856 

64865 

64875 

64885 

64895 

G4904 

64914 

64 924 



446 

4933 

4943 

4953 

4963 

4972 

4982 

4992 

5002 

5011 

5021 



447 

5031 

5040 

5050 

5060 

5070 

5079 

5089 

5099 

5108 

5118 



448 

5128 

5137 

5147 

5157 

5167 

5176 

5186 

5196 

5205 

5215 



449 

5225 

5234 

5244 

5254 

5263 

5273 

5283 

5292 

5302 

5312 



450 

65321 

65331 

65341 

65350 

65360 

653C9 

65379 

65389 

65398 

65408 


9 

451 

5418 

5427 

5487 

5447 

5456 

5466 

5475 

5485 

5495 

5504 

L 

1 

452 

5514 

5523 

5533 

5543 

5552 

5562 

5571 

5581 

6591 

5600 

2 

2 

453 

5610 

5619 

5629 

5639 

5648 

5658 

5667 

5677 

5686 

5696 

3 

3 

454 

5706 

5715 

5725 

5734 

5744 

5753 

5763 

5772 

5782 

5792 

4 

4 

455 

65801 

65811 

65820 

6583.0 

65839 

65849 

65858 

65868 

65877 

65887 

5 

5 

456 

5896 

5906 

5916 

5925 

5935 

5944 

5954 

5963 

5973 

5982 

6 

5 

457 

5992 

6001 

6011 

6020 

6030 

6039 

6049 

6058 

6068 

6077 

i 

6 

458 

6087 

C09G 

6106 

6115 

6124 

6134 

6143 

6153 

6162 

6172 

8 

7 

459 

C181 

6191 

6200 

6210 

6219 

6229 

6238 

6247 

6257 

6266 

9 

8 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 




































190 


Logarithms or Numbers, 


No. 4600 to 5200. Logarithms. Log. 66276 to /1600. 


No.' 

0 1 

1 

2 

3 

4 

5 

6 | 

7 

8 

9 


10 

460 

66276 

66235 

C6295 

66304 

66314 

66323 

66332 

C6342 

66351 

66361 

1 

1 

401 

6370 

6380 

6389 

6398 

6408 

6417 

6427 

6436 

6445 

6455 

2 

2 

462 

6464 

6474 

6483 

6492 

6502 

6511 

6521 

6530 

6539 

6549 

3 

3 

463 

6558 

6567 

6577 

6586 

6596 

6605 

6614 

6624 

€033 

6642 

4 

4 

464 

6652 

6661 

6071 

6680 

6689 

6699 

6708 

6717 

6727 

6736 

5 

5 

46) 

66745 

667 55 

0676 4 

66773 

66783 

66792 

G6801 

G6811 

G6820 

60829 

6 

6 

466 

6839 

6848 

6857 

6867 

6876 

6885 

6894 

6904 

6913 

6922 

7 

7 

467 

6932 

6941 

6950 

6960 

6969 

6978 

6987 

6997 

7006 

7015 

8 

8 

468 

7025 

7034 

7043 

7052 

7062 

7071 

7080 

7089 

7099 

7108 

9 

9 

469 

7117 

7127 

7136 

7145 

7154 

7164 

7173 

7182 

7191 

7201 



470 

67210 

67219 

67228 

67237 

67217 

67256 

67265 

C7274 

67284 

67293 



471 

7302 

7311 

7321 

7330 

7339 

7348 

7357 

7367 

7 37 6 

7385 



472 

7394 

7403. 

7413 

7422 

7431 

7440 

7449 

7459 

7468 

7477 



473 

7486 

7495 

7504 

7514 

7523 

7532 

7541 

7550 

7560 

7569 



474 

7578 

7587 

7596 

7605 

7614 

7624 

7633 

7642 

7651 

7660 



475 

67669 

67679 

67688 

67697 

67700 

67715 

C7724 

67733 

G7742 

67752 



476 

7761 

7770 

7779 

7788 

7797 

7806 

7815 

7825 

7834 

7843 



477 

7852 

7861 

7870 

7879 

7888 

7897 

7906 

7916 

7925 

7934 



473 

7943 

7952 

7961 

7970 

7979 

7988 

7997 

8006 

8015 

8024 



479 

8034 

8043 

8052 

8061 

8070 

8079 

8088 

8097 

8106 

8115 



480 

68124 

68133 

68142 

68151 

68160 

68169 

68178 

68187 

68196 

68205 



481 

8216 

8224 

8233 

8242 

8251 

8260 

82G9 

8278 

8287 

8296 



482 

8305 

8314 

8323 

8332 

8311 

8350 

8359 

8368 

8377 

8386 



483 

8395 

8404 

8113 

8422 

8431 

8440 

8449 

8458 

8467 

&76 



484 

8485 

8494 

8502 

8511 

8520 

8529 

8538 

8547 

8556 

8565 


Q 

485 

68574 

68583 

68592 

68601 

68610 

GSG19 

68628 

68637 

6S646 

68656 

1 

1 

486 

8664 

8673 

8681 

8690 

8699 

8708 

8717 

8726 

8735 

8744 


9 

487 

8753 

8762 

8771 

8780 

8789 

8797 

8806 

8815 

8824 

8833 

9 


488 

8842 

8S51 

8860 

8869 

8t>78 

8886 

8895 

8904 

8913 

8922 

A 

± 

489 

8931 

8940 

8949 

8958 

8966 

8975 

8984 

8993 

9002 

9011 



490 

69020 

69028 

69037 

69046 

69055 

69064 

69073 

C9082 

69090 

69099 



491 

9108 

9117 

9126 

9135 

9144 

9152 

9161 

9170 

9179 

9188 

7 

A 

492 

9197 

9205 

9214 

9223 

9232 

9241 

9249 

9258 

9267 

9276 


7 

493 

9285 

9294 

9302 

9311 

9320 

9329 

9338 

9346 

9355 

9364 

0 

Q 

494 

9373 

9381 

9390 

9399 

9408 

9417 

9425 

9434 

9443 

9452 



495 

69461 

69469 

69478 

69487 

69496 

69504 

69513 

69522 

69531 

69539 



496 

654S 

9557 

9566 

9574 

9583 

9592 

9601 

9609 

9618 

9627 



497 

9636 

9614 

9653 

9662 

9671 

9079 

9688 

9697 

9705 

9714 



498 

9723 

9732 

9740 

9749 

9758 

9767 

9775 

9784 

9793 

9801 



499 

9810 

9819 

9827 

9836 

9845 

9854 

9862 

9871 

9880 

9888 



5<K> 

69897 

69906 

09914 

69923 

69932 

09940 

69949 

69958 

69966 

69975 



501 

9984 

9992 

70001 

70010 

70018 

70027 

70036 

70044 

70053 

70062 



502 

70070 

70079 

0088 

0096 

0105 

0114 

0122 

0131 

0140 

0148 



503 

0157 

01G5 

0174 

0183 

0191 

0200 

0209 

0217 

022 G 

0234 



504 

0243 

0252 

0260 

0269 

0278 

0286 

0295 

0303 

0312 

0321 



505 

70329 

70338 

70346 

70355 

70364 

70372 

70381 

70389 

70398 

70406 



506 

0415 

0424 

0432 

0441 

0449 

0458 

0467 

0475 

0484 

0492 



507 

0501 

0509 

0518 

0526 

0535 

0544 

0552 

0561 

0569 

0578 



508 

0586 

0595 

0603 

0612 

0621 

0629 

0638 

0646 

0655 

0663 



509 

0672 

0680 

0689 

0697 

0706 

0714 

0723 

0731 

0740 

0749 



510 

70757 

70766 

70774 

70783 

70791 

70800 

708U8 

70817 

70825 

70834 


8 

511 

0842 

0851 

0859 

0868 

0876 

0885 

0893 

0902 

0910 

0919 

l 

1 

612 

0927 

0935 

0944 

0952 

0961 

0969 

0978 

0986 

0995 

1003 

2 

2 

513 

1012 

1020 

1029 

1037 

1046 

1054 

1063 

1071 

1079 

1088 

3 

2 

514 

1096 

1105 

1113 

1122 

1130 

1139 

1147 

1165 

1104 

1172 

4 

3 

615 

71181 

71189 

71198 

71206 

71214 

71223 

71231 

71240 

71248 

71257 

0 

4 

516 

1265 

1273 

1282 

1290 

1299 

1307 

1315 

1324 

1332 

1341 

6 

6 

617 

1349 

1357 

1366 

1374 

1383 

1391 

1399 

1408 

1416 

1425 

7 

C 

518 

1433 

1441 

4450 

1458 

1460 

1475 

1483 

1492 

1500 

1508 

8 

G 

519 

1517 

1525 

1533 

1542 

1550 

1559 

1567 

1575 

1584 

1592 

9 

7 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 




U 































Logarithms of Numbers, 


191 


No. 5200 to 5800. 


Logarithms. 

Log. 71600 to 76343. 

No 

. 0 

1 

2 

3 

4 

5 

6 

i 

8 

1 9 

Q 

52 J 

71600 

71609 

71617 

71625 

71634 

71642 

71650 

71659 

71667 

71675 

l i 

521 

1684 

1692 

1700 

1709 

1717 

1725 

1734 

1742 

1750 

1759 

2 2 

522 

1767 

1775 

1784 

1792 

1800 

1809 

1817 

1825 

1834 

1842 

3 3 

523 

1850 

1858 

1867 

1875 

1883 

1892 

1900 

1908 

1917 

1925 

4 4 

524 

1933 

1941 

1950 

1958 

1966 

1975 

1983 

1991 

1999 

2008 

5 5 

525 

72016 

72024 

72032 

72041 

72049 

72057 

72066 

72074 

72082 

72090 

6 5 

52G 

2099 

2107 

2115 

2123 

2132 

2140 

2148 

2156 

2165 

£173 

7 6 

527 

21 S1 

2189 

2198 

2206 

2214 

2222 

2230 

2239 

2247 

2255 

8 7 

528 

2263 

2272 

2280 

2288 

2296 

2304 

2313 

2321 

2329 

2337 

9 8 

529 

2346 

2354 

2362 

2370 

2378 

2387 

2395 

2403 

2411 

2419 


530 

72428 

72436 

72444 

72452 

72460 

72469 

72477 

72485 

72493 

72501 


531 

2509 

2518 

2526 

2534 

2542 

2550 

2558 

2567 

2575 

2583 


532 

2591 

2599 

2607 

2616 

2624 

2632 

2640 

2648 

2656 

2665 


533 

2673 

2681 

2689 

2697 

2705 

2713 

2722 

2730 

2738 

2746 


534 

2754 

2762 

2770 

2779 

2787 

2795 

2803 

2811 

2819 

2827 


635 

72835 

72843 

72852 

72860 

72868 

72876 

72884 

72892 

72900 

72908 


536 

2916 

2925 

2933 

2941 

2949 

2957 

2965 

2973 

2981 

29S9 


537 

2997 

3006 

3014 

3022 

3030 

3038 

3046 

3054 

3062 

3070 


538 

3078 

3086 

3094 

3102 

3111 

3119 

3127 

3135 

3143 

3151 


539 

3159 

3167 

3175 

3183 

3191 

3199 

3207 

3215 

3223 

3231 


540 

73239 

73247 

7 o255 

73263 

73272 

73280 

73288 

73296 

73304 

73312 


541 

3320 

3328 

3336 

3344 

3352 

3360 

3368 

3376 

3384 

3392 


542 

3400 

3408 

3416 

3424 

3432 

3440 

3448 

3456 

3464 

3472 


543 

3480 

3488 

3456 

3504 

3512 

3520 

3528 

3536 

3544 

3552 


514 

3560 

3568 

3576 

3584 

3592 

3600 

3608 

3616 

3624 

3632 

8 

545 

73640 

73648 

73656 

73664 

73672 

73679 

73687 

73695 

73703 

73711 

546 

3719 

3727 

3735 

3743 

3751 

3759 

3767 

3775 

3783 

3791 

1 1 

547 

3799 

3807 

3815 

3823 

3830 

3838 

3846 

3854 

3862 

3870 

2 2 

548 

3878 

3886 

3894 

3902 

3910 

3918 

3926 

3933 

3941 

3949 

o 2 

549 

3957 

3965 

3973 

3981 

3989 

3997 

4005 

4013 

4020 

4028 

4 6 

550 

74036 

74044 

74052 

74060 

74068 

74076 

74084 

74092 

74099 

74107 

5 4 

551 

4115 

4123 

4131 

4139 

4147 

4155 

4162 

4170 

4178 

4186 

o 0 

552 

4194 

4202 

4210 

4218 

4225 

4233 

4241 

4249 

4257 

4265 

7 b 

553 

4273 

42S0 

4288 

4296 

4304 

4312 

4320 

4327 

4335 

4343 

o o 

554 

4351 

4359 

4367 

4374 

4382 

4390 

4398 

4406 

4414 

4421 . 

9 7 

555 

74429 

74437 

74445 

74453 

74461 

74468 

74476 

74484 

74492 

74500 


556 

4507 

4515 

4523 

4531 

4539 

4547 

4554 

4562 

4570 

4578 


557 

4586 

4593 

4601 

4609 

4617 

4624 

4632 

4640 

4648 

4656 


558 

4663 

4671 

4679 

4687 

4695 

4702 

4710 

4718 

4726 

4733 


559 

4741 

4749 

4757 

4764 

4772 

4780 

4788 

4796 

4803 

4811 


560 

74819 

74827 

74834 

74842 

74850 

74858 

74865 

74873 

74881 

74889 


561 

4896 

4904 

4912 

4920 

4927 

4935 

4943 

4950 

4958 

4966 


562 

4974 

4981 

4989 

4997 

5005 

5012 

5020 

5028 

5035 

5043 


563 

5051 

5059 

5066 

5074 

5082 

5089 

5097 

5105 

5113 

5120 


564 

5123 

5136 

5143 

5151 

5159 

5166 

5174 

5182 

51S9 

5197 


565 

75205 

75213 

75220 

75228 

75236 

75243 

75251 

75259 

75266 

75274 


566 

5282 

5289 

5297 

5305 

5312 

5320 

5328 

5335 

5343 

5351 


567 

5358 

5366 

5374 

5381 

5389 

5397 

5404 

5412 

5420 

6427 


568 

5435 

5442 

5450 

5458 

5465 

5473 

5481 

5488 

5496 

5504 


569 

5511 

5519 

5526 

5534 

5542 

5549 

5557 

5565 

5572 

5580 


570 

75587 

75595 

75603 

75610 

75618 

75626 

75633 

75641 

75648 

75656 

7 

571 

5664 

5671 

5679 

5686 

5694 

5702 

5709 

5717 

5724 

6732 

L 1 

572 

5740 

5747 

5755 

5762 

5770 

5778 

5785 

5793 

5800 

5S08 

l 1 

573 

6815 

5823 

5831 

5838 

5846 

5853 

5861 

5868 

5876 

5884 : 

l 2 

574 

5891 

5899 

5906 

5914 

6921 

6929 

5937 

5944 

5952 

5959 

1 3 

575 

75967 

75974 

75982 

75989 

75997 

76005 

76012 

76020 

76027 

76035 

> 4 

576 

6042 

6050 

6057 

6065 

6072 

6080 

6087 

6095 

6103 

6110 

1 4 

577 

6118 

0125 

6133 

6140 

614S 

6155 

6163 

6170 

6178 

6185 

5 

578 

6193 

6200 

6208 

6215 

6223 

6230 

6238 

6245 

6253 

6260 

6 

579 

6268 

6275 

6283 

6290 

6298 

6806 

6313 

6320 

6328 

6335 

i 6 

No. 

0 1 

1 

2 

3 

4 1 

5 

6 

7 

8 

9 












































192 


Logarithms op Numbers 


No. 5800 to 6400. Logarithms. I-iOg. 76343 to 80618. 


No. 

0 

1 

2 

3 

4 

5 

6 

rj 

7 

8 

9 


H 

580 

703,43 

70350 

76358 

76365 

76373 

70380 

76388 

76395 

70403 

70410 

l 

1 

581 

0418 

6425 

6433 

6440 

0448 

6455 

6462 

0 470 

0477 

0485 

2 

2 

582 

6492 

6500 

0507 

6515 

6522 

0530 

. 6537 

0645 

6552 

6559 

3 

2 

5S3 

6567 

6574 

6582 

6589 

6597 

6004 

0012 

0019 

6026 

€634 

4 

3 

581 

6641 

6649 

0056 

6664 

6071 

0678 

0686 

0693 

6701 

0708 

5 

4 

685 

76710 

70723 

76730 

76738 

70745 

76753 

70700 

70708 

76775 

70782 

0 

5 

5 SO 

0790 

0797 

6805 

6812 

0819 

0827 

0834 

0842 

0849 

0850 

7 

6 

587 

6801 

0871 

6879 

6886 

6893 

69ul 

0908 

0910 

0923 

6930 

8 

0 

588 

0938 

6945 

6053 

6960 

0907 

6975 

0982 

6989 

6997 

7004 

9 

7 

589 

7012 

7019 

7026 

7034 

7041 

7048 

7056 

7003 

7070 

7078 



590 

77085 

77093 

77100 

77107 

77115 

77122 

77129 

77137 

77144 

77151 



591 

7159 

7160 

7173 

7181 

7188 

7195 

7203 

7210 

7217 

7225 



592 

7232 

7240 

7247 

7254 

7262 

7269 

7276 

7283 

7291 

7298 



593 

7305 

7313 

7320 

7327 

7335 

7342 

7349 

7357 

7304 

7371 



594 

7379 

7386 

7393 

7401 

7408 

7415 

7422 

7430 

7437 

7444 



595 

77152 

77459 

77406 

77471 

77481 

77488 

77495 

77503 

77510 

77517 



596 

7525 

7532 

7539 

7516 

7554 

7561 

7568 

7576 

7583 

7690 



597 

7597 

7G05 

7012 

7619 

7027 

7034 

7041 

7048 

7656 

7063 



598 

7070 

7677 

7685 

7692 

7699 

7706 

7714 

7721 

7728 

7735 



599 

77 13 

7750 

7757 

7764 

7772 

7779 

7786 

7793 

7801 

7808 



too 

77815 

77822 

77830 

77837 

77844 

77851 

77859 

77SOO 

77873 

77880 



eoi 

7887 

7895 

7902 

7909 

7910 

7924 

7931 

7938 

7945 

7952 



002 

7900 

7967 

7974 

7981 

7988 

7996 

8003 

8010 

8017^ 

8025 



003 

8032 

8039 

8046 

8053 

8061 

8008 

8075 

8082 

8089 

8097 



004 

8104 

8111 

8118 

8125 

8132 

8140 

8147 

8154 

8161 

8108 



605 

78170 

78183 

78190 

78197 

78204 

78211 

78219 

78226 

78233 

78240 


4 

006 

8247 

8254 

8202 

8269 

8276 

3283 

8290 

8297 

8305 

8312 

1 

JL 

007 

8319 

8320 

8333 

8340 

8347 

8355 

8302 

8369 

8376 

8383 

Xd 

1 

008 

8390 

8398 

8405 

8412 

8419 

8420 

8433 

8440 

8447 

8455 

«> 

A 

z 

009 

8402 

8409 

8476 

8183 

8490 

8497 

8504 

8512 

8519 

8520 

4 

o 

010 

78533 

78540 

78547 

78551 

78561 

78509 

78576 

78583 

78590 

78597 

O 

4 

011 

8004 

8011 

8618 

8625 

8033 

8040 

8047 

8054 

8601 

8608 

O 

4 

012 

8075 

8082 

8689 

8696 

8704 

8711 

8718 

8725 

8732 

8739 

4 

O 

013 

8710 

8753 

8760 

8767 

8774 

8781 

8789 

8796 

8803 

8810 

O 

O 

014 

8817 

8824 

8831 

8838 

8845 

8852 

8859 

8866 

8873 

8880 

y 

u 

015 

78888 

78895 

78902 

78909 

78916 

78923 

78930 

78937 

7894 4 

78951 



010 

8958 

8905 

8972- 

8979 

89S6 

899 i 

9000 

9007 

9. il4 

9021 



017 

9029 

9030 

9043 

9050 

9057 

9001 

9071 

9078 

9085 

9092 



018 

9099 

9100 

9113 

9120 

9127 

9134 

9141 

9148 

9155 

9102 



019 

9109 

9170 

9183 

9190 

9197 

9204 

9211 

9218 

9225 

9232 



620 

79239 

79240 

79253 

79260 

79207 

79274 

79281 

79288 

79295 

79302 



021 

9309 

9316 

9323 

9330 

9;37 

9344 

9351 

9358 

9305 

9372 



622 

9379 

9386 

9393 

9400 

9407 

9414 

9421 

9428 

9435 

9442 



623 

9449 

9456 

9463 

9470 

9477 

9184 

9491 

9498 

9605 

9511 



624 

9518 

9525 

9532 

9539 

9546 

9553 

95 CO 

9567 

9574 

9581 



(125 

79588 

79595 

79602 

796 .9 

79610 

79023 

79030 

79(337 

79044 

79650 



(126 

9057 

9664 

9G71 

9678 

9085 

9092 

9693 

9700 

9713 

9720 



627 

9727 

9734 

9741 

9748 

9754 

9701 

9708 

9775 

9782 

9789 



628 

9790 

9803 

9810 

9S17 

9821 

9831 

9837 

9844 

9851 

9858 



029 

9805 

9S72 

9879 

9S86 

9893 

. 9900 

9906 

9913 

9920 

9927 



63 1 

79934 

79911 

7994S 

79955 

79902 

79969 

79975 

79982 

79989 

79996 


6 

631 

HJ003 

80010 

80017 

80024 

80030 

80037 

80044 

80051 

80058 

80005 

i 

1 

632 

0072 

0079 

0085 

0092 

0099 

0106 

0113 

0120 

0127 

0134 

2 

1 

033 

0140 

0117 

0154 

0161 

0103 

0175 

0182 

0188 

0195 

0202 

ft 

2 

031 

0209 

0216 

0223 

0229 

0236 

0243 

0250 

0257 

0204 

0271 

4 

2 

635 

80277 

802S4 

80291 

80298 

80305 

80312 

80318 

80325 

80332 

80339 

5' 

3 

030 

0340 

0353 

0359 

0366 

0373 

0380 

0387 

0393 

0100 

0407 

6 

4 

037 

0414 

0421 

0428 

0434 

0441 

0448 

0455 

0402 

0408 

0475 

7 

4 

038 

0482 

0489 

0496 

0502 

0509 

0516 

0523 

0530 

0530 

0543 

8 

6 

639 

0550 

0557 

0564 

0570 

0577 

0584 

0591 

0598 

0004 

0611 

9 

6 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 































Logarithms of Numbers. 


193 


No. 6400 to 7000. _ iiogaritfimg. Log. 80618 to 84510. 


No 

0 

1 

2 

3 

4 

0 

6 

7 

8 

9 



6+0 

80618 

80625 

80632 

80638 

80645 

80652 

S0659 

80665 

80672 

80679 


7 

641 

0686 

0693 

0699 

0706 

0713 

0720 

0726 

0733 

0740 

0747 

l 

1 

642 

0754 

0760 

0767 

0774 

0781 

0787 

079+ 

0801 

0808 

0814 

2 

1 

643 

0821 

0828 

0835 

0841 

0848 

0855 

0862 

0868 

0875 

0882 

3 

2 

644 

0889 

0895 

0902 

0909 

0916 

0922 

0929 

0936 

0943 

0949 

4 

3 

645 

8095G 

80963 

80969 

80976 

80983 

80990 

80996 

81003 

81010 

81017 

5 

4 

6+6 

1023 

1030 

1037 

1043 

1050 

1057 

1064 

1070 

1077 

1084 

6 

4 

647 

1090 

1097 

1104 

1111 

1117 

1124 

1131 

1137 

1144 

1151 

7 

5 

648 

1158 

1164 

1171 

1178 

1184 

1191 

1198 

1204 

1211 

1218 

8 

6 

649 

1224 

1231 

1238 

1245 

1251 

1258 

1265 

1271 

1278 

1285 

9 

6 

650 

81291 

81298 

81305 

81311 

81318 

81325 

81331 

81338 

81345 

81351 



651 

1358 

1365 

1371 

1378 

1385 

1391 

1398 

1405 

1411 

1418 



(552 

1425 

1431 

1438 

1445 

1451 

1458 

1465 

1471 

1478 

1485 



653 

1491 

1498 

1505 

1511 

1518 

1525 

1531 

1538 

1544 

1551 



654 

1558 

1564 

1571 

1578 

1584 

1591 

1598 

10u4 

1611 

1617 



655 

81624 

81631 

81637 

81644 

81651 

81657 

81664 

81671 

81677 

81684 



656 

1690 

1697 

1704 

1710 

1717 

1723 

1730 

1737 

1743 

1750 



657 

1757 

1763 

1770 

1776 

1783 

1790 

1796 

1803 

1809 

1816 



658 

1823 

1829 

1836 

1842 

1849 

1856 

1862 

1869 

1875 

1882 



659 

18S9 

1895 

1902 

1908 

1915 

1921 

1928 

1935 

1941 

1948 



660 

81954 

81961 

S1968 

81974 

81981 

81987 

81994 

82000 

82007 

82014 



661 

2020 

2027 

2033 

2040 

2046 

2053 

2060 

2066 

2073 

2079 



662 

2086 

2092 

2099 

2106 

2112 

2119 

2125 

2132 

2138 

2145 



063 

2151 

2158 

2164 

2171 

2178 

2184 

2 91 

2197 

2204 

2210 



604 

2217 

2223 

2230 

2236 

22+3 

2249 

2256 

2263 

2269 

2276 



665 

82282 

82289 

82295 

82302 

82308 

82315 

82321 

82328 

82334 

82341 



666 

2347 

2354 

2360 

2367 

2373 

23S0 

2387 

2393 

2400 

2406 



667 

2413 

2419 

2426 

2432 

2439 

2445 

2452 

2458 

2465 

2471 



668 

2478 

2484 

2491 

2497 

2504 

2510 

2517 

2523 

2530 

2536 



669 

2543 

2549 

2556 

2562 

2569 

2575 

2582 

2588 

2595 

2601 



670 

82607 

82614 

82620 

82627 

82633 

82640 

826+6 

82G53 

82659 

82666 



671 

2672 

2679 

2685 

2692 

2698 

2705 

2711 

2718 

2724 

2730 



672 

2737 

2743 

2750 

2756 

2763 

2709 

2776 

2782 

2789 

2795 



673 

2802 

2808 

2814 

2821 

2827 

2834 

2840 

2847 

2853 

2860 



674 

2866 

2872 

2879 

2885 

2892 

2898 

2905 

2911 

2918 

2924 



675 

82930 

82937 

82943 

82950 

82956 

82903 

82969 

82975 

82982 

82988 



676 

2995 

3001 

3008 

3014 

3020 

3027 

3033 

3040 

3046 

3052 



677 

3059 

3065 

3072 

3078 

3085 

3091 

3097 

3104 

3110 

3117 



078 

3123 

3129 

3136 

3142 

3149 

3155 

3161 

3168 

3174 

3181 



679 

3187 

3193 

3200 

3206 

3213 

3219 

3225 

3232 

3238 

3245 



680 

83251 

83257 

83264 

83270 

83276 

83283 

83289 

83296 

83302 

83308 



681 

3315 

3321 

3327 

3334 

3340 

3347 

3353 

3359 

3366 

3372 



682 

3378 

3385 

3391 

3398 

3404 

3+10 

3+17 

3423 

3429 

3436 



683 

3442 

3448 

3455 

3461 

3467 

3474 

3480 

3487 

3493 

3499 



684 

3506 

3512 

3518 

3525 

3531 

3537 

3544 

3550 

3556 

3563 



685 

83569 

83575 

83582 

83588 

83594 

83601 

83607 

83613 

83620 

83626 


G 

686 

3632 

3639 

3645 

3651 

3658 

366+ 

3670 

8677 

3683 

3689 

1 

1 

687 

3696 

3702 

3708 

3715 

3721 

3727 

3734 

3740 

3746 

3753 

2 

1 

688 

3759 

3765 

3771 

3778 

3784 

3790 

3797 

3803 

3809 

3816 

3 

2 

689 

3822 

3828 

3835 

3841 

3847 

3853 

3860 

3866 

3872 

3879 

4 

2 

690 

83885 

83891 

83897 

83904 

83910 

83916 

83923 

83929 

83935 

83942 

5 

3 

691 

3948 

3954 

3960 

3967 

3973 

3979 

3985 

3992 

3998 

4004 

6 

4 

692 

4011 

4017 

4023 

4029 

4036 

4042 

4048 

4055 

4061 

4067 

7 

4 

693 

4073 

4080 

4086 

4092 

4098 

4105 

4111 

4117 

4123 

4130 

8 

5 

694 

4136 

4142 

4148 

4155 

4101 

4167 

4173 

4180 

4186 

4192 

9 

5 

695 

84198 

84205 

84211 

84217 

84223 

84230 

84236 

84242 

84248 

84255 



696 

4261 

4267 

4273 

4280 

4286 

4292 

429S 

4305 

4311 

4317 



697 

4323 

4330 

4336 

4342 

4348 

4354 

4361 

4367 

4373 

4379 



698 

4386 

4392 

4398 

4404 

4410 

4417 

4423 

4429 

4435 

4442 



699 

4448 

4454 

4460 

4466 

4473 

4479 

4485 

4491 

4497 

4504 



No. 

0 

1 

2 1 

3 1 

4 

5 

6 

7 

8 

9 




13 
































194 


Logarithms op Numbers, 


No. 7000 to 7G00. Logarithms. Log. 84510 to 88081. 


No. 

0 

1 

2 

3 

4 

5 

6 

rr 

i 

8 

9 



700 

84510 

84516 

84522 

84528 

84535 

84541 

84547 

84553 

84559 

845G6 


7 

701 

4572 

4578 

4584 

4590 

4597 

4603 

4609 

4615 

4621 

4628 

l 

1 

702 

4634 

4640 

464G 

4652 

4658 

4665 

4671 

4677 

4683 

4689 


1 

703 

4696 

4702 

4708 

4714 

4720 

4726 

4733 

4739 

4745 

4751 

3 

2 

704 

4757 

4763 

4770 

4776 

4782 

4788 

4794 

4800 

4807 

4813 

4 

3 

705 

84819 

84825 

84831 

84837 

84844 

84850 

84856 

84862 

84S68 

84S74 

5 

4 

70G 

4880 

4887 

4893 

4899 

4905 

4911 

4917 

4924 

4930 

4936 

0 

4 

707 

4942 

4948 

4954 

4960 

4967 

4973 

4979 

4985 

4991 

4997 

7 

5 

70S 

5003 

5009 

5016 

5022 

5028 

5034 

5040 

5046 

5052 

5058 

8 

6 

709 

5065 

5071 

5077 

5083 

5089 

5095 

5101 

5107 

5114 

5120 

9 

6 

710 

85126 

85132 

85138 

85144 

85150 

85156 

85103 

85169 

85175 

85181 



711 

5187 

5193 

5199 

5205 

5211 

5217 

5224 

5230 

5236 

5242 



712 

5248 

5254 

5260 

5266 

5272 

5278 

5285 

5291 

5297 

5303 



713 

5309 

5315 

5321 

5327 

5333 

5339 

5345 

5352 

5358 

5364 



714 

5370 

5376 

5382 

5388 

5394 

5400 

5406 

5412 

5418 

5425 



715 

85431 

85437 

85443 

85449 

85455 

85461 

85467 

85473 

85479 

85485 



716 

5491 

5497 

5503 

5509 

551G 

5522 

5528 

5534 

5540 

5546 



717 

5552 

5558 

5564 

5570 

5576 

5582 

5588 

5594 

5600 

5600 



718 

5612 

5618 

5625 

5631 

5637 

5643 

5649 

5655 

5661 

5667 



719 

5673 

5679 

5685 

5691 

5697 

5703 

5709 

5715 

5721 

5727 



720 

85733 

85739 

85745 

S5751 

85757 

85763 

85769 

85775 

85781 

85788 



721 

5794 

5800 

5806 

5812 

5818 

5824 

5S30 

5836 

5842 

5848 



722 

5854 

5800 

5866 

5872 

5878 

5SS4 

5890 

5S96 

5902 

5908 



723 

5914 

5920 

5926 

5932 

5938 

5944 

5950 

5956 

5962 

5968 



724 

5974 

5980 

5986 

5992 

5998 

6004 

6010 

6016 

6022 

6028 


o 

725 

86034 

86040 

86046 

86052 

86058 

86064 

86070 

86076 

8G082 

86088 

1 

1 

726 

6094 

6100 

6106 

6112 

6118 

6124 

6130 

6136 

6141 

0147 

2 

1 

727 

6153 

6159 

6165 

6171 

6177 

6183 

6189 

6195 

6201 

6207 

3 

2 

728 

6213 

6219 

6225 

6231 

6237 

6243 

6249 

6255 

6261 

6267 

4 

2 

729 

6273 

6279 

6285 

6291 

6297 

6303 

6308 

6314 

6320 

6326 

5 


730 

86332 

86338 

86344 

86350 

86356 

86362 

8(5368 

86374 

86380 

86386 

g 

4 

731 

6392 

6398 

6404 

6410 

6415 

6421 

6427 

6433 

6439 

6445 

7 

4 

732 

6451 

6457 

6463 

6469 

6475 

6481 

6487 

6493 

0499 

6504 

g 

ft 

733 

6510 

6516 

6522 

6528 

6534 

6540 

6546 

6552 

6558 

6564 

9 

ft 

734 


6576 

6581 

6587 

6593 

6599 

6605 

6611 

6617 

6623 



735 

86629 

86635 

86641 

86646 

86652 

86658 

86664 

86670 

86676 

86682 



736 

6688 

6694 

6700 

6705 

6711 

6717 

6723 

6729 

6735 

6741 



737 

6747 

6753 

6769 

6764 

6770 

6776 

6782 

6788 

6794 

6800 



738 

6806 

6812 

6817 

6823 

6829 

6835 

6841 

6847 

6853 

6859 



739 

6864 

6S70 

6876 

6882 

6888 

6894 

6900 

6906 

6911 

6917 



740 

86923 

86929 

86935 

86941 

86947 

86953 

86958 

86964 

86970 

86976 



741 

6982 

6988 

6994 

6999 

7005 

7011 

7017 

7023 

7029 

7035 



742 

7040 

7046 

7052 

7058 

7064 

7070 

7075 

7081 

7087 

7093 



743 

7099 

7105 

7111 

7116 

7122 

7128 

7134 

7140 

7146 

7151 



744 

7157 

7163 

7169 

7175 

7181 

7186 

7192 

7198 

7204 

7210 



745 

87216 

87221 

87227 

87233 

87239 

87245 

87251 

87256 

87262 

87268 



746 

7274 

7280 

7286 

7291 

7297 

7303 

7309 

7315 

7320 

7326 



747 

7332 

7338 

7344 

7349 

7355 

73,61 

7367 

7373 

7379 

7384 



748 

7390 

7396 

7402 

7408 

7413 

7419 

7425 

7431 

7437 

7442 



749 

7448 

7454 

7460 

7466 

7471 

7477 

7483 

7489 

7495 

7500 



750 

87506 

87512 

87518 

87523 

87529 

87535 

87541 

87547 

87552 

87558 


6 

751 

7564 

7570 

7576 

7581 

7587 

7593 

7599 

7604 

7610 

7616 

i 

1 

752 

7622 

7628 

7633 

7639 

7645 

7651 

7656 

7662 

7668 

7674 

2 

1 

763 

7679 

7685 

7691 

7697 

7703 

7708 

7714 

7720 

7726 

7731 

3 

2 

754 

7737 

7743 

7749 

7754 

7760 

7766 

7772 

7777 

7783 

7789 

4 

2 

755 

87795 

87800 

87806 

87812 

87818 

87823 

87829 

87835 

87841 

87846 

5 

n 

O 

756 

7852 

7858 

7864 

7869 

7875 

7881 

7887 

7892 

7898 

7904 

6 

3 

757 

7910 

7915 

7921 

7927 

7933 

7938 

7944 

7950 

7955 

7961 

7 

4 

758 

7967 

7973 

7978 

7984 

7990 

7996 

8001 

8007 

8013 

8018 

8 

4 

759 

8024 

8030 

8036 

8041 

8047 

8053 

8058 

8064 

8070 

8076 

9 

5 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 



























Logarithms of Numbers. 195 


No. 7G00 to 8200. 


Logarithms. 

Log. 88081 to 91381 


No. 

0 

1 

9 

O 

O 

4 

5 

1 6 

7 

8 

9 



700 

88081 

88087 

88093 

88098 

88104 

88110 

88116 

88121 

8S127 

88133 

1 

u 

1 

701 

8138 

8144 

8150 

8156 

8161 

8167 

8173 

8178 

8184 

8190 

o 

1 

762 

8195 

8201 

8207 

8213 

8218 

8224 

8230 

8235 

8241 

8247 


9 

76;i 

8252 

8258 

8264 

8270 

8275 

8281 

8287 

8292 

8298 

83 U4 

f 

9 

76*1 

8309 

8315 

8321 

8326 

8332 

8338 

8343 

8349 

8355 

8360 



765 

88366 

88372 

88377 

88383 

88389 

88395 

8S400 

88406 

88412 

88417 

ft 

o 

j. 

766 

8123 

8429 

84:34 

440 

8146 

8451 

8457 

8463 

8468 

8474 

7 

A 

767 

8480 

8485 

8491 

8497 

8502 

8508 

8513 

8519 

8525 

8530 



768 

8536 

8542 

8547 

8553 

8569 

8564 

8570 

8576 

8581 

8587 

0 

5 

.769 

8593 

8598 

8604 

8610 

8615 

8621 

8627 

8632 

8638 

8643 



770 

88619 

S8655 

88660 

88666 

88672 

88677 

88683 

88689 

88694 

88700 



771 

8705 

8711 

8717 

8722 

8728 

8734 

8739 

8745 

8750 

8756 



772 

8762 

8767 

8773 

8779 

8784 

8790 

8795 

8801 

8807 

8812 



773 

8818 

8S24 

8829 

8835 

8840 

8840 

8852 

8857 

8863 

8868 



774 

8874 

8880 

88S5 

8891 

8897 

8902 

8908 

8913 

8919 

8925 



775 

8 >930 

88936 

88941 

88947 

88953 

88958 

88964 

88969 

88975 

88981 



776 

8986 

8992 

8997 

9003 

9009 

9014 

9020 

9025 

9031 

9037 



777 

9042 

9048 

9053 

9(159 

9064 

9070 

9076 

9081 

9087 

9092 



778 

9098 

9104 

9109 

9115 

9120 

9126 

9131 

9137 

9143 

9148 



779 

9154 

9159 

9165 

9170 

9176 

9182 

9187 

9193 

9198 

9204 



780 

89209 

89215 

89221 

89226 

89232 

89237 

89243 

89248 

89254 

89260 



781 

9265 

9271 

9276 

9282 

9287 

9293 

9298 

9304 

9310 

9315 



782 

9321 

9326 

9332 

9337 

9343 

9348 

9354 

9360 

9365 

9371 



783 

9376 

9382 

9387 

9393 

9398 

9404 

9409 

9415 

9421 

9126 



784 

9432 

9437 

9443 

9448 

9454 

9459 

9465 

9470 

9476 

9481 



785 

89487 

89492 

89498 

89504 

89509 

S9515 

89520 

89526 

89531 

89537 



786 

9542 

9548 

9553 

9559 

9564 

9570 

9575 

9581 

9586 

9592 



787 

9597 

9603 

9609 

9614 

9620 

9625 

9631 

9636 

9042 

9647 



788 

9653 

9658 

9664 

9669 

9675 

9680 

9686 

9691 

9097 

9762 



789 

9708 

9713 

9719 

9724 

9730 

9735 

9741 

9746 

9752 

9757 



790 

89763 

89768 

89774 

89779 

89785 

89790 

89796 

89801 

89807 

89812 



791 

9818 

9823 

9829 

9834 

9840 

9845 

9851 

9856 

9862 

9867 



792 

9873 

9878 

9883 

9889 

9894 

9900 

9905 

9911 

9916 

9922 



793 

9927 

9933 

9938 

9944 

9949 

9955 

9960 

9966 

9971 

9977 



794 

9982 

9988 

9993 

9998 

90004 

90U09 

90015 

90020 

90026 

90031 



795 

90037 

90042 

90U48 

90053 

90059 

90064 

90069 

90075 

90080 

90086 



796 

0091 

0097 

0102 

0108 

0113 

0119 

0124 

0129 

0135 

0140 



797 

0146 

0151 

0157 

0162 

0168 

0173 

0179 

0184 

0189 

0195 



798 

0200 

0206 

0211 

0217 

0222 

0227 

0233 

0238 

0244 

0249 



799 

0255 

0260 

0266 

0271 

0276 

0282 

0287 

0293 

0298 

0304 



800 

90309 

90314 

90320 

90325 

90331 

90336 

90342 

90347 

90352 

90358 



801 

0363 

0369 

0374 

0380 

0385 

0390 

0396 

0401 

0407 

0412 



802 

0417 

0423 

0428 

0434 

0439 

0445 

0450 

0455 

0461 

0466 



803 

0172 

0477 

0482 

0488 

0493 

0499 

0504 

0509 

0515 

0520 



804 

0526 

0531 

0536 

0542 

0547 

0553 

0558 

0563 

0569 

0574 



805 

90580 

90585 

90590 

90596 

90601 

90607 

90612 

90617 

90623 

90628 


5 

806 

0634 

0639 

0644 

0650 

0655 

0660 

0666 

0671 

0677 

0682 

l 

1 

807 

0687 

0693 

0698 

0703 

0709 

0714 

0720 

0725 

0730 

0736 

2 

1 

808 

0741 

0747 

0752 

0757 

0763 

0768 

0773 

0779 

0784 

0789 

3 

2 

809 

0795 

0800 

0806 

0811 

0816 

(1822 

0827 

0832 

0838 

0843 

4 

2 

810 

90849 

90854 

90859 

90865 

90870 

90875 

90881 

90886 

90891 

90897 

5 

3 

Sll 

0902 

0907 

0913 

0918 

0924 

0929 

0934 

0940 

0945 

0950 

6 

3 

812 

0956 

0961 

0966 

0972 

0977 

0982 

0988 

0993 

0998 

1004 

7 

4 

813 

1009 

1014 

1020 

1025 

1030 

1036 

1041 

1046 

1052 

1057 

8 

4 

814 

1062 

1068 

1073 

1078 

1084 

1089 

1094 

1100 

1105 

1110 

9 

5 

815 

91116 

91121 

91126 

91132 

91137 

91142 

91148 

91153 

91158 

91164 



816 

1169 

1174 

1180 

1185 

1190 

1196 

1201 

1206 

1212 

1217 



817 

1222 

1228 

1233 

1238 

1243 

1249 

1254 

1259 

1265 

1270 



818 

1275 

1281 

1286 

1291 

1297 

1302 

1307 

1312 

1318 

1323 



819 

1328 

1334 

1339 

1341 

1350 

1355 

1360 

1365 

1371 

1374 



No. 

0 

1 

2 

O 

O 

4 

5 

G 

7 

8 

9 





































196 


Logarithms of Numbers 


No. 8200 to 8800. Logarithms. L/Og. 91381 to 94448. 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

S' 

G 

820 

91381 

91387 

91392 

91397 

91403 

91408 

91413 

91418 

91424 

91429 

1 

1 

821 

1434 

1440 

1445 

1450 

1455 

1461 

1466 

1471 

1477 

1482 

2 

1 

822 

14S7 

1492 

1498 

1603 

1508 

1514 

1519 

1524 

1529 

1535 

3 

2 

823 

1540 

1545 

1551 

1556 

1561 

1566 

1572 

1577 

1582 

1587 

4 

2 

824 

1593 

1598 

1603 

161.9 

1614 

1619 

1624 

1630 

1635 

1640 

5 

3 

825 

91645 

91651 

91656 

91661 

91666 

91672 

91677 

91682 

91687 

91693 

6 

4 

82G 

1698 

1703 

1709 

1714 

1719 

1724 

1730 

1735 

1740 

1745 

7 

4 

827 

1751 

1756 

1761 

1766 

1772 

1777 

1782 

1787 

1793 

1798 

8 

5 

828 

1803 

1808 

1814 

1819 

1824 

1829 

1834 

1840 

1845 

1850 

9 

5 

829 

1855 

1861 

I860 

1871 

1876 

1882 

1887 

1892 

1897 

1903 



83C 

91908 

91913 

91918 

91924 

91929 

91934 

91939 

91944 

91950 

91955 



831 

1960 

1965 

1971 

1976 

1981 

1986 

19J1 

1997 

2002 

2<i07 



832 

2012 

2018 

2023 

2028 

2033 

2038 

2044 

2049 

2054 

2059 



833 

2065 

2()70 

2075 

2080 

2085 

2091 

21)96 

2101 

2106 

2111 



834 

2117 

2122 

2127 

2132 

2137 

2143 

2148 

2153 

2158 

2163 



835 

92169 

92174 

92179 

92184 

92189 

92195 

92200 

92205 

92210 

92215 



836 

2221 

2226 

2231 

2236 

2241 

2247 

2252 

2257 

2262 

2267 



837 

2273 

2278 

2283 

2288 

2293 

2298 

2304 

2309 

2314 

2319 



838 

2324 

2330 

2335 

2340 

2345 

2350 

2355 

2361 

2366 

2371 



839 

2376 

2381 

2387 

2392 

2397 

2402 

2407 

2412 

2418 

2423 



840 

92428 

92433 

92438 

92443 

92449 

92454 

92459 

92464 

92469 

92474 



841 

2480 

2485 

2490 

2495 

2500 

2505 

2511 

2516 

2521 

2526 



842 

2531 

2536 

2542 

2.547 

2552 

2557 

2562 

2567 

2572 

2578 



843 

2583 

2588 

2593 

2598 

2603 

2609 

2614 

2619 

2621 

2629 



844 

2634 

2639 

2645 

2650 

2655 

2660 

2665 

2670 

2675 

2681 


5 

845 

92686 

92691 

92696 

92701 

92706 

92711 

92716 

92722 

92727 

92732 

1 

1 

846 

2737 

2742 

2747 

2752 

2758 

2763 

2768 

2773 

2778 

2783 

2 

1 

847 

2788 

2793 

2799 

2804 

2809 

2814 

2819 

2824 

2829 

2834 

3 

2 

848 

2840 

2845 

2850 

2855 

2860 

2865 

2870 

2875 

2881 

2S86 

4 

2 

849 

2891 

2896 

2901 

2906 

2911 

2916 

2921 

2927 

2932 

2937 

5 

3 

850 

92942 

92947 

92952 

92957 

92962 

92967 

92973 

92978 

92983 

92988 

(5 

3 

851 

2993 

2998 

3003 

3008 

3013 

3018 

3024 

3029 

3034 

3039 

7 

4 

852 

3044 

3049 

3054 

3059 

3064 

3069 

3075 

3080 

3085 

3090 

8 

4 

853 

3095 

3100 

3105 

3110 

3115 

3120 

3125 

3131 

3136 

3141 

9 

5 

854 

3146 

3151 

3156 

3161 

3166 

3171 

3176 

3181 

3186 

3192 



855 

93197 

93202 

93207 

93212 

93217 

93222 

93227 

93232 

93237 

93242 



856 

3247 

3252 

3258 

3263 

3268 

3273 

3278 

3283 

3288 

3293 



857 

3298 

3303 

31308 

3313 

3318 

3323 

<.'328 

3534 

3539 

3344 



858 

3349 

3354 

3359 

3364 

3369 

3374 

3379 

3384 

3389 

3394 



859 

3399 

3404 

3409 

3414 

3420 

3425 

3430 

3435 

3440 

3145 



860 

93450 

93 ,55 

934«3<j 

93465 

93470 

93475 

93480 

93185 

93490 

93495 



861 

3500 

3505 

3510 

3515 

3520 

3526 

3531 

3536 

3541 

3546 



862 

3 <51 

3556 

3561 

3566 

3571 

35 < 0 

3581 

3586 

3591 

3596 



863 

3601 

3606 

3611 

3616 

3621 

3626 

3631 

3636 

3641 

3646 



864 

3651 

3656 

3661 

3666 

3671 

3676 

3682 

3687 

3692 

3697 



865 

93702 

93707 

93712 

93717 

93722 

93727 

93732 

93737 

93742 

93747 



866 

3752 

3757 

3762 

3767 

3772 

3777 

3782 

3787 

3792 

3797 



867 

3802 

3807 

3812 

3817 

3822 

3827 

3832 

3837 

3842 

3847 



868 

3852 

3857 

3862 

3867 

3872 

3877 

3-82 

3887 

3892 

3897 



869 

3902 

3907 

3912 

3917 

3922 

3927 

3932 

3937 

3942 

3947 



870 

93962 

93957 

93962 

93967 

93972 

93977 

93982 

93987 

93992 

93997 


4 

871 

4002 

4007 

4012 

4017 

4022 

4027 

4032 

4037 

4042 

4047 

i 

0 

872 

4052 

4057 

4962 

4067 

4072 

4077 

4082 

4086 

4091 

4096 

2 

1 

873 

4101 

4106 

4111 

4116 

4121 

4126 

4131 

4136 

4141 

4146 

3 

1 

874 

4151 

4156 

4161 

4166 

4171 

4176 

4181 

4186 

4191 

4196 

4 

2 

875 

94201 

94206 

94211 

94216 

9422 L 

9422G 

94231 

94236 

94240 

94245 

5 

2 

876 

4250 

4255 

4260 

4265 

4270 

4275 

4280 

4285 

4290 

4295 

6 

2 

877 

4300 

4305 

4310 

4315 

4320 

4325 

4330 

4335 

4340 

4345 

7 

3 

878 

4349 

4354 

4359 

4364 

4369 

4374 

4379 

4384 

4389 

4394 

8 

3 

879 

4399 

4404 

4109 

4414 

4419 

4424 

4429 

4433 

4438 

4443 

9 

4 

No. 

0 

1 

2 

3 

4 

5 

1 6 

7 

8 

9 































Logarithms op Numbers 


197 


No. 8800 to 9400. Logarithms. I^og. 94448 to 97313. 


No 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 



880 

94448 

94453 

94458 

94463 

94468 

94473 

94478 

94483 

94488 

94493 

1 

J 

1 

881 

4498 

4503 

4507 

4512 

4517 

4522 

4527 

4532 

4537 

4542 

2 

1 

882 

4547 

4552 

4557 

4562 

4567 

4571 

4576 

4581 

4580 

4591 

‘f 

o 

883 

4596 

4601 

4606 

4011 

4016 

4621 

4626 

4030 

4035 

4040 

4 

•> 

881 

4045 

4050 

4055 

4000 

4665 

4670 

4075 

4680 

4085 

4689 

r. 

•A 

885 

94094 

94099 

94704 

94709 

94714 

91719 

94724 

94729 

94734 

94738 

t: 

A 

880 

4743 

4748 

4753 

4758 

4703 

4708 

4773 

4778 

4783 

4787 

7 

A 

887 

4792 

4797 

4802 

4807 

4812 

4817 

4822 

4827 

4832 

4836 


A 

888 

4841 

4840 

4851 

4850 

4861 

4866 

4871 

4876 

4880 

4885 



889 

4890 

4895 

4900 

4905 

4910 

4915 

4919 

4924 

4929 

4934 



890 

94939 

94944 

94949 

94954 

94959 

94963 

94968 

94973 

94978 

94983 



891 

4988 

4993 

4998 

5002 

5007 

5012 

5017 

5022 

5027 

5032 



892 

5030 

5041 

5046 

5051 

5056 

5001 

5060 

5071 

5075 

5080 



893 

5085 

5090 

5095 

5100 

5105 

5109 

5114 

5119 

5124 

5129 



894 

5134 

5139 

5143 

6148 

5153 

5158 

5163 

5108 

5173 

5177 



S95 

95182 

95187 

95192 

95197 

95202 

95207 

95211 

95216 

95221 

95226 



896 

5231 

5230 

5240 

5245 

5250 

5255 

6200 

5205 

5270 

5274 



897 

5279 

5284 

5289 

5294 

5299 

5303 

5308 

5313 

5318 

5323 



898 

5328 

5332 

5337 

5342 

5347 

5352 

5357 

5301 

5306 

5371 



899 

5376 

5381 

5386 

5390 

5395 

5400 

5405 

5410 

5415 

5419 



900 

95424 

95429 

95434 

95439 

95444 

95448 

95453 

95458 

95463 

95468 



901 

5472 

5477 

5482 

5487 

5492 

5497 

5501 

5506 

5511 

5516 



902 

5521 

5525 

5530 

5535 

5540 

5545 

5550 

5554 

5559 

5564 



903 

5509 

5574 

5578 

5583 

5588 

5593 

5598 

5002 

5007 

5612 



904 

5617 

5622 

5026 

5631 

5636 

5641 

5646 

5050 

5655 

5660 



905 

95005 

95070 

95074 

95679 

95684 

95689 

95694 

95698 

95703 

95708 



900 

5713 

5718 

5722 

5727 

5732 

5737 

5742 

5746 

5751 

5756 



907 

5701 

5706 

5770 

5775 

5780 

5785 

5789 

5794 

5799 

5804 



908 

5809 

5813 

5818 

5823 

5828 

5832 

5837 

5842 

5847 

5852 



909 

5856 

5861 

5800 

5871 

5875 

5880 

5885 

5890 

5895 

5899 



910 

95904 

95909 

95914 

95918 

95923 

95928 

95933 

95938 

95942 

95947 



911 

5952 

6957 

5901 

5906 

5971 

5970 

5980 

5985 

5990 

5995 



912 

5999 

6004 

6009 

6014 

6019 

6023 

6028 

6033 

6938 

6042 



913 

6047 

0052 

6057 

6061 

6066 

6071 

6076 

0080 

0085 

GOOD 



914 

6095 

6099 

6104 

6109 

6114 

6118 

6123 

0128 

6133 

6137 



915 

90142 

90147 

96152 

96156 

90161 

96160 

96171 

90175 

90180 

96185 



910 

0190 

6194 

6199 

6204 

6209 

0213 

6218 

0223 

6227 

6232 



917 

0237 

6242 

6246 

6251 

6256 

0261 

6265 

6270 

0275 

6280 



918 

0284 

6289 

6294 

0298 

6303 

0308 

6313 

0317 

6322 

6327 



919 

0332 

6336 

6341 

6340 

6:350 

6355 

6360 

6365 

6309 

6374 



920 

90379 

90384 

96388 

96393 

96398 

96402 

96407 

90412 

90417 

96421 



921 

0426 

0431 

0435 

6440 

6445 

6450 

6454 

0459 

6404 

6468 



922 

0473 

G478 

6483 

0487 

0492 

6497 

6501 

6506 

6511 

6515 



923 

6520 

6525 

6530 

0534 

6539 

6544 

6548 

6553 

6558 

6562 



924 

6567 

6572 

6577 

6581 

6586 

6591 

6505 

0000 

6605 

6609 



925 

96014 

90019 

96024 

96028 

96033 

96638 

96G42 

90047 

96052 

96656 


4 

920 

0661 

0060 

6070 

6675 

6080 

6685 

6089 

6094 

0G99 

6703 

l 

0 

927 

6708 

0713 

0717 

0722 

0727 

6731 

6736 

0741 

6745 

0750 

2 

1 

928 

6755 

6759 

6704 

G7C9 

0774 

6778 

0783 

0788 

0792 

6797 

3 

1 

929 

6802 

6806 

6811 

6816 

6820 

6825 

6830 

0834 

0839 

6844 

4 

2 

930 

96848 

90853 

90858 

90802 

96867 

96872 

96876 

90881 

90880 

90890 

5 

2 

931 

6895 

0900 

0904 

G909 

6914 

0918 

6923 

0928 

6932 

6937 

6 

2 

932 

6942 

0946 

0951 

0956 

6900 

0965 

6970 

0974 

0979 

0984 

7 

3 

933 

6988 

6993 

0997 

7002 

7007 

7011 

7016 

7021 

7025 

7030 

8 

3 

934 

7035 

7039 

7044 

7049 

7053 

7058 

7063 

7007 

7072 

7077 

9 

4 

935 

97081 

97 080 

97090 

97095 

97100 

97104 

97109 

97114 

97118 

9 1 123 



936 

7128 

7132 

7137 

7142 

7146 

7151 

7155 

7160 

7165 

7109 



937 

7174 

7179 

7183 

7188 

7192 

7197 

7202 

7206 

7211 

7210 



938 

7220 

7225 

7230 

7231 

7239 

7243 

7248 

7253 

7257 

7202 



939 

7267 

7271 

7276 

7280 

7285 

7290 

7294 

7299 

7304 

7308 



No. 

0 

1 

2 1 

3 

4 

5 

6 

7 

8 

9 



































193 Logarithms op Numbers. 


No. 9400 to 10000. Logarithms. Log. 9/313 to 99996. 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


5 

940 

97313 

97317 

97322 

97327 

97331 

97336 

97340 

97345 

97350 

97354 

l 

1 

941 

7359 

7364 

7368 

7373 

7377 

7382 

7387 

7391 

7396 

7400 

2 

1 

942 

7405 

7410 

7414 

7419 

7421 

7428 

7433 

•7437 

7442 

7447 

3 

2 

943 

7451 

7456 

7460 

7465 

7470 

7474 

7479 

7483 

7488 

7493 

4 

2 

944 

7497 

7502 

7506 

7511 

7516 

7520 

7525 

7529 

7534 

7539 

5 

3 

945 

97543 

97548 

97552 

97557 

97562 

97566 

97571 

97575 

97580 

97585 

6 

3 

946 

7589 

7594 

7598 

7603 

7607 

7612 

7617 

7621 

7626 

7630 

7 

4 

947 

7635 

7640 

7644 

7649 

7653 

7658 

7663 

7667 

7672 

7676 

8 

4 

948 

7681 

7685 

7690 

7695 

7619 

7704 

7708 

7713 

7717 

7722 

9 

5 

949 

7727 

7731 

7736 

7740 

7745 

7749 

7754 

7759 

7763 

7768 



950 

97772 

97777 

97782 

977S6 

97791 

97795 

97800 

97804 

97809 

97813 



951 

7818 

7823 

7827 

7832 

7836 

7841 

7845 

7850 

7855 

7859 



952 

7864 

7868 

7873 

7877 

7882 

7886 

7s91 

7896 

7900 

7905 



958 

7909 

7914 

7918 

7923 

7928 

7932 

7937 

71141 

7946 

7950 



954 

7955 

7959 

7964 

7968 

7973 

7978 

79S2 

7987 

7991 

7996 



955 

98000 

98005 

98009 

98014 

98019 

98023 

98028 

98032 

98037 

98041 



956 

8046 

8050 

8055 

8U59 

8004 

8068 

8073 

8078 

8082 

8087 



957 

8091 

8096 

8100 

8105 

8109 

8114 

8118 

8123 

8127 

8132 



95S 

8137 

8141 

8146 

8150 

8155 

8159 

8104 

8168 

8173 

8177 



959 

8182 

8186 

8191 

8195 

8200 

8204 

8209 

8214 

8218 

8223 



960 

9S227 

98232 

98236 

98241 

9S245 

98250 

98254 

9S259 

98203 

98268 



961 

8272 

8277 

8281 

8286 

8290 

8295 

8299 

8304 

8308 

8313 



962 

8318 

8322 

8327 

8331 

8336 

8340 

8345 

8349 

8354 

8358 



963 

8363 

8367 

8372 

8376 

8381 

8385 

8390 

8394 

8399 

8403 



964 

8408 

8412 

8417 

8421 

8426 

8430 

8435 

8439 

8444 

8448 



965 

98453 

98457 

98462 

9S466 

98471 

9S475 

98480 

98484 

98489 

98493 



966 

8498 

8502 

8507 

8511 

8510 

8520 

8525 

8529 

8534 

8538 



967 

8543 

8647 

8552 

8556 

8561 

8565 

8570 

8574 

~8579 

8583 



968 

8588 

8592 

8597 

8601 

8005 

8610 

8614 

8619 

8623 

8628 



969 

8632 

8637 

8641 

8646 

8650 

8655 

8059 

8064 

8668 

8673 



970 

98677 

9S682 

986S6 

98691 

98695 

98700 

98704 

98709 

9S713 

98717 



971 

8722 

8726 

8731 

8735 

8740 

8741 

8749 

8753 

8758 

8762 



972 

8767 

8771 

8776 

8780 

8784 

8789 

8793 

8798 

8802 

8807 



973 

8811 

8816 

8820 

8825 

8829 

8834 

8838 

8843 

8847 

8851 



974 

8856 

8S60 

8865 

8869 

8874 

8878 

8883 

8887 

8892 

8896 



975 

98900 

98906 

98909 

98914 

98918 

98923 

98927 

98932 

98936 

98941 



976 

8945 

8949 

8954 

8958 

8963 

8967 

8972 

8976 

8981 

8986 



977 

8989 

8994 

8998 

9003 

9007 

9012 

9016 

9021 

9025 

9029 



978 

9034 

9038 

9043 

9047 

9052 

9050 

9061 

9065 

9069 

9074 



979 

9078 

9083 

9087 

9092 

9096 

9100 

9105 

9109 

9114 

9118 



980 

99123 

99127 

99131 

99130 

99140 

99145 

99149 

99154 

99158* 

99162 



981 

9167 

9171 

9170 

9180 

9185 

9189 

9193 

9198 

9202 

9207 



9S2 

9211 

9216 

9220 

9224 

9229 

9233 

9238 

9242 

9247 

9251 



983 

9255 

9260 

9204 

9269 

9273 

9277 

9282 

9286 

9291 

9295 



984 

9300 

9304 

9308 

9313 

9317 

9322 

9320 

9330 

9335 

9339 



985 

99344 

99348 

99352 

99357 

99361 

99366 

99370 

99374 

99379 

99383 


4 

986 

9388 

9392 

9396 

9401 

9405 

9410 

9414 

9419 

9423 

9427 

1 

0 

987 

9432 

9436 

9441 

9446 

9449 

9454 

9458 

9463 

9467 

9471 

2 

1 

988 

9476 

9480 

9484 

94S9 

9493 

9498 

9502 

9506 

9511 

9515 

3 

1 

989 

9520 

9524 

9528 

9533 

9537 

9542 

9546 

9550 

9555 

9559 

4 

2 

990 

99564 

99568 

99572 

99577 

99581 

99585 

99590 

99594 

99599 

99603 

5 

2 

991 

9607 

9612 

9616 

9621 

9625 

9629 

9634 

9638 

9642 

9647 

6 

2 

992 

9651 

9656 

9660 

9664 

9669 

9673 

9677 

9682 

9686 

9691 

7 

3 

993 

9695 

9699 

9704 

9708 

9712 

9717 

9721 

9720 

9730 

9734 

8 

3 

994 

9739 

9743 

9747 

9752 

9756 

9760 

9765 

9769 

9774 

9778 

9 

4 

995 

99782 

99787 

99791 

99795 

99800 

99804 

99808 

99813 

99817 

99822 



996 

9826 

9830 

9835 

9839 

9843 

9848 

9852 

9856 

9861 

9865 



997 

9870 

9874 

9878 

98 S3 

9887 

9891 

9896 

9900 

9904 

9909 



998 

9913 

9917 

9922 

9926 

9930 

9935 

9939 

9944 

9948 

9952 



999 

9957 

9961 

9965 

9970 

9974 

9978 

9983 

9987 

9991 

9996 



No. 

0 

1 

2 

1 3 

4 

5 

6 

7 

8 

9 


I 


































LoGARifHMS Trigonometric. 


199 


O' 1 

0° 



Logarithms. 


179 c 

ll h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

00 

0 

Inf.Neg. 

Infinite. 

Inf. Nee;. 

Infinite. 

10.00000 

10.00000 

60 

CO 

4 

l 

6.40373 

13.53027 

0.46373 

13.53027 

00000 

OOOOO 

59 

50 

8 

2 

70470 

23524 

70470 

23524 

00000 

ooooo 

58 

52 

12 

3 

94085 

05915 

94085 

05915 

00000 

ooooo 

57 

48 

10 

4 

7.00579 

12.93421 

7.06579 

12.93421 

ooooo 

ooooo 

56 

44 

20 

5 

7.10270 

12.83730 

7.10270 

12.83730 

10.00000 

10.00000 

55 

40 

24 

0 

24188 

75812 

24188 

75812 

ooooo 

ooooo 

54 

36 

28 

7 

30882 

09118 

30882 

69118 

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53 

32 

32 

8 

30082 

63318 

30082 

63318 

ooooo 

ooooo 

52 

28 

30 

9 

41797 

58203 

41797 

58203 

ooooo 

ooooo 

51 

24 

40 

10 

7.40373 

12.53027 

7.46373 

12.53627 

10.00000 

10.00000 

50 

20 

44 

11 

50512 

49488 

50512 

49488 

ooooo 

ooooo 

49 

10 

48 

12 

64291 

45709 

54291 

45709 

ooooo 

ooooo 

48 

12 

52 

13 

57707 

42233 

57707 

42233 

ooooo 

ooooo 

47 

8 

50 

14 

00985 

39015 

60986 

39014 

ooooo 

OOOOO 

46 

4 

1 

15 

7.03982 

12.30018 

7.03982 

12.30018 

10.00000 

10.00000 

45 

59 

4 

10 

00784 

33210 

00785 

33215 

ooooo 

ooooo 

44 

50 

8 

17 

09417 

305S3 

09418 

30582 

OOiXll 

9.99999 

43 

52 

12 

18 

71900 

28100 

71900 

28100 

ooooi 

99999 

42 

48 

10 

19 

74248 

25752 

74248 

25752 

00001 

99999 

41 

44 

20 

20 

7.70475 

12.23525 

7.76470 

12.23524 

10.00001 

9.99999 

40 

40 

24 

21 

78594 

21400 

78595 

21405 

00001 

99999 

39 

36 

28 

22 

80015 

19385 

80015 

19385 

ooooi 

99999 

38 

32 

32 

23 

82545 

17455 

82546 

17454 

00001 

99999 

37 

28 

30 

24 

84393 

15607 

84394 

15606 

OOOOI 

99999 

30 

24 

40 

25 

7.8G1GG 

12.13834 

7.86167 

12.13833 

10.00001 

9.9 Q 999 

35 

20 

44 

20 

87S70 

12130 

87871 

12129 

OOOOI 

99999 

34 

10 

48 

27 

89509 

10491 

89510 

10490 

OOOOI 

99999 

33 

12 

52 

28 

91088 

08912 

91089 

08911 

00001 

99999 

32 

8 

50 

29 

92612 

07388 

92013 

07387 

00002 

99998 

31 

4 

2 

30 

7.94084 

12.05910 

7.94086 

12.05914 

10.00002 

9.99998 

30 

58 

4 

31 

95508 

04492 

95510 

04490 

00002 

99998 

29 

56 

8 

32 

90887 

03113 

96889 

03111 

00002 

9999S 

28 

52 

12 

33 

98223 

01777 

98225 

01775 

00002 

99998 

27 

48 

10 

34 

99520 

00480 

99522 

00478 

00002 

99998 

20 

44 

20 

35 

8.00779 

11.99221 

8.00781 

11.99219 

10.00002 

9.99998 

25 

40 

24 

30 

02002 

97998 

02004 

97996 

00002 

99998 

24 

3G 

28 

37 

03192 

90808 

03194 

90800 

00003 

99997 

23 

32 

32 

38 

04350 

95050 

04353 

95047 

00003 

99997 

22 

28 

30 

39 

05478 

94522 

05481 

94519 

00003 

99997 

21 

24 

40 

40 

8.00578 

11.93422 

8.00581 

11.93419 

10.00003 

9.99997 

20 

20 

44 

41 

07050 

92350 

07053 

92347 

00003 

99997 

19 

16 

48 

42 

08090 

91304 

08700 

91300 

00003 

99997 

18 

12 

52 

43 

09718 

90282 

09722 

90278 

00003 

99997 

17 

8 

5G 

41 

10717 

89283 

10720 

89280 

00004 

99990 

10 

4 

3 

45 

8.11093 

11.88307 

8.11096 

11.88304 

10.00004 

9.99990 

15 

57 

4 

40 

12047 

87353 

12651 

87349 

00004 

99990 

14 

56 

8 

47 

13581 

86419 

13585 

80415 

00004 

99990 

13 

52 

12 

48 

14495 

85505 

14500 

85500 

00004 

99996 

12 

48 

10 

49 

15391 

84609 

15395 

84005 

00004 

99990 

11 

44 

20 

50 

8.10208 

11.83732 

8.10273 

11.83727 

10.00005 

9.99995 

10 

40 

24 

51 

17128 

82872 

17133 

82807 

00005 

99995 

9 

36 

28 

52 

17971 

82029 

17976 

82024 

00005 

99995 

8 

32 

32 

53 

18798 

81202 

18804 

81190 

00005 

99995 

7 

28 

30 

54 

19010 

80390 

19610 

80384 

00005 

99995 

6- 

24 

40 

55 

8.20407 

11.79593 

8.20413 

11.79587 

10.00000 

9.99994 

5 

20 

44 

50 

21189 

78811 

21195 

78805 

ooooo 

99994 

4 

10 

48 

57 

21958 

78042 

21904 

78036 

ooooo 

99994 

3 

12 

62 

58 

22713 

77287 

22720 

77280 

ooooo 

99994 

2 

8 

50 

59 

23450 

76544 

23462 

76538 

ooooo 

99994 

1 

4 

4 

00 

24180 

75814 

24192 

75808 

00007 

99993 

0 

56 

M. 8. 

M 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

\r. s. 

GM 

£)0° 





89° 

5 h 




















200 Logarithms Trigonometric. 


0 h 

1° 



Logarithms. 


178° 

ll h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

4 

0 

8.24186 

11.75814 

8.24192 

11.75808 

10.00007 

9.99993 

60 

50 

4 

1 

24903 

75097 

24910 

75090 

00007 

99993 

59 

56 

8 

2 

25609 

74391 

25616 

74384 

00007 

99993 

58 

52 

12 

3 

26304 

73696 

26312 

73688 

00007 

99993 

57 

48 

16 

4 

26988 

73012 

26996 

. 73004 

00008 

99992 

56 

44 

20 

5 

8.27661 

11.72339 

8.27669 

11.72331 

10.00008 

9.99992 

55 

40 

24 

(i 

28324 

71676 

28332 

71668 

00008 

99992 

54 

36 

28 

7 

28977 

71023 

28986 

71014 

00008 

99992 

53 

32 

32 

8 

29621 

70379 

29629 

70371 

00008 

99992 

52 

28 

36 

9 

30255 

69745 

30263 

69737 

00009 

99991 

51 

24 

4U 

10 

8.30879 

11.69121 

8.30888 

11.69112 

10.00009 

9.99991 

50 

20 

44 

11 

31495 

68505 

31505 

6S495 

00009 

99991 

49 

16 

48 

12 

32103 

67897 

32112 

67888 

00010 

99990 

48 

12 

52 

13 

32702 

67298 

32711 

67289 

00010 

99990 

47 

8 

56 

14 

33292 

66708 

33302 

66698 

00010 

99990 

46 

4 

5 

15 

8.33875 

11.66125 

8.33886 

11.66114 

10.00010 

9.99990 

45 

55 

4 

16 

34450 

65550 

34461 

65539 

00011 

99989 

44 

56 

8 

17 

35018 

64982 

35029 

64971 

00011 

99989 

43 

52 

12 

18 

35578 

64422 

35590 

64410 

00011 

99989 

42 

48 

16 

19 

36131 

63869 

36143 

63857 

00011 

99989 

41 

44 

20 

20 

8.36678 

11.63322 

8.36689 

11.63311 

10.00012 

9.99988 

40 

40 

24 

21 

37217 

62783 

37229 

62771 

00012 

999S8 

39 

36 

28 

22 

37750 

62250 

37762 

62238 

00012 

99988 

38 

32 

32 

23 

38276 

61724 

38289 

61711 

00013 

99987 

37 

28 

36 

24 

38796 

61204 

38809 

61191 

00013 

, 99987 

36 

24 

40 

25 

8.39310 

11.60690 

8.39323 

11.60677 

10.00013 

9.99987 

35 

20 

41 

26 

39818 

60182 

39832 

60168 

00014 

99986 

34 

16 

48 

27 

40320 

59680 

40334 

59666 

00014 

99986 

33 

12 

52 

28 

40816 

59184 

40830 

59170 

00014 

99986 

32 

8 

56 

29 

41307 

58693 

41321 

58679 

00015 

99985 

31 

4 

G 

30 

8.41792 

11.58208 

8.41807 

11.58193 

10.00015 

9.999S5 

30 

54 

4 

31 

42272 

57728 

42287 

57713 

00015 

99985 

29 

56 

8 

32 

42746 

57254 

42762 

57238 

00016 

99984 

28 

52 

12 

33 

43216 

56784 

43232 

56768 

00016 

99984 

27 

48 

16 

34 

43680 

56320 

43696 

50304 

00016 

99984 

26 

44 

20 

35 

8.44139 

11.55861 

8.44156 

11.55844 

10.00017 

9.99983 

25 

40 

24 

36 

44594 

55406 

44611 

55389 

00017 

99983 

24 

36 

28 

37 

45044 

54956 

45061 

54939 

00017 

99983 

23 

32 

32 

38 

45489 

54511 

45507 

54493 

00018 

99982 

22 

28 

36 

39 

45930 

54070 

45948 

54052 

00018 

99982 

21 

24 

40 

40 

8.46366 

11.53634 

8.4638b- 

11.53615 

10.00018 

9.99982 

20 

20 

44 

41 

46799 

53201 

46817 

53183 

00019 

99981 

19 

16 

48 

42 

47226 

52774 

47245 

521 »X) 

00019 

99981 

18 

12 

52 

43 

47650 

52350 

47669 

52331 

00019 

99981 

17 

8 

56 

44 

48069 

51931 

48089 

51911 

00020 

99980 

16 

4 

7 

45 

8.48485 

11.51515 

8.48505 

11.51495 

10.00020 

9.99980 

15 

53 

4 

46 

48896 

51104 

48917 

51083 

00021 

99979 

14 

56 

8 

47 

49304 

50696 

49325 

50675 

00021 

99979 

13 

52 

12 

48 

49708 

50292 

49729 

50271 

00021 

99979 

12 

48 

16 

49 

50108 

49892 

50130 

49870 

00022 

99978 

11 

14 

20 

50 

8.50504 

11.49496 

8.50527 

11.49473 

10.00022 

9.99978 

10 

40 

24 

51 

50897 

49103 

50920 

49080 

00023 

99977 

9 

36 

28 

52 

51287 

48713 

51310 

48690 

00023 

99977 

8 

32 

32 

53 

51673 

48327 

51696 

48304 

00023 

99977 

7 

28 

36 

54 

52055 

47945 

52079 

47921 

00024 

99976 

6 

24 

40 

55 

8.52434 

11.47566 

8.52459 

11.47541 

10.00024 

9.99976 

5 

20 

44 

56 

52810 

47190 

62835 

47165 

00025 

99975 

4 

16 

48 

57 

53183 

46817 

53208 

46792 

00025 

99975 

3 

12 

52 

58 

53552 

46448 

63578 

46422 

00026 

99974 

2 

8 

56 

59 

53919 

46081 

53945 

46055 

00026 

99974 

1 

4 

8 

60 

54282 

45718 

54308 

45692 

00026 

99974 

0 

5a 

Al.S. 

6 h 

M 

91° 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

o 

a ao 

CO 

M.S. 

5* 


















Logarithms Trigonometric. 201 


0 h 

2° 



Logarithms. 


177 C 

ll h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

8 

0 

8.54282 

11.45718 

8.54308 

11.45692 

10.00026 

9.99974 

60 

5i 

4 

1 

54642 

45358 

54669 

45331 

00027 

99973 

59 

56 

8 

2 

54999 

45001 

55027 

44973 

00027 

99973 

58 

52 

12 

3 

55354 

44646 

55382 

44618 

00028 

99972 

57 

48 

10 

4 

55705 

44295 

55734 

44266 

00028 

99972 

56 

44 

20 

5 

8.56054 

11.43946 

8.56083 

11.43917 

10.00029 

9.99971 

55 

40 

24 

6 

56400 

43600 

56429 

43571 

00029 

99971 

54 

36 

28 

7 

56743 

43257 

56773 

43227 

00030 

99970 

53 

32 

32 

8 

57084 

42916 

57114 

42886 

00030 

99970 

52 

28 

30 

9 

57421 

42579 

57452 

42548 

00031 

99909 

51 

24 

40 

10 

8.57757 

11.42243 

8.57788 

11.42212 

10.00031 

9.99969 

50 

20 

44 

11 

58089 

41911 

58121 

41879 

00032 

99908 

49 

16 

48 

12 

58419 

41581 

58451 

41549 

00032 

99968 

48 

12 

52 

13 

58747 

41253 

58779 

41221 

0003:) 

99967 

47 

8 

50 

14 

59072 

40928 

59105 

40895 

00033 

99967 

46 

4 

9 

15 

8.59395 

11.40005 

8.59428 

11.40572 

10.00033 

9.99967 

45 

51 

4 

16 

59715 

40285 

59749 

40251 

00034 

99966 

44 

56 

8 

17 

00033 

39967 

60068 

39932 

00034 

99966 

43 

52 

12 

18 

00349 

39651 

60384 

39616 

p 00035 

99965 

42 

48 

10 

19 

60062 

39:338 

60698 

39302 

00036 

99964 

41 

44 

20 

20 

8.60973 

11.39027 

8.61009 

11.38991 

10.00036 

9.99964 

40 

40 

24 

21 

61282 

38718 

61319 

38681 

00037 

99963 

39 

36 

28 

22 

01589 

38411 

61626 

38374 

00087 

99903 

38 

32 

32 

23 

01894 

38106 

61931 

38009 

00038 

99962 

37 

28 

36 

24 

02196 

37804 

62234 

37766 

00038 

99962 

36 

24 

40 

25 

8.62497 

11.37503 

8.62535 

11.37465 

10.00039 

9.99961 

35 

20 

44 

26 

62795 

37205 

62834 

37166 

00039 

99961 

34 

16 

48 

27 

63091 

36909 

63131 

36869 

00040 

99960 

33 

12 

52 

28 

63385 

36615 

63426 

36574 

00040 

99960 

32 

8 

50 

29 

63678 

36322 

63718 

36282 

00041 

99959 

31 

4 

10 

30 

8.03968 

11.36032 

8.64009 

11.35991 

10.00041 

9.99959 

30 

50 

4 

31 

64256 

35744 

64298 

35702 

00042 

99958 

29 

56 

8 

32 

64543 

35457 

64585 

35415 

00042 

99958 

28 

52 

12 

33 

64827 

35173 

64870 

35130 

00043 

99957 

27 

48 

10 

34 

65110 

34890 

65154 

34846 

00044 

99956 

26 

44 

20 

35 

8.65391 

11.34609 

8.65435 

11.34565 

10.00044 

9.99956 

25 

40 

24 

36 

65070 

34330 

65715 

34285 

00045 

99955 

24 

36 

28 

37 

65947 

34053 

65993 

34007 

00045 

99955 

23 

32 

32 

38 

66223 

33777 

66269 

33731 

00046 

99954 

22 

28 

36 

39 

66497 

33503 

60543* 

33457 

00046 

99954 

21 

24 

40 

40 

8.66769 

11.33231 

8.66816 

11.33184 

10.00047 

9.99953 

20 

20 

44 

41 

67039 

32961 

67087 

32913 

00048 

99952 

19 

16 

48 

42 

67308 

32692 

67356 

32644 

00048 

99952 

18 

12 

52 

43 

67575 

32425 

67624 

32376 

00049 

99951 

17 

8 

50 

44 

67841 

32159 

6785)0 

32110 

00049 

99951 

16 

4 

11 

45 

8.68104 

11.31896 

8.68154 

11.31846 

10.00050 

9.99950 

15 

19 

4 

40 

68367 

31633 

68417 

31583 

00051 

99949 

14 

56 

8 

47 

68627 

31373 

68678 

31322 

00051 

99949 

13 

52 

12 

48 

68886 

31114 

68938 

31062 

00052 

99948 

12 

48 

10 

49 

69144 

30856 

69196 

30804 

00052 

99948 

11 

44 

20 

50 

8.69400 

11.30600 

8.09453 

11.30547 

10.00053 

9.99947 

10 

40 

24 

51 

69654 

30346 

69708 

30292 

00054 

99946 

9 

36 

28 

52 

69907 

30093 

69962 

30038 

00054 

99946 

8 

32 

32 

53 

70159 

29841 

70214 

29786 

00055 

99945 

7 

28 

30 

54 

70409 

29591 

7of?>5 

29535 

00056 

99944 

6 

24 

40 

55 

8.70058 

11.29342 

8.70714 

11.29286 

10.00056 

9.99944 

5 

20 

44 

50 

70905 

29095 

70962 

29038 

00057 

99943 

4 

16 

48 

57 

71151 

28849 

71208 

28792 

00058 

99942 

3 

12 

52 

58 

71395 

28605 

71453 

28547 

00058 

99942 

2 

8 

50 

59 

71638 

28362 

71697 

28303 

00059 

99941 

1 

4 

12 

00 

71880 

28120 

71940 

28060 

00060 

99940 

0 

48 

>1. S. 
6 h 

M 

92° 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

87° 

M.S. 

5 h 



















202 Logarithms Trigonometric. 


0 h 

3° 



Logarithms. 


170° 


M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

12! 

0 

8.71880 

11.28120 

8.71940 

11.28060 

10.00060 

9.99940 

GO 

-18 

4 

1 

72120 

27880 

72181 

27819 

00060 

99940 

59 

56 

8 

2 

* 72869 

27641 

72420 

27580 

- 0006t 

99939 

58 

52 

12 

3 

72597 

27403 

72059 

275541 

00002 

99938 

57 

48 

16 

4 

72834 

27 ICO 

72890 

27104 

00062 

99938 

50 

44 

20 

5 

8.73069 

11.20931 

8.73132 

11.20808 

10.00003 

9.99937 

55 

40 

24 

6 

73303 

20097 

73300 

266554 

00064 

99936 

54 

556 

28 

7 

73535 

20405 

73000 

26400 

00064 

999:50 

53 

32 

82 

8 

73707 

26233 

73832 

26108 

00005 

999555 

52 

28 

30 

9 

73997 

20003 

740655 

25937 

OOOOG 

999554 

51 

24 

40 

10 

8.74220 

11.25774 

8.74292 

11.25708 

10.00060 

9.99034 

50 

20 

44 

11 

74454 

25540 

74521 

25479 

00007 

99933 

49 

16 

48 

12 

74G80 

25320 

74748 

25252 

00008 

99932 

48 

12 

52 

13 

74900 

25094 

74974 

25020 

000G8 

99932 

47 

8 

56 

14 

75130 

24870 

75199 

24801 

00069 

99931 

4G 

4 

13 

15 

8.75353 

11.24047 

8.754255 

11.24577 

10.00070 

9.99930 

45 

47 

4 

10 

75575 

24425 

75(545 

245555 

00071 

99929 

44 

56 

8 

17 

75795 

24205 

75807 

2415555 

00071 

99929 

43 

52 

12 

18 

76015 

23985 

*76087 

23913 

00072 

99928 

42 

48 

10 

19 

76234 

23760 

705106 

23094 

00073 

99927 

41 

44 

20 

20 

8.70451 

11.23549 

8.76525 

11.2:5475 

10.00074 

9.99926 

40 

40 

24 

21 

70G07 

23333 

70742 

23258 

00074 

99920 

39 

36 

28 

22 

70883 

23117 

70958 

23042 

00075 

99925 

38 

32 

32 

23 

77097 

22903 

77173 

22827 

00070 

99924 

87 

28 

30 

24 

77310 

22690 

77387 

22613 

00077 - 

99923 

36 

24 

10 

25 

8.77522 

11.22478 

8.77G00 

11.22400 

10.00077 

9.99923 

35 

20 

44 

20 

77733 

22207 

77811 

22189 

00078 

99922 

34 

16 

48 

27 

77943 

22057 

78022 

21978 

00079 

99921 

33 

12 

52 

28 

78152 

21848 

782552 

21768 

000 v 0 

99920 

32 

8 

50 

29 

78300 

21040 

78441 

21559 

00080 

99920 

31 

4 

14 

30 

8.78508 

11.21452 

8.78049 

11.215551 

10.00081 

9.99919 

30 

4(1 

. 4 

31 

78774 

21220 

78855 

21145 

00082 

99918 

29 

56 

8 

32 

78979 

21021 

79001 

20939 

00083 

99917 

28 

52 

12 

33 

79183 

20817 

792GG 

20734 

00083 

99917 

27 

48 

10 

34 

79380 

20014 

79470 

20530 

00084 

99916 

26 

44 

20 

35 

8.79688 

11.20412 

8.79073 

11.20327 

10.00085 

9.99915 

25 

40 

24 

30 

79789 

20211 

79875 

20125 

00986 

99914 

24 

55G 

28 

37 

79990 

20010 

80070 

19924 

00087 

99913 

23 

32 

32 

38 

80189 

19811 

80277 

19723 

00087 

99913 

22 

28 

30 

39 

80388 

19612 

80476 

10524 

00(588 

99912 

21 

24 

40 

40 

8.80585 

11.19415 

8.80074 

11.19326 

10.00089 

9.99911 

20 

20 

44 

41 

80782 

19218 

80872 

19128 

00090 

99910 

19 

1G 

48 

42 

80978 

19022 

81008 

18932 

00091 

99909 

18 

12 

52 

43 

81173 

18827 

812(54 

187550 

00091 

99909 

17 

8 

50 

44 

81307 

18633 

81459 

18541 

00092 

99908 

16 

4 

15 

45 

8.81500 

11.18440 

8.81653 

11.18347 

10.00093 

9.99907 

15 

45 

4 

40 

81752 

18248 

81840 

18154 

00094 

99900 

14 

56 

8 

47 

8194-1 

18050 

82038 

17902 

00095 

99905 

13 

52 

12 

48 

82134 

178GG 

822550 

17770 

00096 

99904 

12 

48 

10 

49 

82324 

17070 

82420 

17580 

00096 

99904 

11 

44 

20 

50 

8.82513 

11.17487 

8.82010 

11.17:590 

10.00097 

9.99903 

10 

40 

24 

51 

82701 

17299 

82799 

17201 

00098 

99902 

9 

36 

28 

52 

82888 

17112 

82987 

17013 

00099 

99901 

8 

32 

32 

63 

83075 

10925 

83175 

16825 

09100 

99900 

7 

28 

36 

54 

83201 

10739 

83301 

16039 

00101 

99899 

6 

24 

40 

55 

8.83446 

11.16554 

8.855547 

11.16453 

10.00102 

9.99898 

6 

20 

44 

50 

83030 

10370 

83732 

10208 

00102 

99898 

4 

16 

48 

57 

83813 

10187 

83910 

1G0S4 

00103 

99S97 

3 

12 

52 

58 

83996 

16004 

84100 

15900 

00104 

99890 

2 

8 

50 

59 

84177 

15823 

84282 

15718 

00105 

99895 

1 

4 

10 

60 

84358 

15042 

84404 

15530 

00106 

99804 

0 

44 

M.S. 

6 h 

M 

93° 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

86 ° 

M.S. 

6 h 























Logarithms Trigonometric. 


203 




Logarithms. 


175° 

ll h 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M. S 

8.8-4358 

11.15642 

8.84464 

11.15536 

10.00106 

9.99894 

60 

44 

84539 

15461 

84646 

15354 

00107 

99893 

59 

56 

84718 

15282 

84826 

]5174 

00108 

99892 

58 

52 

84897 

15103 

85006 

14994 

00109 

99891 

57 

48 

85075 

14925 

85185 

14815 

00109 

99891 

56 

44 

8.85252 

11.14743 

8.S5363 

11.14637 

10.00110 

9.99890 

55 

40 

85429 

14571 

85540 

14460 

00111 

99889 

54 

36 

85605 

14395 

85717 

14283 

00112 

99888 

53 

32 

85780 

14220 

85893 

14107 

00113 

99887 

52 

28 

85955 

14045 

86069 

13931 

00114 

99886 

51 

24 

8.8G128 

11.13872 

8.86243 

11.13757 

10.00115 

9.99885 

50 

20 

86301 

13699 

86417 

13583 

00116 

99884 

49 

16 

8G474 

13526 

86591 

13409 

00117 

99883 

48 

12 

86645 

13355 

86763 

13237 

00118 

998S2 

47 

8 

8681G 

13184 

86935 

13065 

00119 

998^1 

46 

4 

8.80987 

11.13013 

8.87106 

11.12894 

10.00120 

9.99880 

45 

43 

87156 

12844 

87277 

12723 

00121 

99879 

44 

56 

87325 

12675 

87447 

12553 

00121 

9987a 

43 

52 

87494 

12506 

87616 

12384 

00122 

99878 

42 

48 

87661 

12339 

87785 

12215 

00123 

99877 

41 

44 

8.87829 

11.12171 

8.87953 

11.12047 

10.00124 

9.99876 

40 

40 

87995 

12005 

88120 

11880 

00125 

99875 

39 

36 

88161 

11839 

8S287 

11713 

00126 

99874 

38 

32 

88320 

11674 

88453 

11547 

00127 

99573 

37 

28 

8S490 

11510 

88618 

11382 

00128 

99872 

36 

24 

8.88654 

11.11346 

8.88783 

11.11217 

10.00129 

9.99871 

35 

20 

88817 

11183 

8S943 

11052 

00130 

99870 

34 

16 

88980 

11020 

89111 

10889 

00131 

99869 

33 

12 

89142 

10858 

89274 

10726 

00132 

99868 

32 

8 

89304 

10G9G 

89437 

10663 

00133 

99867 

31 

4 

8.89464 

11.10536 

8.S9598 

11.10402 

10.00134 

9.99866 

30 

42 

89625 

10375 

89760 

10240 

00135 

99865 

29 

56 . 

89784 

10216 

89920 

10080 

00136 

99864 

28 

52 

89943 

10057 

900SO 

09920 

00137 

99S63 

27 

48 

90102 

09898 

90240 

09760 

00138 

99862 

26 

44 

8.90260 

11.09740 

8.90399 

11.09601 

10.00139 

9.99861 

25 

40 

90417 

09583 

90557 

09443 

00140 

99860 

24 

36 

90574 

09426 

90715 

09285 

00141 

99859 

23 

32 

90730 

09270 

90872 

09128 

00142 

99858 

22 

28 

90885 

09115 

91029 

08971 

00143 

99857 

21 

24 

8.91040 

11.08960 

8.91185 

11.08815 

10.00144 

9.99856 

20 

20 

91195 

08805 

91340 

08660 

00145 

99855 

19 

16 

91349 

08651 

91495 

08505 

00146 

99854 

18 

12 

91502 

08498 

91650 

0S350 

00147 

99853 

17 

8 

91655 

08345 

91803 

08197 

00148 

99852 

16 

4 

8.91807 

11.0S193 

8.91957 

11.08043 

10.00149 

9.99S51 

15 

41 

91959 

08041 

92110 

07890 

00150 

99850 

14 

56 

92110 

07890 

92262 

07738 

• 00152 

99843 

13 

52 

92261 

07739 

92414 

07586 

00153 

99847 

12 

48 

92411 

07589 

925.65 

07435 

00154 

99846 

11 

44 

8.92561 

11.07439 

8.92716 

11.07284 

10.00155 

9.99845 

10 

40 

92710 

07290 

92866 

07134 

00156 

99S44 

9 

36 

92859 

07141 

93016 

06984 

00157 

99843 

8 

32 

93007 

06993 

93165 

06835 

00158 

99842 

7 

28 

93154 

06840 

93313 

06687 

00159 

99841 

6 

24 

8.9330 L 

11.06699 

8.93462 

11.06538 

10.00160 

9.99840 

5 

20 

93448 

06552 

93609 

06391 

00161 

99839 

4 

10 

93594 

06406 

93756 

06244 

00162 

99s38 

3 

12 

93740 

06260 

93903 

06097 

00163 

99837 

2 

8 

93885 

06115 

94049 

05951 

00164 

99836 

1 

4 

94030 

05970 

94195 

05805 

OOlGG 

99834 

0 

40 

Cosine. 

Secant. 'Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

85° 

M.S. 

5 h 




















204 


Logarithms Trigonometric. 


0 b 

5° 



Logarithms. 



M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

20 

0 

8.94030 

11.05970 

8.94195 

11.05805 

10.00166 

9.99834 

4 

1 

94174 

05826 

94340 

05660 

00167 

99833 

8 

2 

94317 

05683 

94485 

05515 

001G8 

99832 

12 

3 

94461 

05539 

94630 

05370 

00169 

- 99831 

1G 

4 

94603 

05397 

94773 

05227 

00170 

99830 

20 

5 

8.94746 

11.05254 

8 94917 

11.05083 

10.00171 

9.99829 

21 

6 

94887 

05113 

95060 

04940 

00172 

99828 

28 

7 

95029 

04971 

95202 

04798 

00173 

99827 

32 

8 

95170 

04830 

95344 

04656 

00175 

99825 

3G 

9 

95310 

04690 

95486 

04514 

00176 

99824 

40 

10 

8.95450 

11.04550 

8.95627 

11.0-1373 

10.00177 

9.99823 

44 

11 

95589 

04111 

95767 

04233 

00178 

99822 

48 

12 

95728 

04272 

95908 

041.92 

00179 

99821 

52 

13 

95867 

04133 

96047 

03953 

00180 

99820 

6G 

14 

96005 

03995 

96187 

03813 

00181 

99819 

21 

15 

8.96143 

11.03857 

8.96325 

11.03675 

10.00183 

9.99817 

4 

16 

96280 

03720 

96464 

03536 

00184 

99816 

8 

17 

96417 

03583 

96602 

03398 

00185 

99815 

12 

18 

96553 

03447 

96739 

03261 

00186 

99814 

1G 

19 

96689 

03311 

96877 

03123 

00187 

99813 

20 

20 

8.96825 

11.03175 

8.97013 

11.02987 

10.00188 

9.99812 

24 

21 

96960 

03040 

97150 

02850 

00190 

99810 

28 

22 

97095 

02905 

97285 

02715 

00191 

99809 

32 

23 

97229 

02771 

97421 

02579 

00192 

99808 

36 

24 

97363 

02637 

97556 

02444 

00193 

99807 

40 

25 

8.97496 

11.02504 

8.97691 

11.02309 

10.00194 

9.99806 

44 

26 

97629 

02371 

97825 

02175 

00196 

99804 

48 

27 

97762 

02238 

97959 

62041 

00197 

99803 

52 

28 

97894 

02106 

98092 

01908 

00198 

99802 

56 

29 

98026 

01974 

98225 

01775 

00199 

99801 

22 

30 

8.98157 

11.01843 

8.98358 

11.01642 

10.00200 

9.99800 

4 

31 

98288 

01712 

98490 

01510 

00202 

99708 

8 

32 

98419 

01581 

98622 

01378 

00203 

99797 

12 

33 

98549 

01451 

98753 

01247 

00204 

99796 

1G 

34 

98679 

01321 

98884 

01116 

00205 

99795 

20 

35 

8.98808 

11.01192 

8.99015 

11.00985 

10.00207 

9.99793 

24 

3G 

98987 

01063 

99145 

00855 

00208 

99792 

28 

37 

99006 

00934 

99275 

00725 

00209 

99791 

32 

38 

99194 

00806 

99405 

00595 

00210 

99790 

3G 

39 

99322 

00678 

99534 

00466 

00212 

99788 

40 

40 

8.99460 

11.00550 

8.99662 

11.00338 

10.00213 

9.99787 

41 

41 

99577 

00423 

99791 

00209 

00214 

99786 

48 

42 

99704 

00296 

99919 

00081 

00215 

99785 

52 

43 

99830 

00170 

9.00046 

10 99954 

00217 

99783 

56 

41 

99956 

00044 

00174 

99826 

00218 

99782 

23 

45 

9.00082 

10.99918 

9.00301 

10.99699 

10.00219 

9.99781 

4 

46 

00207 

99793 

00427 

99573 

00220 

99780 

8 

47 

00332 

996<1l 

00653 

99447 

00222 

99778 

12 

48 

00456 

99544 

00679 

99321 

00223 

99777 

16 

49 

00581 

99419 

00805 

99195 

00224 

99776 

20 

50 

9.00704 

10.99296 

9.00930 

10.99070 

10.00225 

9.99775 

24 

51 

00828 

99172 

01055 

98945 

00227 

99773 

28 

52 

00951 

99049 

01179 

98821 

00228 

99772 

32 

63 

01074 

98926 

01303 

98697 

00229 

99771 

36 

54 

01196 

98804 

01427 

98573 

00231 

99769 

40 

55 

9.01318 

10.98682 

9.01550 

10.98450 

10.00232 

9.99768 

44 

56 

01440 

98560 

01673 

98327 

00233 

99767 

48 

57 

01561 

98439 

01796 

98204 

00235 

99765 

52 

58 

01682 

98318 

01918 

98082 

00236 

99764 

56 

59 

01803 

98197 

02040 

97960 

00237 

99763 

24 

60 

01923 

98077 

02162 

97838 

00239 

99761 

M.S. 

G h 

M 

95° 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 




















Logarithms Trigonometric. 


205 


O'* 

6° 



Logarithms. 


173° 

11“ 

M.S. 

M 

Sine. 

Cosocant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 


0 

9.01923 

10.98077 

9.02162 

10.97838 

10.00239 

9.99761 

60 

30 

4 

1 

02043 

97957 

02283 

97717 

00240 

99760 

59 

56 

8 

2 

02163 

97837 

02404 

97596 

00241 

99759 

58 

52 

12 

3 

02283 

97717 

02525 

97475 

00243 

99757 

57 

48 

10 

4 

02402 

97598 

02645 

97355 

00244 

99756 

56 

44 

20 

5 

9.02520 

10.97480 

9.02766 

10.97234 

10.00245 

9.99755 

55 

40 

24 

6 

02639 

97361 

02885 

97115 

00247 

99753 

54 

36 

28 

7 

02757 

97243 

03005 

96995 

00248 

99752 

53 

32 

32 

8 

02874 

97126 

03124 

96S76 

00249 

99751 

52 

28 

30 

9 

02992 

97008 

03242 

90758 

00251 

99749 

61 

24 

40 

10 

9.03109 

10.96891 

9.03361 

10.96039 

10.00252 

9.99748 

50 

20 

44 

11 

03226 

96774 

03479 

96521 

00253 

99747 1 

49 

16 

48 

12 

03342 

96658 

03597 

90403 

00255 

99745 

48 

12 

62 

13 

03458 

96542 

03714 

96286 

00256 

99744 

47 

8 

60 

14 

03574 

96426 

03832 

90108 

00258 

99742 

46 

4 

23 

15 

9.03690 

10.96310 

9.03948 

10.96052 

10.00259 

9.99741 

45 

35 

4 

16 

03805 

96196 

04065 

95935 

00260 

99740 

44 

56 

8 

17 

03920 

96080 

04181 

95819 

00262 

99738 

43 

62 

12 

18 

04034 

95966 

04297 

95703 

00263 

99737 

42 

48 

10 

19 

04149 

95851 

04413 

95587 

00264 

99736 

41 

44 

20 

20 

9.04262 

10.95738 

9.04528 

10.95472 

10.00266 

9.99734 

40 

40 

24 

21 

04376 

95624 

04643 

95357 

00267 

99733 

39 

36 

28 

22 

04490 

95510 

04758 

95242 

00269 

99731 

38 

32 

32 

23 

04603 

95397 

04873 

95127 

00270 

99730 

37 

28 

30 

24 

04715 

95285 

04987 

95013 

00272 

99728 

30 

24 

40 

25 

9.04828 

10.95172 

9.05101 

10.94899 

10.00273 

9.99727 

35 

20 

44 

26 

04940 

96060 

05214 

94786 

00274 

99726 

34 

16 

48 

27 

05052 

94948 

05328 

94672 

00276 

99724 

33 

\2 

52 

28 

05164 

94836 

05441 

94559 

00277 

99723 

32 

8 

60 

29 

05275 

94725 

05553 

94447 

00279 

99721 

31 

4 

tiO 

30 

9.05386 

10.94614 

9.05666 

10.94334 

10.00280 

9.99720 

30 


4 

31 

05497 

94503 

05778 

94222 

00282 

99718 

29 

66 

8 

32 

05607 

94393 

05890 

94110 

00283 

99717 

28 

52 

12 

33 

05717 

94283 

00002 

93998 

00284 

99716 

27 

48 

16 

34 

05827 

94173 

06113 

93887 

00286 

99714 

26 

11 

20 

36 

9.05937 

10.94063 

9.06224 

10.93776 

10.00287 

9.99713 

25 

40 

24 

36 

06046 

93954 

06335 

93665 

00289 

99711 

24 

36 

28 

37 

06155 

93845 

06445 

93555 

00290 

99710 

23 

32 

82 

38 

06264 

93736 

06550 

93444 

00292 

99708 

22 

28 

30 

39 

06372 

9362S 

06606 

93334 

00293 

99707 

21 

24 

40 

40 

9.06481 

10.93519 

9.00775 

10.93225 

10.00295 

9.99705 

20 

20 

44 

41 

06589 

93411 

06885 

93115 

00296 

99704 

19 

16 

48 

42 

00696 

93304 

06994 

93006 

00298 

99702 

18 

12 

62 

43 

06804 

93196 

07103 

92S97 

00299 

99701 

17 

8 

66 

44 

00911 

93089 

07211 

92789 

00301 

97699 

16 

4 

tit 

46 

9.07018 

10.92982 

9.07320 

10.92680 

10.00302 

9.99698 

15 

33 

4 

46 

07124 

92876 

07428 

92572 

00304 

99696 

14 

66 

8 

47 

07231 

92769 

07536 

92464 

00305 

99695 

13 

52 

12 

48 

07337 

92663 

07643 

92357 

00307 

99693 

12 

48 

16 

49 

07442 

92558 

07751 

92249 

00308 

99692 

11 

44 

20 

50 

9.07548 

10.92452 

9.07858 

10.92142 

10.00310 

8.99690 

10 

40 

24 

61 

07653 

92347 

07904 

92036 

00311 

99689 

9 

36 

28 

52 

07758 

92242 

08071 

91929 

00313 

99687 

8 

32 

32 

53 

07863 

92137 

08177 

91823 

00314 

99686 

7 

28 

36 

54 

07968 

92032 

08283 

91717 

00316 

996S4 

6 

24 

40 

65 

9.08072 

10.91928 

9.08389 

10.91611 

10.00317 

9.99683 

5 

20 

44 

56 

08176 

91824 

08495 

91505 

00319 

99681 

4 

16 

48 

57 

08280 

91720 

08000 

91400 

00320 

99680 

3 

12 

62 

58 

08383 

91617 

08705 

91295 

00322 

99678 

2 

8 

66 

69 

08486 

91514 

08810 

91190 

00323 

99677 

1 

4 

88 

60 

08589 

91411 

08914 

91086 

00325 

99675 

0 

3 a 

Jl.S. 

6“ 

M 

9G 0 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

cc 

1 

M.S. 

5“ 


















206 Logarithms Trigonometric. 


o h 

7° 



Logarithms. 


172° 

ll b 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

>38 

0 

9.08589 

10.91411 

9.08914 

10.910 v 6 

10.00325 

9.99675 

60 

32 

4 

1 

08692 

91308 

09019 

90981 

00326 

99674 

59 

56 

8 

2 

08795 

91205 

09123 

90877 

00328 

99672 

58 

52 

12 

3 

08S97 

91103 

09227 

90773 

00330 

99670 

57 

48 

16 

4 

08999 

91001 

09330 

90070 

00331 

99669 

56 

41 

20 

5 

9.09101 

10.90899 

9.094:34 

10.90566 

10.00333 

9.99667 

55 

40 

24 

6 

09202 

90798 

09537 

90403 

00334 

99666 

54 

3t> 

28 

7 

09304 

90696 

09640 

90360 

00336 

99664 

53 

32 

32 

8 

09405 

90595 

09742 

90258 

00337 

99603 

52 

28 

36 

9 

09506 

90494 

09845 

90155 

• 00339 

99661 

51 

24 

40 

10, 

9.09606 

10.90394 

9.09947 

10.90053 

10.00341 

9.99659 

50 

20 

44 

11 1 

09707 

90293 

10049 

89951 

00342 

99658 

49 

16 

48 

12 

09807 

90193 

10150 

89850 

00344 

99050 

48 

12 

52 

13 

09907 

90093 

10252 

89748 

00345 

99655 

47 

8 

56 

14 

10006 

89994 

10353 

89647 

00347 

99653 

46 

4 

39 

15 

9.10106 

10.89894 

9.10454 

10.89546 

10.00349 

9.99651 

45 

31 

4 

16 

10205 

89795 

10555 

89445 

00350 

99650 

44 

50 

8 

17 

10304 

89696 

10650 

89344 

00352 

9964S 

43 

52 

12 

18 

10402 

89598 

10756 

89244 

00353 

99647 

42 

48 

16 

19 

1050 L 

89499 

10856 

89144 

00355 

9964.3 

41 

44 

20 

20 

9.10599 

10.89401 

9.10956 

10.89044 

10.00357 

9.99643 

40 

40 

24 

21 

10697 

89303 

11056 

88944 

00358 

99642 

39 

36 

28 

22 

10795 

89205 

11155 

88845 

00360 

99640 

38 

32 

32 

23 

14)893 

89107 

11254 

88746 

00362 

99638 

37 

28 

36 

24 

10990 

89010 

11353 

88647 

00363 

99637 

36 

24 

40 

25 

9.11087 

10.88913 

9.11452 

10.88548 

10.00365 

9.99635 

35 

20 

44 

26 

11184 

88816 

11551 

88449 

00367 

99633 

34 

16 

48 

27 

11281 

88719 

11649 

88351 

00368 

99632 

33 

12 

52 

28 

11377 

88623 

11747 

88253 

00370 

99630 

32 

8 

56 

29 

11474 

88526 

11845 

88155 

00371 

99629 

31 

4 

30 

30 

9.11570 

10.88430 

9.11943 

10.88057 

10.00373 

9.99627 

30 

30 

4 

31 

11666 

88334 

12040 

87960 

00375 

99625 

29 

66 

8 

32 

11761 

88239 

12138 

87862 

00376 

99624 

28 

52 

12 

33 

11857 

88143 

12235 

87765 

00378 

99622 

27 

48 

16 

34 

11952 

88048 

12332 

87668 

00380 

99620 

26 

44 

20 

35 

9.12047 

10.87953 

9.12428 

10.87572 

10.00382 

9.99618 

25 

40 

24 

36 

12142 

87858 

12525 

87175 

00383 

99617 

24 

36 

2S 

37 

12236 

87764 

12621 

87379 

00385 

99015 

23 

32 

32 

38 

12331 

87669 

12717 

87283 

00387 

99613 

22 

28 

36 

39 

12425 

87575 

12813 

87187 

00388 

99612 

21 

24 

40 

40 

9.12519 

10.87481 

9.12909 

10.87091 

10.00390 

9.99610 

20 

20 

44 

41 

12612 

87388 

13004 

86996 

00392 

99608 

19 

16 

48 

42 

12706 

87294 

13099 

80901 

00393 

99607 

18 

12 

52 

43 

12799 

87201 

13194 

86806 

00395 

99605 

17 

8 

56 

44 

12892 

87108 

13289 

86711 

09397 

99603 

16 

4 

31 

45 

9.12985 

10.87015 

9.13384 

10.8661G 

10.00399 

9.99601 

15 

29 

4 

46 

13078 

86922 

13478 

86522 

00400 

99600 

14 

56 

8 

47 

13171 

86829 

13573 

86127 

00402 

99598 

13 

52 

12 

48 

13263 

86737 

13667 

86:133 

00404 

99596 

12 

18 

16 

49 

13355 

86645 

13761 

86239 

00405 

99595 

11 

44 

29 

50 

9.13447 

10.86553 

9.13854 

10.86146 

10.00407 

9.99593 

10 

40 

24 

51 

13539 

86461 

13948 

86052 

00409 

99591 

9 

30 

28 

52 

13630 

86370 

14041 

85959 

00411 

99589 

8 

32 

32 

53 

13722 

86278 

14134 

85866 

00112 

99588 

7 

28 

36 

54 

13813 

86187 

14227 

85773 

00414 

99586 

6 

24 

40 

55 

9.13904 

10.86096 

9.14320 

10.85080 

10.00416 

9.99584 

5 

20 

44 

56 

13994 

86006 

14412 

85588 

00418 

99582 

4 

16 

48 

57 

14085 

85915 

14504 

85496 

00419 

99581 

3 

12 

52 

58 

14175 

85825 

14597 

85403 

00421 

99579 

2 

8 

56 

59 

14266 

85734 

14688 

85312 

00423 

99577 

1 

4 

33 

60 

14350 

S5644 

14780 

85220 

00425 

99575 

0 

28 

M.S. 

6 U 

M 

97° 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

CC 

£ 

o 

M.S. 

5 h 





















Logarithms Trigonometric. 


207 


0 * 

1 GO 

o 



Logarithms. 


171° 

ll h 

M.S. 

M 

Sine. 

Cosecant. 

Taugeut. 

Cotangent. 

Secant. 

Cosine. 

M 

M. S. 

33 

0 

9.14356 

10.85644 

9.14780 

10.85220 

10.00425 

9.99575 

60 

38 

4 

1 

14445 

bo55o 

14872 

85128 

00426 

99574 

59 

56 

8 

2 

14535 

85465 

14963 

85037 

00428 

99572 

58 

52 

12 

3 

14624 

85376 

15054 

84946 

00430 

99570 

57 

48 

16 

4 

14714 

85286 

15145 

84855 

00432 

99568 

56 

44 

20 

5 

9.14803 

10.85197 

9.15236 

10.84761 

10.00434 

9.99566 

55 

40 

24 

6 

14891 

85109 

15327 

84673 

00415 

99565 

54 

36 

2S 

7 

14980 

85020 

15417 

84583 

00437 

99563 

53 

32 

32 

8 

15069 

84931 

15508 

84492 

00439 

995C1 

52 

28 

36 

9 

15157 

84843 

15598 

84402 

00441 

99559 

51 

24 

40 

10 

9.15245 

10.84755 

9.156S8 

10.84312 

10.00443 

9.99557 

50 

20 

44 

11 

15333 

84667 

15777 

84223 

00444 

99556 

49 

16 

48 

12 

15421 

84579 

15867 

84133 

00446 

99554 

48 

12 

52 

13 

15508 

84492 

15956 

84044 

00448 

99552 

47 

8 

56 

14 

15590 

84404 

16046 

83954 

00450 

99550 

46 

4 

33 

15 

9.15683 

10.84317 

9.16136 

10.83865 

10.00452 

9.9954S 

45 

27 

4 

16 

15770 

84230 

16224 

83776 

00454 

99546 

44 

56 

8 

17 

15857 

84143 

16312 

83688 

00455 

99545 

43 

52 

12 

18 

15944 

84056 

16401 

83599 

00457 

99543 

42 

48 

16 

19 

16030 

83970 

16489 

83511 

00459 

99541 

41 

44 

20 

20 

9.16116 

10.83884 

9.16577 

10.83423 

10.00461 

9.99539 

40 

40 

24 

21 

16203 

83797 

1 6665 

83335 

00463 

99537 

39 

36 

28 

22 

16289 

83711 

16753 

83247 

00405 

99535 

38 

32 

32 

23 

16374 

83626 

16841 

83159 

00467 

99533 

37 

28 

36 

24 

16460 

83540 

16928 

83072 

00468 

99532 

36 

24 

40 

25 

9.16545 

10.83455 

9.17016 

10.82984 

10.00470 

9.99530 

35 

20 

44 

26 

16631 

83369 

17103 

82897 

00472 

99528 

34 

16 

48 

27 

16716 

83284 

17190 

82810 

00474 

99526 

33 

12 

52 

28 

16801 

83199 

17277 

82723 

00476 

99524 

32 

8 

56 

29 

16886 

83114 

17363 

82637 

00478 

99522 

31 

4 

34 

30 

9.16970 

10.83030 

9.17450 

10.82550 

10.00480 

9.99520 

30 

36 

4 

31 

17055 

82945 

17536 

82464 

00482 

99518 

29 

56 

8 

32 

17139 

82861 

17622 

82378 

00483 

99517 

28 

52 

12 

33 

17223 

82777 

17708 

82292 

00485 

99515 

27 

48 

16 

34 

17307 

82693 

17794 

82206 

00487 

99513 

26 

44 

20 

35 

9.17391 

10.82609 

9.17880 

10.82120 

10.00489 

9.99511 

25 

40 

24 

36 

17474 

82526 

17965 

82035 

00491 

99509 

24 

36 

28 

37 

17558 

82442 

18051 

81949 

00493 

99507 

23 

32 

32 

38 

17641 

82359 

18136 

81864 

00495 

99505 

22 

28 

36 

39 

17724 

82276 

18221 

81779 

00497 

99503 

21 

24 

40 

40 

9.17807 

10.82193 

9.18306 

10.81694 

10.00499 

6.99501 

20 

20 

44 

41 

17890 

82110 

18891 

81609 

00501 

99499 

19 

16 

48 

42 

17973 

82027 

18475 

81525 

00503 

99497 

18 

12 

52 

43 

18055 

81945 

18560 

81440 

00505 

99495 

17 

8 

56 

44 

18137 

81863 

18644 

81356 

00506 

99494 

16 

4 

35 

45 

9.18220 

10.81780 

9.18728 

10.81272 

10.00508 

9.99492 

15 

35 

4 

46 

18302 

81698 

18812 

81188 

00510 

99490 

14 

56 

8 

47 

18383 

81617 

18896 

81104 

00512 

99488 

13 

52 

12 

48 

18405 

81535 

18979 

81021 

00514 

99486 

12 

48 

16 

49 

1K547 

81453 

19063 

80937 

00516 

99484 

11 

44 

20 

50 

9.18628 

10.81372 

9.19146 

10.80854 

10.00518 

9.99482 

10 

40 

24 

51 

18709 

81291 

19229 

80771 

00520 

99480 

9 

36 

28 

52 

18790 

81210 

19312 

80C8S 

00522 

99478 

8 

32 

32 

53 

18871 

81129 

19395 

80605 

00524 

99476 

7 

28 

36 

54 

18952 

81048 

'19478 

80522 

00526 

99474 

6 

24 

40 

55 

9.19033 

1080967 

9.19561 

10.80439 

10.00528 

9.99472 

5 

20 

44 

56 

19113 

80887 

19643 

80357 

00530 

99470 

4 

16 

48 

57 

19193 

80807 

19725 

80275 

00532 

99468 

3 

12 

52 

58 

19273 

80727 

19807 

80193 

00534 

99466 

2 

8 

56 

59 

19353 

80647 

19889 

80111 

00536 

99164 

1 

4 

30 

60 

19433 

80567 

19971 

80029 

00538 

99462 

0 

34 

M. S. 

t) h 

1 C? 2 

Cosine. 

Secant. 

Cotangent 

Taugeut. 

Cosecant. 

Sine. 

M 

81° 

M. S. 

5“ 
























208 Logarithms Trigonometric. 


0 h 

9° 



Logarithms. 


O 

o 

11“ 

MS. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

30 

0 

9.19433 

10.S0567 

9.19971 

10.80029 

10.00538 

9.99462 

60 

24 

4 

1 

19513 

S0487 

20053 

79947 

00540 

99400 

59 

5G 

8 

2 

19592 

80408 

20134 

79800 

00542 

99458 

58 

62 

12 

3 

19072 

80328 

20216 

79784 

00544 

99450 

57 

48 

10 

4 

19751 

80249 

20297 

79703 

00540 

99454 

56 

44 

20 

5 

9.19830 

10.80170 

9.20378 

10.79022 

10.00548 

9.99452 

55 

40 

24 

6 

19909 

80091 

20459 

79541 

00560 

99460 

54 

30 

28 

7 

19988 

80012 

20540 

79460 

00552 

99448 

53 

32 

32 

8 

20007 

79933 

20021 

79379 

00554 

99440 

52 

28 

3G 

9 

20145 

79855 

20701 

79299 

00556 

99444 

51 

24 

40 

10 

9.20223 

10.79777 

9.20782 

10.79218 

10.00558 

9.99442 

50 

20 

44 

11 

20302 

79698 

20802 

79138 

00500 

99440 

49 

10 

48 

12 

20380 

79620 

20942 

79058 

00502 

99438 

48 

12 

52 

13 

20458 

79542 

21022 

78978 

00564 

99430 

47 

8 

50 

14 

20535 

794(35 

21102 

78898 

00500 

99434 

40 

4 

37 

15 

9.20013 

10.79387 

9.21182 

10.78818 

10.00568 

9.99432 

45 

2 .J 

4 

10 

20091 

795309 

21201 

78739 

00571 

99429 

44 

50 

8 

17 

20708 

79232 

21341 

78059 

00573 

99427 

43 

52 

12 

18 

20845 

79155 

21420 

78580 

00575 

99425 

42 

48 

10 

19 

20922 

79078 

21499 

78501 

00577 

99423 

41 

44 

20 

20 

9.20999 

10.79001 

9.21578 

10.78422 

10.00579 

9.99421 

40 

40 

24 

21 

21070 

78924 

21657 

78343 

00581 

99419 

39 

30 

28 

22 

21153 

78847 

21736 

78201 

00583 

99417 

38 

32 

32 

23 

21229 

78771 

21814 

78180 

00585 

99415 

37 

28 

30 

24 

21300 

78694 

21893 

78107 

005S7 

99413 

30 

24 

40 

25 

9.21382 

10.78618 

9.21971 

10.7S029 

10.00589 

9.99411 

35 

20 

44 

26 

21458 

78542 

22049 

77961 

00591 

99409 

34 

10 

48 

27 

215:34 

78466 

22127 

77873 

00593 

99407 

33 

12 

52 

28 

21010 

78390 

22205 

77795 

00590 

99404 

32 

8 

50 

29 

21085 

78315 

22283 

77717 

00598 

99402 

31 

4 

38 

30 

9.21701 

10.78239 

9.22361 

10.77639 

10.00000 

9.99400 

30 

22 

4 

31 

21836 

78164 

22438 

77562 

00002 

99398 

29 

50 

8 

32 

21912 

78088 

22510 

77484 

00004 

99390 

28 

52 

12 

33 

21987 

78013 

22593 

77407 

00006 

99394 

27 

48 

10 

34 

22002 

77938 

22070 

77330 

00008 

99392 

20 

44 

20 

36 

9.22137 

10.77863 

9.22747 

10.77263 

10.00010 

9.99390 

25 

40 

24 

30 

22211 

77789 

22824 

77176 

00012 

99388 

24 

36 

28 

37 

22286 

77714 

22901 

77099 

00015 

99385 

23 

32 

32 

38 

22361 

77039 

22977 

77023 

00017 

99383 

22 

28 

30 

39 

22435 

77505 

23054 

76946 

00019 

99381 

21 

24 

40 

40 

9.22509 

10.77491 

9.23130 

10.70870 

10.00021 

9.99379 

20 

20 

44 

41 

22583 

77417 

23206 

76794 

00023 

99377 

19 

10 

43 

42 

22657 

77343 

232S3 

70717 

00625 

99375 

18 

12 

52 

43 

22731 

77209 

23359 

70041 

00628 

99372 

17 

8 

50 

44 

22805 

77195 

23135 

70565 

00030 

99370 

10 

4 

3 9 

45 

9.22878 

10.77122 

9.23510 

10.76490 

10.00632 

9.99368 

15 

21 

4 

40 

22952 

77048 

23586 

76414 

00634 

99360 

14 

50 

8 

47 

23025 

70975 

23661 

70339 

00630 

99364 

13 

52 

12 

48 

23098 

70902 

23737 

76203 

00038 

99362 

12 

48 

10 

49 

23171 

70829 

23812 

76188 

00041 

99359 

11 

44 

20 

50 

9.23244 

10.70750 

9.23887 

10.76113 

10.00643 

9.99357 

10 

40 

24 

51 

23317 

7G683 

23902 

70038 

00045 

99355 

9 

30 

28 

52 

2:3390 

70010 

24037 

75963 

00047 

99353 

8 

32 

32 

63 

23462 

70538 

24112 

75888 

00049 

99351 

7 

28 

30 

54 

23535 

70405 

24186 

75814 

00652 

99348 

0 

24 

40 

55 

9.23607 

10.70393 

9.24261 

10.75739 

10.00G54 

9.99340 

5 

20 

44 

50 

23679 

70321 

24335 

75605 

00056 

99344 

4 

16 

48 

57 

23762 

70248 

24410 

75590 

00058 

99342 

3 

12 

52 

58 

23823 

70177 

24484 

75516 

00000 

99340 

2 

8 

50 

59 

23895 

70105 

24558 

75442 

00003 

99337 

1 

4 

40 

60 

23907 

70033 

24032 

75368 

00005 

99336 

0 

20 

M.S. 

6 h 

o 

05 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

80° 

M.S. 

5 h 















Logarithms Trigonometric. ’ 209 


o h 

10° 



Logarithms. 


169° 

ll h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

40 

0 

9.23967 

10.76033 

9.24632 

10.75368 

10.00665 

9.99335 

60 

30 

4 

1 

24039 

75961 

24706 

75294 

00667 

99333 

59 

56 

8 

2 

24110 

75890 

21779 

75221, 

00669 

99331 

58 

52 

12 

3 

24181 

75819 

24853 

75147 

00672 

99328 

57 

48 

lb 

4 

24253 

75747 

24926 

75074 

00674 

99326 

56 

44 

20 

5 

0.24324 

10.75676 

9.25000 

10.75000 

10.00676 

9.99324 

55 

40 

24 

6 

24395 

75605 

25073 

74927 

00678 

99322 

54 

36 

28 

7 

24466 

75534 

25146 

74854 

00681 

99319 

53 

32 

82 

8 

24536 

75464 

25219 

74781 

00683 

99317 

52 

28 

86 

9 

24607 

75393 

25292 

74708 

00685 

99315 

51 

24 

40 

10 

9.24677 

10.75323 

9.25365 

10.74635 

10.00687 

9.99313 

50 

20 

44 

11 

24748 

75252 

25437 

74563 

00690 

99310 

49 

16 

48 

12 

24818 

75182 

25510 

74490 

00692 

99308 

48 

12 

52 

13 

24888 

75112 

25582 

74418 

00694 

99306 

47 

8 

56 

14 

24958 

75042 

25655 

74345 

00696 

99304 

46 

4 

41 

15 

9.25028 

10.74972 

9.25727 

10.74273 

10.00699 

9.99301 

45 

19 

4 

16 

25098 

74902 

25799 

74201 

00701 

99299 

44 

56 

8 

17 

25168 

74832 

25871 

74129 

00703 

99297 

43 

52 

12 

18 

25237 

74763 

25943 

74057 

00706 

99294 

42 

48 

16 

19 

25307 

74693 

26015 

73985 

00708 

99292 

41 

44 

20 

20 

9.25376 

10.74624 

9.26086 

10.73914 

10.00710 

9.99290 

40 

40 

24 

21 

25445 

74555 

26158 

73842 

00712 

99288 

39 

36 

28 

22 

25514 

74486 

26229 

73771 

00715 

99285 

38 

32 

32 

23 

25583 

74417 

26301 

73699 

00717 

99283 

37 

28 

36 

24 

25652 

74348 

26372 

73628 

00719 

99281 

36 

24 

40 

25 

9.25721 

10.74279 

9.26443 

10.73557 

10.00722 

9.99278 

35 

20 

44 

26 

25790 

74210 

26514 

73486 

00724 

99276 

34 

16 

48 

27 

25858 

74142 

26585 

73415 

00726 

99274 

33 

12 

52 

28 

25927 

74073 

26655 

73345 

00729 

99271 

32 

8 

56 

29 

25995 

74005 

26726 

73274 

00731 

9926J 

31 

4 

44 

30 

9.26063 

10.73937 

9.26797 

10.73203 

10.00733 

9.99267 

30 

18 

4 

31 

26131 

73869 

26867 

73133 

00736 

99264 

29 

56 

8 

32 

26199 

73801 

26937 

73063 

00738 

99262 

28 

52 

12 

33 

26267 

73733 

27008 

72992 

00740 

99260 

27 

48 

16 

34 

26335 

73665 

27078 

72922 

00743 

99257 

26 

44 

20 

35 

9.26403 

10.73597 

9.27148 

10.72852 

10.00745 

9.99255 

25 

40 

24 

36 

26470 

73530 

27218 

72782 

00748 

99252 

24 

36 

28 

37 

26538 

73462 

27288 

72712 

00750 

99250 

23 

32 

32 

38 

26605 

73395 

27357 

72643 

00752 

99248 

22 

28 

36 

39 

26072 

73328 

27427 

72573 

00755 

99245 

21 

24 

40 

40 

9.26739 

10.73261 

9.27496 

10.72504 

10.00757 

9.99243 

20 

20 

44 

41 

20806 

73194 

27566 

72434 

00759 

99241 

19 

16 

48 

42 

26873 

73127 

27635 

72365 

00762 

99238 

18 

12 

52 

43 

26940 

73060 

27704 

72296 

00764 

99236 

17 

8 

56 

44 

27007 

72993 

27773 

72227 

00767 

99233 

16 

4 

43 

45 

9.27073 

10.72927 

9.27842 

10.72158 

10.00769 

9.99231 

15 

17 

4 

46 

27140 

72860 

27911 

72089 

00771 

99229 

14 

56 

8 

47 

27206 

72794 

27980 

72020 

00774 

99226 

13 

52 

12 

48 

27273 

72727 

28049 

71951 

00776 

99224 

12 

48 

16 

49 

27339 

72661 

28117 

71883 

00779 

99221 

11 

44 

20 

50 

9.27405 

10.72595 

9.28186 

10.71814 

10.00781 

9.99219 

10 

40 

24 

51 

27471 

72529 

28254 

71746 

00783 

99217 

9 

36 

28 

52 

27537 

72463 

28323 

71677 

00786 

99214 

8 

32 

32 

53 

27602 

72398 

28391 

71609 

00788 

99212 

7 

28 

36 

54 

27668 

72332 

28459 

71541 

00791 

99209 

6 

24 

40 

55 

9.27734 

10.72266 

9.28527 

10.71473 

10.00793 

9.99207 

5 

20 

44 

56 

27799 

72201 

28595 

71405 

00796 

99204 

4 

16 

48 

57 

27864 

72136 

28662 

71338 

00798 

99202 

3 

12 

52 

58 

27930 

72070 

28730 

71270 

00800 

99200 

2 

8 

56 

59 

27995 

72005 

28798 

71202 

00803 

99197 

1 

4 

44 

60 

28060 

71940 

28865 

71135 

00805 

99195 

0 

16 

M.S. 

M 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

M.S. 

6 h 

100 c 







79° 

5 h 


14 




















210 ' Logarithms Trigonometric. 


(p 

11° 



Logarithms. 


168° 

ll h 

M.S. 

M 

Sine. 

Coisecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

44 

0 

9.28000 

10.71940 

9.28865 

10.71135 

10.00805 

9.99195 

60 

1G 

4 

1 

28125 

71875 

28933 

71067 

00808 

99192 

50 

56 

8 

2 

28190 

71810 

£9000 

29067 

71)900 

00810 

99190 

58 

52 

12 

3 

2S254 

71746 

70933 

00813 

99187 

57 

48 

lb 

4 

28319 

71681 

29134 

70866 

00815 

99185 

56 

41 

20 

5 

9.28384 

10.71616 

9.29201 

10.70799 

10.00818 

9.99182 

55 

40 

24 

6 

28448 

71552 

29268 

70732 

00820 

99180 

54 

36 

28 

7 

28512 

71488 

29335 

70665 

00823 

99177 

53 

32 

32 

8 

28577 

71423 

29402 

70598 

00825 

99175 

52 

28 

3b 

9 

2S641 

71359 

29468 

70532 

00828 

99172 

51 

24 

40 

10 

9.2S705 

10.71295 

9.29535 

10.70465 

10.00830 

9.99170 

50 

20 

44 

11 

28769 

71231 

29601 

70399 

00833 

99167 

49 

16 

48 

12 

28833 

71167 

29668 

70332 

00835 

99165 

48 

12 

52 

13 

28896 

71104 

29734 

70266 

0083S 

99162 

47 

8 

6G 

14 

289G0 

71040 

29800 

70200 

00840 

99160 

46 

4 

45 

15 

9.29024 

10.70976 

9.29866 

10.70134 

10.00843 

9.99157 

45 

15 

4 

lb 

29087 

70913 

29932 

70068 

00845 

99155 

44 

5(3 

8 

17 

29150 

70850 

29998 

70002 

00848 

99152 

43 

52 

12 

18 

29214 

70786 

30064 

69936 

00850 

99150 

42 

48 

lb 

19 

29277 

70723 

30130 

69870 

00853 

99147 

41 

44 

20 

20 

9.29340 

10.70660 

9.30195 

10.69805 

10.00855 

9.99145 

40 

40 

24 

21 

29403 

70597 

30261 

G9739 

00858 

99142 

39 

36 

28 

22 

29406 

70534 

30326 

69674 

00860 

99140 

38 

32 

32 

23 

29529 

70471 

30391 

69609 

00863 

99137 

37 

28 

36 

24 

29591 

70400 

30457 

69543 

00865 

99135 

3(3 

24 

40 

25 

9.29654 

10.70346 

9.30522 

10.09478 

10.00868 

9.99132 

35 

20 

44 

26 

29716 

70284 

30587 

69413 

00870 

99130 

34 

16 

48 

27 

29779 

70221 

30652 

G9348 

00873 

99127 

33 

12 

52 

28 

29841 

70159 

30717 

G9283 

00876 

99124 

32 

8 

5G 

29 

29903 

70097 

30782 

69218 

00878 

99122 

31 

4 

4G 

30 

9.2996b 

10.701)34 

9.30846 

10.69154 

10.00881 

9.99119 

30 

14 

4 

31 

30028 

69972 

30911 

69089 

00883 

99117 

29 

56 

8 

32 

30090 

69910 

30975 

69025 

00886 

99114 

28 

52 

12 

33 

30151 

69849 

31040 

G8960 

00888 

99112 

27 

48 

lb 

34 

30213 

69787 

31104 

68S96 

00891 

99109 

26 

41 

20 

35 

9.30275 

10.69725 

9.31168 

10.68832 

10.00894 

9.90106 

25 

40 

24 

36 

30336 

69664 

31233 

68767 

00896 

90104 

24 

36 

28 

37 

30398 

69602 

31297 

68703 

00899 

99101 

23 

32 

32 

38 

30459 

69541 

31361 

68639 

00901 

90090 

22 

28 

3b 

39 

30521 

69479 

31425 

68575 

00904 

99096 

21 

24 

40 

40 

9.30582 

10.69418 

9.31489 

10.68511 

10.00907 

9.99093 

20 

20 

44 

41 

30643 

69357 

31552 

68448 

00909 

99091 

19 

16 

48 

42 

30704 

69296 

31016 

683S4 

00912 

99088 

18 

12 

52 

43 

30765 

C9235 

31679 

68321 

00914 

99086 

17 

8 

5b 

41 

30826 

G9174 

31743 

68257 

00917 

99083 

16 

4 

47 

45 

9.30S87 

10.69113 

9.31806 

10.68194 

10.00920 

9.99080 

15 

13 

4 

46 

30947 

69053 

31870 

68130 

00922 

99078 

14 

56 

8 

47 

31008 

68992 

31933 

68067 

00925 

99075 

13 

52 

12 

48 

31068 

C8932 

31996 

68004 

00928 

99072 

12 

48 

lb 

49 

31129 

68871 

32059 

67941 

00930 

99070 

11 

44 

20 

50 

9.31189 

10.68811 

9.32122 

10.67878 

10.00933 

9.99067 

10 

4u 

24 

51 

31250 

68750 

32185 

67815 

00936 

99064 

9 

36 

28 

52 

31310 

68690 

32248 

67752 

00938 

99062 

8 

32 

32 

53 

31370 

68630 

32311 

67689 

00941 

99059 

7 

28 

36 

54 

31430 

68570 

32373 

67627 

00944 

99056 

6 

24 

40 

55 

9.31490 

10.68510 

9.32436 

10.67564 

10.00940 

9.99054 

5 

20 

44 

5b 

31549 

68451 

32498 

67502 

00949 

99051 

4 

16 

48 

57 

31609 

68301 

32561 

67439 

00952 

99048 

3 

12 

52 

58 

31669 

68331 

32623 

67377 

•00954 

99046 

2 

8 

5b 

59 

31728 

68272 

32685 

67315 

00957 

99043 

1 

4 

48 

60 

31788 

G8212 

32747 

67253 

00960 

99040 

0 

13 

M.S. 

6 h 

M 

ior 

Cosine. 

5 

Secant. 

Cotangeut 

Tangent. 

Cosecant. 

Sine. 

M 

78° 

M.S. 

5 h 

















Logarithms Trigonometric. 211 


0 h 

12 c 



Logarithms. 


167° 

ll h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

48 

0 

9.31788 

10.68212 

9.32747 

10.67253 

10.00960 

9.99040 

60 

13 

4 

1 

31847 

68153 

32810 

67190 

00962 

99038 

59 

56 

8 

2 

31907 

68093 

32872 

67128 

00965 

99035 

58 

52 

12 

3 

31966 

68034 

32933 

67067 

00968 

99032 

57 

48 

16 

4 

32025 

67975 

32995 

‘ 67005 

00970 

99030 

56 

44 

20 

5 

9.32084 

10.67916 

9.33057 

10.66913 

10.00973 

9.99027 

55 

40 

24 

6 

32143 

67857 

33119 

66S81 

00976 

99024 

54 

36 

28 

7 

32202 

67798 

33180 

66820 

00978 

99022 

53 

32 

32 

8 

32261 

67739 

33212 

66758 

00981 

99019 

52 

28 

36 

0 

32319 

67681 

33303 

66697 

00984 

99016 

51 

24 

40 

10 

9.82378 

10.67622 

9.33365 

10.66635 

10.00987 

9.99013 

50 

20 

44 

11 

32437 

67563 

33426 

66574 

00989 

99011 

49 

16 

48 

12 

32495 

67505 

33487 

66513 

00992 

99008 

48 

12 

52 

13 

32553 

67447 

83548 

66452 

00995 

99005 

47 

8 

56 

14 

32612 

67388 

33609 

66391 

00998 

99002 

46 

4 

4‘.) 

15 

9.32670 

10.67330 

9.33670 

10.66330 

10.01000 

9.99000 

45 

11 

4 

16 

32728 

67272 

33731 

66269 

01003 

98997 

44 

56 

8 

17 

32786 

67214 

33792 

66208 

01006 

98994 

43 

52 

12 

18 

32844 

67156 

33853 

66147 

01009 

98991 

42 

48 

16 

19 

32902 

67098 

33913 

66087 

01011 

98989 

41 

44 

20 

20 

9.32960 

10.67040 

9.33974 

10.66026 

10.01014 

9.98986 

40 

40 

24 

21 

33018 

66982 

34034 

65966 

01017 

98983 

39 

36 

28 

22 

33075 

66925 

34095 

65905 

01020 

9S980 

38 

32 

32 

23 

33133 

66867 

34155 

65845 

01022 

98978 

37 

28 

36 

24 

33190 

66810 

34215 

65785 

01025 

98975 

36 

24 

40 

25 

9.33248 

10.66752 

9.34276 

10.65724 

10.01028 

9.98972 

35 

20 

44 

26 

33305 

66695 

34336 

65664 

01031 

98969 

34 

16 

48 

27 

83362 

66638 

34396 

65604 

01033 

98967 

33 

12 

52 

28 

33420 

665S0 

34456 

65544 

01036 

98964 

32 

8 

56 

29 

33477 

66523 

34516 

65484 

01039 

98961 

31 

4 

50 

30 

9.33534 

10.66466 

9.34576 

10.65424 

10.01042 

9.98958 

30 

10 

4 

31 

33591 

66409 

34635 

65365 

01045 

98955 

29 

56 

8 

32 

33647 

66353 

34695 

65305 

01047 

98953 

28 

52 

12 

33 

33704 

66296 

34755 

65245 

01050 

98950 

27 

48 

16 

34 

33761 

66239 

34814 

65186 

01053 

98947 

26 

44 

20 

35 

9.33818 

10.66182 

9.34874 

10.65126 

10.01056 

9.98944 

25 

40 

24 

36 

33874 

66126 

34933 

65067 

01059 

98941 

24 

36 

28 

37 

33931 

66069 

34992 

65008 

01062 

98938 

23 

32 

32 

38 

33987 

66013 

35051 

64949 

01064 

98936 

22 

28 

36 

39 

34043 

65957 

35111 

64889 

01067 

98933 

21 

24 

40 

40 

9.34100 

10.65900 

9.35170 

10.64830 

10.01070 

9.98930 

20 

20 

44 

41 

84156 

65844 

35229 

64771 

01073 

98927 

19 

16 

48 

42 

34212 

65788 

35288 

64712 

01076 

98924 

18 

12 

52 

43 

34268 

65732 

35347 

64653 

01079 

98921 

17 

8 

56 

44 

34324 

65676 

35405 

64595 

01081 

9S919 

16 

4 

51 

45 

9.84880 

10.65620 

9.35464 

10.64536 

10.01084 

9.98916 

15 

9 

4 

46 

34436 

65564 

35523 

64477 

01087 

98913 

14 

56 

8 

47 

34491 

65509 

35581 

64419 

01090 

98910 

13 

52 

12 

48 

34547 

65453 

35640 

64360 

01093 

98907 

12 

48 

16 

49 

84602 

65398 

35698 

64302 

01096 

98904 

11 

44 

20 

50 

9.34658 

10.65342 

9.35757 

10.64243 

10.01099 

9.98901 

10 

40 

24 

51 

34713 

65287 

35815 

64185 

01102 

98898 

9 

36 

28 

52 

34769 

65231 

35873 

64127 

01104 

98896 

8 

32 

32 

53 

34824 

65176 

35931 

64069 

01107 

98893 

7 

28 

36 

54 

34879 

65121 

35989 

64011 

OHIO 

98890 

6 

24 

40 

55 

9.34934 

10 65066 

9.36047 

10.63953 

10.01113 

9.98887 

5 

20 

44 

56 

34989 

65011 

36105 

63895 

01116 

98884 

4 

16 

48 

57 

35044 

Cl 956 

36163 

63837 

01119 

98881 

3 

12 

52 

68 

35099 

64901 

36221 

63779 

01122 

98878 

2 

8 

56 

69 

35154 

64846 

36279 

63721 

01125 

9S875 

1 

4 

52 

60 

35209 

64791 

36336 

63664 

01128 

98872 

0 

8 

A!. S. 

G h 

M 

102 * 

Cosine. 

5 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

77° 

M. S. 

5* 



















212 Logarithms Trigonometric. 


o b 

13° 



Logarithms. 


166° 

ll h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

554 

0 

9.35209 

10.64791 

9.36336 

10.63604 

10.01128 

9.98872 

60 

8 

4 

1 

35263 

64737 

86394 

63606 

01131 

98869 

59 

56 

8 

2 

35318 

64682 

36152 

63548 

01133 

98867 

58 

52 

12 

3 

35373 

64627 

36509 

63491 

01136 

98864 

57 

48 

16 

4 

35427 

64573 

36566 

63434 

01139 

98861 

56 

44 

20 

5 

9.35481 

10.64519 

9.36624 

10.63376 

10.01142 

9.98858 

55 

40 

24 

6 

35536 

64164 

36081 

63319 

01145 

98855 

54 

36 

28 

7 

35590 

64410 

36738 

63262 

01148 

98852 

53 

32 

32 

8 

35644 

64356 

36795 

63205 

01151 

98849 

52 

28 

36 

9 

35698 

64302 

36852 

63148 

01154 

98846 

51 

24 

40 

10 

9.35752 

10.64248 

9.36909 

10.63091 

10.01157 

9.98843 

50 

20 

44 

11 

35806 

64194 

36966 

63034 

01160 

98840 

49 

16 

48 

12 

35860 

64140 

37023 

62977 

01163 

98837 

48 

12 

62 

13 

35914 

64086 

37080 

62920 

01166 

98834 

47 

8 

56 

14 

35968 

64032 

37137 

62863 

01169 

9S831 

46 

4 

53 

15 

9.36022 

10.63978 

9.37193 

10.62807 

10.01172 

9.98828 

45 

7 

4 

16 

36075 

63925 

37250 

62750 

01175 

98825 

44 

56 

8 

17 

36129 

63871 

37306 

62694 

01178 

98822 

43 

52 

12 

18 

36182 

63818 

37363 

62637 

01181 

98819 

42 

48 

16 

19 

36236 

63764 

37419 

62581 

01184 

98810 

41 

44 

20 

26 

9.36289 

10.63711 

9.37476 

10.62524 

10.01187 

9.98813 

40 

40 

24 

21 

36342 

63658 

37532 

62468 

01190 

98810 

39 

36 

28 

22 

36395 

63605 

37588 

62412 

01193 

98807 

38 

32 

32 

23 

36449 

63551 

37644 

62356 

01196 

98804 

37 

28 

36 

24 

36502 

63498 

37700 

62300 

01199 

98801 

36 

24 

40 

25 

9.36555 

10.63445 

9.37756 

10.62244 

10.01202 

9.98798 

35 

20 

44 

26 

36608 

63392 

37812 

62188 

01205 

98795 

34 

16 

48 

27 

36660 

63340 

37868 

62132 

012U8 

98792 

33 

12 

52 

28 

36713 

63287 

37924 

62076 

01211 

98789 

32 

8 

56 

29 

36766 

63234 

37980 

02020 

01214 

98786 

31 

4 

54 

30 

9.36819 

10.63181 

9.38035 

10.61965. 

10.01217 

9.98783 

30 

6 

4 

31 

36871 

63129 

38091 

61909 

01220 

98780 

29 

56 

8 

32 

36924 

63076 

38147 

61853 

01223 

98777 

28 

52 

12 

33 

36976 

03024 

38202 

61798 

01226 

98774 

27 

48 

16 

34 

37028 

62972 

38257 

61743 

01229 

98771 

26 

44 

20 

35 

9.37081 

10.62919 

9.38313 

10.61687 

10.01232 

9.98768 

25 

40 

24 

36 

37133 

62867 

38368 

61632 

01235 

98765 

24 

36 

28 

37 

37185 

62815 

38423 

61577 

01238 

98702 

23 

3-2 

32 

38 

37237 

62703 

38479 

61521 

01241 

98759 

22 

28 

36 

39 

37289 

62711 

38534 

61466 

01244 

98756 

21 

24 

40 

40 

9.37341 

10.62659 

9.38589 

10.61411 

10.01247 

9.98753 

20 

20 

44 

41 

37393 

62607 

38644 

61356 

01250 

98750 

19 

16 

48 

42 

37445 

62555 

38699 

61301 

01254 

98746 

18 

12 

62 

43 

37497 

62503 

38754 

61246 

01257 

98743 

17 

8 

56 

44 

37549 

62451 

38808 

61192 

01260 

98740 

16 

4 

55 

45 

9.37600 

10.62400 

9.38863 

10.61137 

10.01263 

9.98737 

15 

5 

4 

46 

37652 

62348 

38918 

61082 

01266 

98734 

14 

56 

8 

47 

37703 

62297 

38972 

61028 

01269 

98731 

13 

52 

12 

48 

37755 

62245 

39027 

60973 

01272 

98728 

12 

48 

16 

49 

37806 

62194 

39082 

60918 

01275 

98725 

11 

44 

20 

60 

9.37858 

10.62142 

9.39136 

10.60864 

10.01278 

9.98722 

10 

40 

24 

51 

37909 

62091 

39190 

60810 

01281 

98719 

9 

36 

28 

52 

37960 

62040 

39245 

60755 

01285 

98715 

8 

32 

32 

53 

38011 

61989 

39299 

60701 

01288 

98712 

7 

28 

36 

54 

38062 

61938 

39353 

60647 

01291 

98709 

6 

24 

40 

55 

9.38113 

1061887 

9.39407 

10.60593 

10.01294 

9.98706 

5 

20 

44 

56 

38164 

01836 

39461 

60539 

01297 

98703 

4 

16 

4S 

57 

38215 

01785 

39515 

60485 

01300 

98700 

3 

12 

52 

58 

38266 

61734 

39569 

60431 

01303 

98697 

2 

8 

56 

59 

38317 

61683 

39623 

60377 

01306 

98694 

1 

4 

50 

60 

38368 

61632 

39677 

60323 

01310 

98690 

0 

4 

M.S. 

6 b 

M 

103 

Cosine. 

0 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

76° 

M.S. 

5 h 
















Logarithms Trigonometric. 


213 


o h 

14° 



Logarithms. 


165° 

ll b 

M.S 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

50 

0 

9.38368 

10.61632 

9.39677 

10.60323 

10.01310 

9.98690 

60 

4 

4 

1 

38418 

61582 

39731 

60209 

01313 

98687 

59 

56 

8 

2 

38169 

61531 

39785 

60215 

01316 

98684 

58 

52 

12 

3 

38519 

61181 

39838 

60162 

01319 

98681 

57 

48 

16 

4 

38570 

61430 

39892 

60108 

01322 

98678 

56 

44 

20 

5 

9.38620 

10.61380 

9.39945 

10.60055 

10.01325 

9.98675 

55 

40 

24 

6 

38670 

61330 

39999 

60001 

01329 

98671 

54 

36 

28 

7 

38721 

61279 

40052 

59948 

01332 

98668 

53 

32 

32 

8 

3S771 

61229 

40106 

59894 

01335 

98665 

52 

28 

36 

9 

38821 

61179 

40159 

59841 

01338 

98662 

51 

24 

40 

10 

9.38871 

10.61129 

9.40212 

10.59788 

10.01341 

9.98659 

50 

20 

44 

11 

38921 

61079 

40266 

59734 

01344 

98656 

49 

16 

48 

12 

38971 

61029 

40319 

59681 

01348 

98652 

48 

12 

52 

13 

39U21 

60979 

40372 

59628 

01351 

98649 

47 

8 

56 

14 

39071 

60929 

40427} 

59575 

01354 

98646 

46 

4 

57 

15 

9.39121 

10.60879 

9.40478 

10.59522 

10.01357 

9.98643 

45 

3 

4 

16 

39170 

60830 

40531 

59469 

01360 

98640 

44 

5l3 

8 

17 

39220 

60780 

40584 

59416 

01364 

98636 

43 

52 

12 

18 

39270 

60730 

40636 

£9364 

01367 

98633 

42 

48 

16 

19 

39319 

60681 

40689 

59311 

01370 

98630 

4 1 

44 

20 

20 

9.39209 

10.60531 

9.40742 

10.59258 

10.01373 

9.98627 

40 

4o 

24 

21 

39418 

60582 

40795 

59205 

01377 

98623 

39 

36 

28 

22 

394ii7 

60533 

40847 

59153 

01380 

98620 

38 

32 

32 

23 

39517 

60483 

40900 

59100 

01383 

98617 

37 

28 

36 

24 

39566 

60434 

40952 

59048 

01386 

98814 

36 

24 

40 

25 

9.39615 

10.60385 

9.41005 

10.58995 

10.01390 

9.98610 

35 

20 

44 

26 

39601 

60336 

41057 

58943 

01393 

98607 

34 

16 

48 

27 

39713 

60287 

41109 

58841 

01396 

98604 

33 

12 

52 

28 

39702 

60238 

41161 

58839 

01399 

98601 

32 

8 

56 

29 

39811 

60189 

41214 

58786 

01403 

98597 

31 

4 

58 

30 

9.39860 

10.60140 

9.41266 

10.587:34 

10.01403 

9.98594 

30 

a 

4 

31 

39909 

60091 

41318 

58682 

01409 

98591 

29 

56 

8 

32 

39958 

60042 

41370 

58630 

014 >2 

98588 

28 

52 

12 

33 

4 >006 

59994 

41422 

58578 

01416 

98584 

27 

48 

16 

34 

40055 

59945 

41474 

58526 

01419 

98581 

26 

41 

20 

35 

9.40103 

10.:' 9897 

9.41526 

10.58474 

10.01422 

9.98578 

25 

40 

24 

36 

40152 

59848 

41578 

58422 

01426 

98574 

24 

36 

28 

37 

40200 

59SU0 

41629 

58371 

01429 

98571 

23 

32 

32 

38 

40249 

59751 

41681 

58319 

01432 

98568 

22 

28 

36 

39 

40297 

59703 

41733 

58267 

01435 

98565 

21 

24 

40 

40 

9.40346 

10.59654 

9.41784 

10.58216 

10.01439 

9.98561 

20 

20 

44 

41 

40194 

59606 

41836 

58164 

01442 

98558 

19 

16 

48 

42 

40442 

59558 

41887 

58113 

01445 

98555 

18 

12 

52 

43 

40490 

59510 

41939 

58061 

01449 

98551 

17 

8 

56 

44 

4053S 

59462 

41990 

58010 

01452 

98548 

16 

4 

59 

45 

9.40586 

10.59414 

9.42041 

10.57959 

10.01455 

9.98545 

15 

1 

4 

46 

40614 

59366 

42093 

57907 

01459 

98541 

14 

56 

8 

47 

40682 

59318 

42144 

57856 

01462 

98538 

13 

62 

12 

48 

40730 

59270 

42195 

57805 

01465 

98535 

12 

48 

16 

49 

40778 

59222 

42246 

57754 

01469 

98531 

11 

ii 

20 

50 

9.40825 

10.59175 

9.42297 

10.57,03 

10.01472 

9.98528 

10 

10 

24 

51 

40873 

59127 

42348 

57652 

(>1475 

98525 

9 

36 

28 

52 

40921 

59079 

42399 

57601 

01479 

98521 

8 

32 

32 

53 

40968 

59032 

42450 

57550 

01482 

98518 

7 

28 

36 

54 

41016 

58981 

42501 

57499 

01485 

98515 

6 

24 

40 

55 

9.41063 

10.58937 

9.42552 

10.57418 

10.01489 

9.98511 

5 

20 

44 

56 

41111 

58889 

42603 

57397 

01492 

98508 

4 

16 

48 

57 

41158 

58842 

42653 

57347 

01495 

98505 

3 

12 

52 

58 

41205 

58795 

42704 

57296 

01499 

98501 

2 

8 

56 

59 

41252 

58748 

42755 

57245 

01502 

9849 S 

1 

4 

GO 

60 

41300 

58700 

42805 

57195 

01506 

98494 

0 

O 

MS. 

M 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

M.S. 

t> h 

104° 






75° 

5 h 
-- 























214 


Logarithms Trigonometric. 


l h 

15° 



Logarithms. 


164° 

— 

10 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

0 

0 

9.41300 

10.58700 

9.42805 

10.57195 

10.01566 

9.98494 

60 

GO 

4 

1 

41347 

58653 

42856 

57144 

01509 

98491 

59 

66 

8 

2 

41394 

686 '6 

42906 

57094 

01512 

984S8 

58 

62 

12 

3 

41441 

58559 

42957 

57043 

01516 

98484 

67 

48 

16 

4 

41488 

58512 

43007 

56993 

01-519 

9S481 

66 

44 

20 

5 

9.41535 

10.58405 

9.43057 

10.56943 

10.01523 

9.98477 

55 

40 

24 

6 

41582 

58418 

43108 

66892 

01526 

98474 

54 

36 

28 

7 

41628 

68372 

43158 

56842 

01529 

98471 

53 

32 

32 

8 

41675 

68325 

43208 

66792 

01533 

98467 

52 

28 

36 

9 

41722 

58278 

43258 

56742 

01536 

98464 

51 

24 

40 

10 

9.41768 

10.5*232 

9.43308 

10.56692 

10.01540 

9.98460 

50 

20 

44 

11 

41815 

58185 

43358 

56642 

01543 

98457 

49 

16 

48 

12 

41861 

58139 

43408 

56592 

01547 

98453 

48 

12 

52 

13 

41908 

58092 

43458 

56542 

• 01550 

98450 

47 

8 

56 

14 

41954 

58046 

43508 

56492 

01553 

98447 

46 

4 

1 

15 

9.42001 

10.57999 

9.43558 

10.56442 

10.01557 

9.98443 

45 

50 

4 

16 

42i >47 

67953 

43007 

56393 

01560 

98440 

44 

56 

8 

17 

42093 

67907 

43657 

56343 

01564 

98436 

43 

52 

12 

18 

42140 

57860 

43707 

50293 

01567 

98433 

42 

48 

16 

19 

42186 

57814 

43756 

56244 

01571 

98429 

41 

44 

20 

20 

9.422 2 

10.57768 

9.43806 

10.56194 

10.01574 

9.98426 

40 

40 

24 

21 

42278 

67722 

471855 

56145 

01578 

98422 

39 

36 

28 

22 

42324 

57676 

43905 

66095 

01581 

98419 

38 

32 

32 

23 

42370 

67630 

43954 

56046 

01585 

98415 

37 

28 

36 

24 

42416 

57584 

44004 

55996 

01588 

98412 

36 

24 

40 

25 

9.42461 

10.57539 

9.44053 

10.55947 

10.01591 

9.98409 

35 

20 

44 

26 

42507 

57493 

44102 

55898 

01595 

98405 

34 

16 

48 

27 

42553 

57447 

44151 

55849 

01598 

98402 

33 

12 

52 

28 

42599 

57401 

44201 

65769 

01602 

98398 

32 

8 

56 

29 

42644 

57356 

44250 

55750 

01605 

98395 

31 

4 

a 

30 

9.42690 

10.57310 

9.44299 

10.55701 

10.01609 

9.98391 

30 

58 

4 

31 

42735 

57265 

44348 

55652 

01612 

98388 

29 

56 

8 

32 

42781 

57219 

44397 

55603 

01616 

98384 

28 

52 

12 

33 

42826 

67174 

44446 

55554 

01619 

98381 

27 

48 

16 

34 

42872 

67128 

44495 

55505 

01623 

98377 

26 

44 

20 

35 

9.42917 

10.57083 

9.44544 

10.55456 

10.01627 

9.98373 

25 

40 

24 

36 

42962 

57038 

44592 

55408 

01630 

98370 

2-4 

36 

28 

37 

43008 

56992 

44641 

55359 

01634 

98366 

23 

32 

32 

3S 

43053 

60947 

44690 

65310 

01637 

98363 

22 

28 

36 

39 

43098 

56902 

44738 

55262 

01641 

9S359 

21 

24 

40 

40 

9.43143 

10.56857 

9.44787 

10.56213 

10.01644 

9.98356 

20 

20 

44 

41 

43188 

56812 

44836 

55164 

01618 

98352 

19 

16 

48 

42 

43233 

56767 

4-1884 

55116 

01651 

98349 

18 

12 

52 

43 

43278 

66722 

44933 

55067 

01655 

98345 

17 

8 

56 

44 

43323 

56677 

44981 

55019 

01658 

98342 

16 

4 

3 

45 

9.4:3367 

10.56633 

9.45029 

10.54971 

10.01662 

9.98338 

15 

57 

4 

46 

43412 

56588 

45078 

54922 

01666 

98334 

14 

56 

8 

47 

43457 

56543 

45126 

54874 

01669 

0833 L 

13 

62 

12 

48 

43502 

56498 

45174 

64826 

01673 

98327 

12 

48 

16 

49 

43546 

56454 

46222 

54778 

01676 

98324 

11 

44 

20 

50 

9.43501 

10.56409 

9.45271 

10.54729 

10.01680 

9.98320 

10 

40 

24 

51 

43635 

56365 

45319 

64681 

01683 

98317 

9 

36 

28 

52 

43680 

56320 

45367 

54633 

01687 

98313 

8 

32 

32 

53 

43724 

66276 

45415 

54585 

01691 

98309 


28 

36 

54 

43769 

66231 

45463 

54537 

01034 

98306 

6 

24 

40 

55 

9.43813 

10.56187 

9.45511 

10.54489 

10.01698 

9.98302 

6 

20 

44 

56 

43857 

56143 

45559 

54-441 

01701 

98299 

4 

16 

48 

57 

43901 

66099 

45606 

54304 

01705 

98295 

3 

12 

52 

58 

43946 

56054 

45654 

54346 

01709 

98291 

2 

8 

56 

59 

43990 

56010 

45702 

54298 

01712 

98288 

1 

4 

4: 

60 

44034 

55966 

45750 

54250 

01716 

98284 

0 

50 

M.S. 

M 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

M.S. 

7 h 

105 

D 






74° 

4“ 























Logarithms Trigonometric. 


215 


l b 


M.S. 

4 

4 

8 

12 

16 

20 

24 

28 

32 

36 

40 

44 

48 

52 

56 

5 
4 
8 

12 

16 

20 

24 

28 

32 

36 

40 

44 

48 

52 

56 

G 

4 

8 

12 

16 

20 

24 

28 

32 

36 

40 

44 

48 

52 

56 

7 
4 

8 

12 

16 

20 

24 

28 

32 

36 

40 

44 

48 

52 

56 

8 

M.S. 

7“ 


16° 



Logarithms. 


163° 

10 h 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangeut. 

Secant. 

Cosine. 

M 

M.S. 

0 

9.44084 

10.55966 

9.45750 

10.54250 

10.01716 

9.9S284 

60 

5G 

l 

44078 

55922 

45797 

54203 

01719 

98281 

59 

56 

2 

44122 

55878 

45845 

54155 

01723 

98277 

58 

52 

3 

44166 

65834 

45892 

54108 

01727 

98273 

57 

48 

4 

44210 

55790 

45940 

54060 

01730 

98270 

56 

44 

6 

9.44253 

10.55747 

9.45987 

10.54013 

10.01734 

9.98266 

55 

40 

6 

44297 

56703 

46035 

53965 

01738 

98262 

54 

36 

7 

44341 

55659 

46082 

53918 

01741 

98259 

53 

32 

8 

44385 

65615 

46130 

53870 

01745 

98255 

52 

28 

9 

44428 

55572 

46177 

53823 

01749 

98251 

51 

24 

10 

9.44472 

10.55528 

9.46224 

10.53776 

10.01752 

9.98248 

50 

20 

11 

44516 

55484 

46271 

53729 

01756 

98244 

*9 

16 

12 

44559 

65441 

46319 

53681 

01760 

98240 

48 

12 

13 

44602 

65398 

46366 

53634 

01763 

98237 

47 

8 

14 

44646 

55354 

46413 

53587 

01767 

98233 

46 

4 

15 

9.446S9 

10.56311 

9.46460 

10.53540 

10.01771 

9.98229 

45 

5 5 

16 

44733 

55267 

46507 

53493 

01774 

98226 

44 

56 

17 

44776 

65224 

46554 

63446 

01778 

98222 

43 

52 

18 

44819 

65181 

46601 

53399 

01782 

98218 

42 

48 

19 

44862 

55138 

46648 

53352 

01785 

98215 

41 

44 

20 

9.44905 

10.55095 

9.46694 

10.53306 

10.01789 

9.98211 

40 

40 

21 

44948 

55052 

46741 

53259 

01793 

98207 

39 

36 

22 

44992 

65008 

46788 

53212 

01796 

98204 

38 

32 

23 

45035 

64965 

46835 

53165 

01800 

98200 

37 

28 

24 

45077 

64923 

46881 

53119 

01804 

98196 

36 

24 

25 

9.45120 

10.54880 

9.4692S 

10.53072 

10.01808 

9.98192 

35 

20 

26 

45163 

64837 

46975 

53025 

01811 

98189 

34 

16 

27 

45206 

54794 

47021 

52979 

01815 

98185 

33 

12 

28 

45249 

54751 

47068 

52931 

01819 

98181 

32 

8 

29 

45292 

54708 

47114 

52886 

01823 

98177 

31 

4 

30 

9.45334 

10.54666 

9.47160 

10.52840 

10.01826 

9.98174 

30 

54 

31 

45377 

54623 

47207 

52793 

01830 

98170 

29 

56 

32 

45419 

64581 

47253 

52747 

01834 

98166 

28 

52 

33 

45462 

54538 

47299 

52701 

01838 

98162 

27 

48 

34 

45504 

54496 

47346 

62654 

01841 

98159 

26 

44 

35 

9.45547 

10.54453 

9.47392 

10.52608 

10.01845 

9.98155 

25 

40 

36 

45589 

64411 

47438 

52562 

01849 

98151 

24 

36 

37 

45632 

64368 

47484 

52516 

01S53 

98147 

23 

32 

38 

45674 

54326 

47530 

52470 

01856 

98144 

22 

28 

39 

45716 

54284 

47576 

52424 

01860 

98140 

21 

24 

40 

9.45758 

10.54242 

9.47622 

10.52378 

10.01864 

9.98136 

20 

20 

41 

45801 

54199 

47668 

52332 

01868 

9S132 

19 

16 

42 

45843 

54157 

47714 

62286 

01871 

98129 

18 

12 

43 

45885 

54115 

47760 

52240 

01875 

98125 

17 

8 

44 

45927 

54073 

47806 

52194 

01879 

98121 

16 

4 

45 

9.45969 

10.54031 

9.47852 

10.52148 

10.01883 

9.98117 

15 

53 

46 

46011 

53989 

47897 

52103 

01887 

98113 

14 

56 

47 

46053 

63947 

47943 

52057 

01890 

98110 

13 

52 

48 

46095 

63905 

47989 

62011 

01894 

98106 

12 

48 

49 

46136 

53864 

48035 

51965 

01898 

98102 

11 

44 

50 

9.46178 

10.53822 

9.48080- 

10.51920 

10.01902 

9.98098 

10 

40 

51 

46220 

63780 

48126 

61874 

01906 

98094 

9 

36 

52 

46262 

53738 

48171 

61829 

01910 

98U90 

8 

32 

53 

46303 

53697 

48217 

51783 

01913 

98087 

7 

28 

54 

46345 

53655 

48262 

51738 

01917 

98083 

6 

24 

55 

9.46386 

10 53614 

9.48307 

•10.51693 

10.01921 

9.98079 

5 

20 

56 

46428 

53572 

48353 

51647 

01925 

98075 

4 

16 

57 

46469 

63531 

48398 

61602 

01929 

98U71 

3 

12 

58 

46511 

53489 

48443 

61557 

01933 

98067 

2 

8 

59 

46552 

63448 

48489 

51511 

01937 

98063 

1 

4 

60 

46594 

53406 

48534 

51466 

01940 

98060 

0 

52 

M 

10G 

Cosine. 

0 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

73° 

M. S. 

4 h 

















216 Logarithms Trigonometric. 


l b 

17° 



Logarithms. 


162° 

10 b 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

8 

0 

9.46594 

10.53406 

9.48534 

10.51466 

10.01940 

9.98060 

60 

5'i 

4 

1 

46635 

53365 

48579 

51421 

01944 

9SU56 

59 

56 

8 

2 

46676 

63324 

48624 

51376 

01948 

98052 

58 

52 

12 

3 

46717 

53283 

48669 

51331 

01952 

98048 

57 

48 

10 

4 

46758 

53242 

48714 

51286 

01956 

98044 

56 

44 

20 

5 

9.46800 

10.53200 

9.48759 

10.51241 

10.01960 

9.98040 

55 

40 

24 

6 

46841 

53159 

48804 

51196 

01964 

98036 

54 

36 

28 

7 

46882 

53118 

48849 

51151 

01968 

98032 

53 

32 

32 

8 

46923 

£3077 

48894 

51106 

01971 

98029 

52 

28 

36 

9 

46964 

53036 

48939 

51061 

01975 

98025 

51 

24 

40 

10 

9.47005 

10.52995 

9.48984 

10.51016 

10.01979 

9.98021 

50 

20 

44 

11 

47045 

62955 

49029 

50971 

01983 

98017 

49 

16 

48 

12 

47086 

52914 

49073 

50927 

01987 

98013 

48 

12 

52 

13 

47127 

52873 

49118 

50882 

01991 

98009 

47 

8 

66 

14 

47168 

52832 

49163 

50837 

01995 

98005 

46 

4 

9 

15 

9.47209 

10.52791 

9.49207 

10.50793 

10.01999 

9.98001 

45 

51 

4 

16 

47249 

52751 

49252 

50748 

02003 

97997 

44 

56 

8 

17 

47290 

52710 

49296 

50704 

02007 

97993 

43 

52 

12 

18 

47330 

52670 

49341 

50659 

02011 

97989 

42 

48 

16 

19 

47371 

52629 

49385 

50615 

02014 

97986 

41 

44 

20 

20 

9.47411 

10.52589 

9.49430 

10.50670 

10.02018 

9.97982 

40 

40 

24 

21 

47452 

52548 

49474 

50526 

02022 

9797S 

39 

36 

28 

22 

47492 

52508 

49519 

50481 

02026 

97974 

38 

32 

32 

23 

47533 

52467 

49563 

50437 

02030 

97970 

37 

28 

36 

24 

47573 

52427 

49607 

50393 

02034 

97966 

36 

24 

40 

25 

9.47613 

10.52387 

9.49652 

10.50348 

10.02038 

9.97962 

35 

20 

44 

26 

47654 

52346 

49696 

50304 

02042 

97958 

34 

16 

48 

27 

47694 

52306 

49740 

60260 

02046 

97954 

33 

12 

52 

28 

47734 

52266 

49784 

50216 

021/50 

97950 

32 

8 

66 

29 

47774 

52226 

49828 

50172 

02054 

97946 

31 

4 

10 

30 

9.47814 

10.52186 

9.49872 

10.50128 

10.02058 

9.97942 

30 

50 

4 

31 

47854 

52146 

49916 

50084 

02062 

97938 

29 

56 

8 

32 

47894 

52106 

49960 

50040 

02066 

97934 

28 

52 

12 

33 

47934 

52066 

50004 

49996 

02070 

97930 

27 

48 

16 

34 

47974 

62026 

50i48 

49952 

02074 

97926 

26 

44 

20 

35 

9.48014 

10.51986 

9.50092 

10.49908 

10.02078 

9.97922 

25 

40 

24 

36 

48054 

51946 

50136 

49864 

02082 

97918 

24 

36 

28 

37 

48094 

51906 

50180 

49820 

02086 

97914 

23 

32 

32 

38 

48133 

51867 

50223 

49777 

02090 

97910 

22 

28 

36 

39 

48173 

51827 

50267 

49733 

02094 

97906 

21 

24 

40 

40 

9.48213 

10.51787 

9.50311 

10.49689 

10.02098 

9.97902 

20 

20 

44 

41 

48252 

51748 

50355 

49645 

02102 

97898 

19 

16 

48 

42 

48292 

51708 

50398 

49602 

02106 

97894 

18 

12 

52 

43 

48332 

51668 

50442 

49558 

02110 

97890 

17 

8 

66 

44 

48371 

51629 

50485 

49515 

02114 

97886 

16 

4 

11 

45 

9.48411 

10.51589 

9.50529 

10.49471 

10.02118 

9.97882 

15 

±‘J 

4 

46 

48450 

51550 

50572 

49428 

02122 

97878 

14 

56 

8 

47 

48490 

61610 

50616 

49384 

02126 

97874 

13 

52 

12 

48 

48529 

51471 

50659 

49341 

02130 

97870 

12 

48 

16 

49 

48568 

51432 

50703 

49297 

02134 

97866 

11 

44 

20 

50 

9.48607 

10.51393 

9.50746 

10.49254 

10.02139 

9.97861 

10 

40 

24 

51 

48647 

61353 

50789 

49211 

02143 

97857 

9 

36 

28 

52 

48686 

51314 

60833 

49167 

02147 

97853 

8 

32 

32 

53 

48725 

51275 

50876 

49124 

02151 

97849 

7 

28 

36 

54 

48764 

51236 

50919 

49081 

02155 

97845 

6 

24 

40 

55 

9.48803 

10.51197 

9.50962 

10.49038 

10.02159 

9.97841 

6 

20 

44 

56 

48842 

51158 

51005 

48995 

02163 

97837 

4 

16 

48 

57 

48881 

51119 

61048 

48952 

02167 

97833 

3 

12 

52 

58 

48920 

51080 

61092 

48908 

02171 

97829 

2 

8 

56 

59 

48959 

51041 

51135 

48865 

02175 

97825 

1 

4 

1» 

60 

48998 

51002 

51178 

48822 

02179 

97821 

0 

L8 

MS. 

M 

Cosine. 

Secant. 

Cotangent 

Taugeut. 

Cosecant. 

Sine. 

M 

M. S. 

7 b 

107 

o 






72° 

4 b 















Logarithms Trigonometric. 


217 


l h 

18° 



Logarithms. 


161° 

10 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

12 

0 

9.48998 

10.51002 

9.51178 

10.48822 

10.02179 

9.97821 

60 

48 

4 

1 

49037 

50963 

51221 

48779 

02183 

97817 

59 

56 

8 

2 

49076 

50024 

51204 

48736 

02188 

97812 

58 

52 

12 

3 

49115 

50885 

51306 

48694 

02192 

97808 

57 

48 

16 

4 

49153 

50847 

51349 

48651 

02196 

97804 

56 

44 

20 

5 

9.49192 

10.50308 

9.51392 

10.48608 

10.02200 

9.97800 

55 

40 

24 

6 

49231 

50769 

51435 

48565 

02204 

97796 

54 

36 

28 

7 

49269 

50731 

51478 

48522 

02203 

97792 

53 

32 

32 

8 

49308 

50692 

51520 

48480 

02212 

97788 

52 

28 

36 

9 

49347 

50653 

51563 

48437 

02216 

97784 

51 

24 

40 

10 

9.49385 

10.50615 

9.51606 

10.48394 

10.02221 

9.97779 

50 

20 

44 

11 

49424 

50576 

51648 

48352 

02225 

97775 

49 

16 

48 

12 

49462 

50538 

51691 

48309 

02229 

97771 

48 

12 

52 

13 

49500 

50500 

51734 

48266 

02233 

97767 

47 

8 

56 

14 

49539 

50461 

51776 

48224 

02237 

97763 

46 

4 

13 

15 

9.49577 

10.50423 

9.51819 

10.48181 

10.02241 

9.97759 

45 

47 

4 

16 

49615 

50385 

51861 

48139 

02246 

97754 

44 

56 

8 

17 

49654 

50346 

51903 

48097 

02250 

97750 

43 

52 

12 

18 

49692 

50308 

51946 

48054 

02254 

9774(5 

42 

48 

16 

19 

49730 

50270 

51988 

48012 

02258 

97742 

41 

4-4 

20 

20 

9.49768 

10.50232 

9.52031 

10.47969 

10.02262 

9.97738 

40 

40 

24 

21 

49806 

50194 

52073 

47927 

02266 

97734 

39 

36 

28 

22 

49844 

50156 

52115 

47885 

02271 

97729 

38 

32 

32 

23 

49882 

50118 

52157 

47843 

02275 

97725 

37 

28 

36 

24 

49920 

50080 

52200 

47800 

02279 

97721 

36 

24 

40 

25 

9.49958 

10.50042 

9.52242 

10.47758 

10.02283 

9.97717 

35 

20 

44 

26 

49996 

50004 

52284 

47716 

02287 

97713 

34 

16 

48 

27 

50034 

49960 

52326 

47674 

02292 

97708 

33 

12 

52 

28 

50072 

49928 

52368 

47632 

02296 

97704 

32 

8 

56 

29 

50110 

49890 

52410 

47590 

02300 

97700 

31 

4 

14 

30 

9.50148 

10.49852 

9.52452 

10.47548 

10.02304 

9.97(596 

30 

4G 

4 

31 

50185 

49815 

52494 

47506 

02309 

97691 

29 

56 

8 

32 

50223 

49777 

52536 

47464 

02313 

97687 

28 

52 

12 

33 

50261 

49739 

52578 

47422 

02317 

97683 

27 

48 

16 

34 

50298 

49702 

52620 

47380 

02321 

97679 

26 

44 

20 

35 

9.50336 

10.49664 

9.52661 

10.47339 

10.02326 

9.97674 

25 

40 

24 

36 

60374 

49626 

62703 

47297 

02330 

97670 

24 

36 

28 

37 

50411 

49589 

52745 

47255 

02334 

97666 

23 

32 

32 

38 

50449 

49551 

52787 

47213 

02338 

97662 

22 

28 

36 

39 

50486 

49514 

52829 

47171 

02343 

97657 

21 

24 

40 

40 

9.50523 

10.49477 

9.52870 

10.47130 

10.02347 

9.97653 

20 

20 

44 

41 

50561 

49439 

52912 

47088 

02351 

97649 

19 

16 

48 

42 

50598 

49402 

52953 

47047 

02355 

97645 

18 

12 

52 

43 

50635 

49365 

52995 

47005 

02360 

97640 

17 

8 

56 

44 

50673 

49327 

53037 

46963 

02364 

97636 

16 

4 

15 

45 

9.50710 

10.49290 

9.53078 

10.46922 

10.02368 

9.97632 

15 

45 

4 

46 

50747 

49253 

53120 

46880 

02372 

97628 

14 

56 

8 

47 

50784 

49216 

53161 

46839 

02377 

97623 

13 

52 

12 

48 

50821 

49179 

53202 

46798 

02381 

97619 

12 

48 

16 

49 

50858 

49142 

53244 

46756 

02385 

97615 

11 

44 

20 

50 

9.50896 

10.49101 

9.53285 

10.46715 

10.02390 

9.97610 

10 

40 

24 

51 

50933 

49007 

53327 

46673 

02394 

97606 

9 

36 

28 

52 

50970 

49030 

53368 

46632 

02398 

97602 

8 

32 

32 

53 

51007 

48993 

53409 

46591 

02403 

97597 

7 

28 

36 

54 

51043 

48957 

53450 

46550 

02407 

97593 

6 

24 

40 

55 

9.51080 

10.48920 

9.53492 

10.46508 

10.02411 

9.97589 

5 

20 

44 

56 

51117 

48883 

53533 

46467 

02416 

97584 

4 

16 

48 

57 

51154 

48846 

53574 

46426 

02420 

97580 

3 

12 

52 

58 

51191 

48809 

53615 

46385 

02424 

97576 

2 

8 

56 

59 

51227 

48773 

53656 

46344 

02429 

97571 

1 

4 

10 

60 

51264 

48736 

53697 

46303 

02433 

97567 

0 

44 

M. S. 
.1* 

M 

108 c 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

71° 

M. S 

4 h 






















Logarithms Trigonometric. 


213 


1 

l h 

19° 



Logarithms. 


1G0° 

10 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. I 

M 

M.S.j 

16 

0 

9.51264 

10.48730 

9.53697 

10.40303 

10.02433 

9.97507 

GO 

44: 

4 

1 

51301 

48099 

53738 

46262 

02437 

97563 

59 

56 

8 

2 

51338 

48002 

53779 

40221 

02442 

97558 

58 

52 

12 

3 

51374 

48020 

53820 

40180 

02440 

97554 

57 

48 

16 

4 

51411 

48589 

63801 

40139 

02450 

97550 

56 

44 

20 

5 

9.51447 

10.4S553 

9.53902 

10.46098 

10.02455 

9.97545 

55 

40 

24 

6 

51484 

48516 

63943 

40057 

02459 

97541 

54 

30 

28 

7 

51520 

48480 

63984 

4GO10 

02404 

97530 

53 

32 

32 

8 

51557 

48443 

54025 

45975 

02408 

97532 

52 

28 

30 

9 

51593 

48407 

54065 

45935 

02472 

97528 

51 

24 

40 

10 

9.51029 

10.48321 

9.64106 

10.45894 

10.02477 

9.97523 

50 

20 

44 

11 

51006 

48334 

54147 

45853 

02481 

97519 

49 

10 

4.8 

12 

51702 

48298 

54187 

45813 

02485 

97515 

48 

12 

52 

13 

51738 

48202 

54228 

45772 

02490 

97510 

47 

8 

50 

14 

51774 

48220 

64209 

45731 

02494 

97506 

40 

4 

17 

15 

9.51811 

10.48189 

9.54309 

10.45091 

10.02499 

9.97501 

45 

43 

4 

10 

51847 

48153 

54350 

45050 

02503 

97497 

44 

56 

8 

17 

51883 

4S117 

54390 

45010 

02508 

97492 

43 

52 

12 

18 

51919 

48081 

54431 

45509 

02512 

97488 

42 

48 

10 

19 

51955 

48045 

54471 

45529 

02516 

97484 

41 

44 

20 

20 

9.51991 

10.48009 

9.64512 

10.45488 

10.02521 

9.07479 

40 

40 

24 

21 

52027 

47973 

54552 

45448 

02525 

9.475 

39 

30 

28 

22 

62063 

47937 

54593 

45407 

02530 

97470 

38 

32 

32 

23 

52099 

47901 

54033 

45307 

02534 

9740(5 

37 

28 

30 

24 

52135 

47865 

54073 

45327 

02539 

97401 

36 

24 

40 

25 

9.52171 

10.47829 

9.54714 

10.45286 

10.02543 

9.97457 

35 

20 

44 

26 

52207 

47793 

54754 

45240 

02517 

97453 

34 

10 

48 

27 

52242 

47758 

64794 

45200 

02552 

97448 

33 

12 

52 

28 

52278 

47722 

64835 

45105 

02550 

97444 

32 

8 

50 

29 

52314 

47080 

54s75 

45125 

02501 

97439 

31 

4 

18 

30 

9.52350 

10.47650 

9.54915 

10.45085 

10.02505 

9.97435 

30 

43 

4 

31 

52385 

47616 

54955 

45045 

02570 

97430 

29 

56 

8 

32 

52421 

47579 

54995 

45005 

02574 

97420 

28 

52 

12 

33 

52450 

47544 

56035 

44965 

02579 

97421 

27 

48 

16 

34 

62492 

47508 

55075 

44925 

02583 

97417 

•26 

4A 

20 

35 

9.52527 

10.47473 

9.55115 

10.44885 

10.02588 

9.97412 

25 

40 

24 

30 

52568 

47437 

55155 

4184) 

02592 

97408 

24 

36 

28 

37 

52598 

47402 

6519.5 

44805 

02597 

97403 

23 

32 

32 

38 

62034 

47300 

65235 

447 65 

02601 

97390 

22 

28 

30 

39 

52609 

47331 

65275 

44725 

0200(3 

97394 

21 

24 

40 

40 

9.52705 

10.47295 

9.55315 

10.44085 

10.02010 

9.97390 

20 

20 

44 

41 

52740 

47260 

65355 

44045 

02615 

97385 

19 

10 

48 

42 

52775 

47225 

65395 

44005 

02019 

9T381 

18 

12 

52 

43 

52811 

47189 

65434 

44500 

02024 

97370 

17 

8 

50 

44 

52840 

47154 

55474 

44526 

' 02028 

97372 

10 

4 

111 

45 

9.52881 

10.47119 

9.55514 

10.44486 

10.02033 

9.97307 

15 

41 

4 

40 

62910 

47084 

65554 

44446 

02037 

97303 

14 

56 

8 

47 

52951 

47040 

65593 

44407 

02042 

97358 

13 

52 

12 

48 

62986 

47014 

65033 

41307 

02647 

97353 

12 

48 

10 

49 

53021 

40970 

65073 

44327 

026;)! 

97349 

11 

44 

20 

50 

9.53056 

10.40944 

9.55712 

10.44288 

10.02650 

9.97344 

10 

40 

24 

51 

53092 

40908 

65752 

44248 

02660 

97340 

9 

36 

28 

52 

53126 

40874 

55791 

44209 

02605 

97335 

8 

32 

32 

53 

53101 

46839 

65831 

44109 

026G9 

97331 

7 

28 

30 

54 

53190 

46804 

55870 

44130 

02074 

9732G 

G 

24 

40 

55 

9.53231 

10.46769 

9.55910 

10.-14090 

10.02678 

9.97322 

5 

20 

44 

66 

53206 

40734 

55949 

44051 

02633 

97317 

4 

1C 

48 

57 

53301 

40099 

55989 

44011 

02688 

97312 

3 

12 

52 

58 

53330 

4600-4 

60028 

43972 

02692 

97308 

2 

8 

50 

59 

63370 

46030 

60007 

43933 

02097 

97303 

1 

4 

30 

60 

53405 

40595 

50107 

43893 

02701 

97299 

0 


M.S. 

7 h 

M 

109 

Cosine. 

0 

Secant. 

Cotaugeut 

Taugont. 

Cosecant. 

1 Sine. 

M 

70° 

M.S. 

4“ 

























Logarithms Trigonometric. 219 


p 

20° 


Logarithms. 


159° 

10 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M. S. 

30 

0 

9.534)5 

10.46595 

9.56107 

10.43893 

10.02701 

9.97299 

60 

4:0 

4 

1 

53440 

46560 

56146 

43854 

02706 

97291 

59 

56 

8 

2 

53475 

46525 

56185 

43815 

02711 

97289 

58 

52 

12 

3 

53509 

46491 

56224 

43776 

02715 

97285 

57 

48 

16 

4 

53544 

46456 

56264 

43736 

02720 

97280 

56 

44 

20 

5 

9.53578 

10.46422 

9.56303 

10.43697 

10.02721 

9.97276 

55 

40 

24 

6 

53613 

46387 

66342 

43658 

02729 

97271 

54 

36 

28 

7 

53647 

46353 

56381 

43619 

02734 

972G6 

53 

32 

32 

8 

53682 

46318 

56420 

43580 

02738 

97262 

52 

28 

36 

9 

53716 

46284 

56159 

43541 

02743 

97257 

51 

24 

40 

10 

9.53751 

10.46249 

9.56498 

10.43502 

10.0274S 

9.97252 

50 

20 

44 

11 

53785 

46215 

56537 

43463 

02752 

97248 

49 

16 

48 

12 

53819 

46181 

56576 

43424 

02757 

97243 

48 

12 

52 

13 

53854 

46146 

56615 

43385 

02762 

97238 

47 

8 

56 

14 

53888 

46112 

56654 

43316 

02766 

97234 

46 

4 

31 

15 

9.53922 

10.46078 

9.56693 

10.43307 

10.02771 

9.97229 

45 

30 

4 

16 

53957 

46043 

56732 

43268 

02776 

97224 

44 

56 

8 

17 

53991 

46009 

56771 

43229 

02780 

97220 

43 

52 

12 

18 

54025 

45975 

66810 

43190 

02785 

97215 

42 

48 

16 

19 

54059 

45941 

56849 

43151 

02790 

97210 

41 

44 

20 

20 

9.54093 

10.45907 

9.56887 

10.43113 

10.02794 

9.97206 

40 

40 

24 

21 

54127 

45873 

56926 

43074 

02799 

97201 

39 

30 

28 

22 

5416 L 

45839 

56965 

43035 

02804 

97196 

38 

32 

32 

23 

54195 

45805 

57004 

42996 

02808 

97192 

37 

28 

36 

24 

54229 

45771 

57042 

42958 

02813 

97187 

36 

24 

40 

25 

9.54263 

10.45737 

9.57081 

10.42919 

10.02818 

9.97182 

35 

20 

4-1 

26 

54297 

45703 

57120 

42880 

02822 

97178 

34 

16 

48 

27 

54331 

45669 

57158 

42842 

02827 

97173 

33 

12 

52 

28 

54365 

45635 

57197 

42803 

02532 

97168 

32 

8 

56 

29 

54399 

45601 

57235 

42765 

02837 

97163 

31 

4 

33 

30 

9.54433 

10.45567 

9.57274 

10.42726 

10.02841 

9.97159 

30 

38 

4 

31 

54466 

45534 

57312 

42688 

02846 

97154 

29 

56 

8 

32 

54500 

45500 

57351 

42649 

02851 

97119 

28 

52 

12 

33 

54534 

46466 

57389 

42611 

02855 

97145 

27 

48 

16 

34 

54567 

45433 

57428 

42572 

02860 

97140 

26 

44 

20 

35 

9.54601 

10.45399 

9.57466 

10.42534 

10.02865 

9.97135 

25 

40 

24 

36 

54635 

45365 

57504 

42496 

02570 

97130 

24 

30 

28 

37 

54668 

45332 

57543 

42457 

02871 

97126 

23 

32 

32 

38 

54702 

45298 

57581 

42419 

02879 

97121 

22 

28 

36 

39 

54735 

45265 

57619 

42381 

02884 

97116 

21 

24 

40 

40 

9.54769 

10.45231 

9.57658 

10.42312 

1C.02889 

9.97111 

20 

20 

44 

41 

54802 

45198 

57696 

42304 

02593 

97107 

19 

16 

48 

42 

54836 

45164 

57734 

42266 

02898 

97102 

18 

12 

52 

43 

54869 

45131 

57772 

42228 

02903 

97097 

17 

8 

56 

44 

54903 

45097 

57810 

42190 

02908 

97092 

16 

4 

3.1 

45 

9.54936 

10.45064 

9.57819 

10.42151 

10.02913 

9.97087 

15 

37 

4 

46 

54969 

45031 

57887 

42113 

02917 

97083 

14 

56 

8 

47 

55003 

44997 

57925 

42075 

02922 

97078 

13 

52 

12 

48 

55036 

41964 

57963 

42037 

02927 

97073 

12 

48. 

16 

49 

55069 

44931 

58001 

41999 

02932 

97068 

11 

44 

20 

50 

9.55102 

10.44898 

9.68039 

10.41961 

10.02937 

9.97063 

10 

40 

24 

51 

55136 

44864 

58077 

41923 

02941 

97059 

9 

36 

28 

52 

55169 

44831 

58115 

41885 

02946 

97054 

8 

32 

32 

53 

55202 

4-1798 

58153 

41847 

02951 

97049 

7 

28 

36 

54 

55235 

41765 

58191 

41809 

02956 

97044 

6 

24 

40 

55 

9.55268 

10.44732 

9.58229 

10.41771 

10.02961 

9.97039 

5 

20 

44 

56 

55301 

44699 

68267 

41733 

02965 

97035 

4 

16 

48 

57 

56334 

44666 

58304 

41696 

02970 

97030 

3 

12 

52 

58 

55367 

44633 

58342 

41658 

02975 

97025 

2 

8 

56 

59 

65400 

44600 

58380 

41020 

02980 

97020 

1 

4 

31 

60 

65433 

44567 

58418 

41582 

029S5 

97015 

0 

30 

M.S. 

7 \ 

M 

110 ' 

Cosine. 

D 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

69° 

M. S. 

4 h 



























220 Logarithms Trigonometric. 


l b 

21° 



Logarithms. 


]58° 

10 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

34 

0 

9.55433 

10.44567 

9.58418 

10.41582 

10.02985 

9.97015 

60 

36 

4 

1 

55466 

44534 

58455 

41545 

02990 

97010 

59 

56 

8 

2 

55499 

44501 

58493 

41507 

02995 

97005 

58 

52 

12 

3 

55532 

44468 

58531 

41469 

02999 

97001 

57 

48 

1G 

4 

55564 

41436 

68569 

41431 

03004 

96996 

56 

44 

20 

5 

9.55597 

10.44403 

9.58606 

10.41394 

10.03009 

9.96991 

55 

40 

24 

6 

65630 

44370 

58644 

41356 

03014 

96986 

54 

36 

28 

7 

65663 

44337 

58681 

41319 

03019 

96981 

53 

32 

32 

8 

55695 

44305 

58719 

41281 

03024 

96976 

52 

28 

36 

9 

55728 

44272 

58757 

41243 

03029 

96971 

51 

24 

40 

10 

9.55761 

10.44239 

9.58794 

10.41206 

10.03034 

9.96966 

50 

20 

4-1 

11 

55793 

44207 

58832 

41168 

08038 

96962 

49 

16 

48 

12 

55826 

44174 

58869 

41131 

03043 

96957 

48 

12 

62 

13 

55858 

44142 

58907 

41093 

03048 

96952 

47 

8 

60 

14 

55891 

44109 

58944 

41056 

03053 

96947 

46 

4 

35 

15 

9.55923 

10.44077 

9.58981 

10.41019 

10.03058 

9.96942 

45 

35 

4 

16 

55956 

44044 

59019 

40981 

03063 

96937 

44 

56 

8 

17 

55988 

44012 

59056 

40944 

03068 

96932 

43 

52 

12 

18 

56021 

43979 

59094 

40906 

03073 

96927 

42 

48 

16 

19 

56053 

43947 

59131 

40869 

03078 

96922 

41 

44 

20 

20 

9.56085 

10.43915 

9.59168 

10.40832 

10.030S3 

9.96917 

40 

40 

24 

21 

56118 

43882 

59205 

40795 

03088 

96912 

39 

36 

28 

22 

56150 

43850 

59243 

40757 

03093 

96907 

38 

32 

32 

23 

56182 

43818 

59280 

4O720 

03097 

96903 

37 

28 

36 

24 

56215 

43785 

59317 

40683 

03102 

96898 

36 

24 

40 

25 

9.56247 

10.43753 

9.59354 

10.40646 

10.03107 

9.96893 

35 

20 

44 

26 

56279 

43721 

59391 

40609 

03112 

96888 

34 

16 

48 

27 

56311 

43689 

59429 

40571 

03117 

96883 

33 

12 

62 

28 

66343 

43657 

59466 

40634 

03122 

96878 

32 

8 

56 

29 

56375 

43625 

59503 

40497. 

03127 

96873 

31 

4 

20 

30 

9.56408 

10.43592 

9.59540 

10.40460 

10.03132 

9.96868 

30 

34 

4 

31 

56440 

48560 

59577 

40423 

03137 

96863 

29 

50 

8 

32 

56472 

43628 

59614 

40386 

03142 

96858 

28 

52 

12 

33 

56504 

43496 

59651 

40349 

03147 

96853 

27 

48 

16 

34 

56536 

43464 

59688 

40312 

03152 

96848 

26 

44 

20' 

35 

9.56568 

10.43432 

9.59725 

10.40275 

10.03157 

9.96843 

25 

40 

24 

36 

56599 

48401 

59762 

40238 

03162 

96838 

24 

36 

28 

37 

56631 

43369 

59799 

40201 

03167 

96833 

23 

32 

32 

38 

56663 

431337 

59835 

40165 

03172 

96828 

22 

28 

36 

39 

56695 

43305 

69872 

40128 

03177 

96823 

21 

24 

40 

40 

9.56727 

10.43273 

9.59909 

10.40091 

10.03182 

9.96818 

20 

20 

44 

41 

56759 

48241 

59946 

40054 

03187 

96813 

19 

16 

48 

42 

56790 

48210 

59983 

40017 

03192 

96808 

18 

12 

52 

43 

56822 

43178 

60019 

39981 

03197 

96803 

17 

8 

56 

44 

56854 

43146 

60056 

39944 

03202 

96798 

16 

4 

27 

45 

9.56886 

10.43114 

9.60093 

10.39907 

10.03207 

9.96793 

15 

33 

4 

46 

56917 

43U83 

60130 

39870 

03212 

96788 

14 

56 

8 

47 

56949 

43051 

60166 

39834 

03217 

96783 

13 

52 

12 

48 

56980 

43020 

60203 

39797 

03222 

96778 

12 

48 

16 

49 

57012 

42988 

60240 

39760 

03228 

96772 

11 

44 

20 

50 

9.57044 

10.42956 

9.60276 

10.39724 

10.03233 

9.96767 

10 

40 

24 

51 

57075 

42925 

60313 

39687 

03238 

96762 

9 

36 

28 

52 

57107 

42893 

60349 

39651 

03243 

96757 

8 

32 

32 

53 

57188 

42862 

60386 

39614 

03248 

96752 

7 

28 

36 

54 

57169 

42831 

60422 

39578 

03253 

96747 

6 

24 

40 

55 

9.57201 

10.42799 

0.60459 

10.39541 

10.03268 

9.96742 

5 

20 

44 

56 

57232 

42768 

60495 

39505 

03263 

96737 

4 

16 

48 

57 

57264 

42736 

60532 

39468 

03268 

96732 

3 

12 

52 

58 

57295 

42705 

60568 

39432 

03273 

96727 

2 

8 

56 

59 

57326 

42674 

60605 

39395 

03278 

96722 

1 

4 

28 

60 

57358 

42642 

60641 

39359 

03283 

96717 

0 

33 

M.S. 

7 11 

M 

111 

Cosine. 

o 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

G8° 

M. S. 

4 h 
















-Logarithms Trigonometric. 


221 


l h 

22° 



Logarithms. 


157° 

J0 h 

M.S 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

as 

0 

9.57358 

10.42642 

9.60641 

10.39359 

10.03283 

9.96717 

60 

3a 

4 

1 

57389 

42611 

60677 

39323 

03289 

96711 

59 

56 

8 

2 

57420 

42580 

60714 

39286 

03294 

96706 

58 

52 

12 

3 

57451 

42549 

60750 

39250 

03299 

96701 

57 

48 

16 

4 

57482 

42518 

60786 

39214 

03304 

96696 

56 

44 

20 

5 

9.57514 

10.42486 

9.60823 

10.39177 

10.03309 

9.96691 

55 

40 

24 

6 

57545 

42455 

60859 

39141 

03314 

96686 

54 

36 

28 

7 

67576 

42424 

60895 

39105 

03319 

96681 

53 

32 

32 

8 

57607 

42393 

„ 60931 

39069 

03324 

96676 

52 

28 

36 

9 

57638 

42362 

60967 

39033 

03330 

96670 

51 

24 

40 

10 

9.57669 

10.42331 

9.61004 

10.38996 

10.03335 

9.96665 

50 

20 

44 

11 

57700 

42300 

61040 

38960 

03340 

96660 

49 

16 

48 

12 

67731 

42269 

61076 

38924 

03345 

96655 

48 

12 

52 

13 

57762 

42238 

61112 

38888 

08350 

96650 

47 

8 

56 

14 

57793 

42207 

61148 

38852 

03355 

96645 

46 

4 

ay 

15 

9.57824 

10.42176 

9.61184 

10.38S16 

10.03360 

9.96640 

45 

31 

4 

16 

67855 

42145 

61220 

38780 

03366 

96634 

U 

56 

8 

17 

57885 

42115 

61256 

38744 

03371 

96629 

43 

52 

12 

18 

57916 

42084 

61292 

38708 

03376 

96624 

42 

48 

16 

19 

57947 

42053 

61328 

38672 

08381 

96619 

41 

44 

20 

20 

9.57978 

10.42022 

9.61364 

10.38636 

10.03386 

9.96614 

40 

40 

24 

21 

58008 

41992 

61400 

38600 

03392 

9660S 

39 

36 

28 

22 

68039 

41961 

61436 

38564 

03397 

96603 

38 

32 

32 

23 

58070 

41930 

61472 

38528 

03402 

96598 

37 

28 

36 

24 

58101 

41899 

61508 

38492 

09407 

9659! 

36 

24 

40 

25 

9.58131 

10.41869 

9.61544 

10.38456 

10.03412 

9.96588 

35 

20 

44 

26 

58162 

41833 

61579 

38421 

03418 

96582 

34 

16 

48 

27 

58192 

41808 

61615 

38385 

08423 

96577 

33 

12 

52 

28 

5S223 

41777 

61651 

38349 

03428 

96572 

32 

8 

56 

29 

58253 

41747 

61687 

38313 

03483 

96567 

31 

4 

30 

30 

9.58284 

10.41716 

9.61722 

10.38278 

10.03438 

9.96562 

30 

30 

4 

31 

58314 

41686 

61758 

38242 

03444 

96556 

29 

56 

8 

32 

58345 

41655 

61794 

38206 

03449 

96551 

28 

52 

12 

33 

58375 

41625 

61830 

38170 

08454 

96546 

27 

48 

16 

34 

58406 

41594 

61865 

38135 

03459 

96541 

26 

44 

20 

35 

9.58436 

10.41564 

9.61901 

10.38099 

10.03465 

9.96535 

25 

40 

24 

36 

58467 

41533 

61936 

88064 

03470 

96530 

24 

36 

28 

37 

58497 

41503 

61972 

38028 

03475 

96525 

23 

32 

32 

38 

58527 

41473 

62008 

37992 

03480 

96520 

'22 

28 

36 

39 

58557 

41443 

62043 

87957 

03486 

96514 

21 

24 

40 

40 

9.58588 

10.41412 

9.62079 

10.37921 

10.03491 

9.96509 

20 

20 

44 

41 

58618 

41382 

62114 

37886 

03496 

96504 

19 

16 

48 

42 

58648 

41352 

62150 

37850 

03502 

96498 

18 

12 

52 

43 

58678 

41322 

62185 

37815 

035O7 

96493 

17 

8 

56 

44 

58709 

41291 

62221 

37779 

03512 

96488 

16 

4 

31 

45 

9.58739 

10.41261 

9.62256 

10.37744 

10.03517 

9.96483 

15 

ao 

4 

46 

58769 

41231 

62292 

37708 

03523 

96477 

14 

56 

8 

47 

58799 

41201 

62327 

37673 

03528 

96472 

13 

52 

12 

48 

58829 

41171 

62362 

37638 

03533 

96467 

12 

48 

16 

49 

58859 

41141 

62398 

37602 

03539 

96461 

11 

44 

20 

50 

9.5S889 

10.41111 

9.62433 

10.87567 

10.03544 

9.96456 

10 

40 

24 

51 

58919 

41081 

62468 

37532 

03519 

96451 

9 

36 

28 

52 

58949 

41051 

62504 

37496 

03555 

96415 

8 

32 

32 

53 

58979 

41021 

62539 

37461 

03560 

96410 

7 

28 

36 

54 

59009 

40991 

62574 

37426 

03565 

96435 

6 

24 

40 

55 

9.59039 

10.40961 

9.62609 

10.87391 

10.03571 

9.96429 

5 

20 

44 

56 

59069 

40931 

62645 

3 < 355 

03576 

96424 

4 

16 

48 

57 

59098 

40902 

62680 

37320 

03581 

96419 

3 

12 

52 

58 

59128 

40872 

62715 

37285 

03587 

96413 

2 

8 

66 

59 

59158 

40842 

62750 

37250 

03592 

96408 

1 

4 

3a 

60 

59188 

40812 

62785 

37215 

03597 

96403 

0 

as 

M.S. 

7 L 

M 

112° 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

67° 

M.S. 

4 h 





















222 Logarithms Trigonometric. 


l h 

23° 



Logarithms. 


156° 

10 b 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

33 

0 

9.59188 

10.40812 

9.02785 

10.37215 

10.03597 

9.96403 

60 

38 

4 

1 

59218 

40782 

02820 

37180 

03003 

96397 

59 

56 

8 

2 

59247 

40753 

62855 

37145 

030(18 

96392 

58 

52 

12 

3 

59277 

40723 

02890 

37110 

03013 

96387 

57 

48 

16 

4 

59307 

40093 

62926 

37074 

03019 

96381 

56 

44 

20 

5 

9.59:336 

10.40064 

9.02961 

10.37039 

10.03024 

9.96376 

55 

40 

24 

0 

59306 

40634 

62996 

37004 

03630 

96370 

54 

36 

28 

7 

59396 

40604 

63031 

30969 

03635 

96365 

53 

32 

82 

8 

59425 

40575 

63066 

36934 , 

03040 

96360 

52 

28 

80 

9 

59455 

40545 

03101 

36899 

03046 

96354 

51 

24 

40 

10 

9.59484 

10.40510 

9.03135 

10.30865 

10.03051 

9.96349 

50 

20 

44 

11 

59514 

40480 

03170 

30830 

03057 

96343 

49 

16 

48 

12 

59543 

40457 

63205 

30795 

03062 

96338 

48 

12 

52 

13 

59573 

40427 

63240 

30700 

03607 

963:13 

47 

8 

50 

14 

59002 

40398 

63275 

30725 

03673 

96327 

46 

4 

33 

15 

9.59632 

10.40308 

9.63310 

10.36090 

10.03678 

9.96322 

45 

37 

4 

16 

59601 

40339 

63345 

36055 

03684 

96316 

44 

50 

8 

17 

59690 

40310 

63379 

36621 

03689 

96311 

43 

52 

12 

18 

59720 

40280 

63414 

30586 

03695 

96305 

42 

48 

10 

19 

59749 

40251 

63449 

36551 

03700 

96300 

41 

44 

20 

20 

9.59778 

10.40222 

9.03484 

10.30516 

10.03706 

9.96294 

40 

40 

24 

21 

59808 

40192 

63519 

30481 

03711 

96289 

39 

36 

28 

22 

59S37 

40103 

63553 

30447 

03716 

96284 

38 

32 

82 

23 

59866 

40134 

63588 

30412 

03722 

96278 

37 

28 

30 

24 

59895 

40105 

63623 

36377 

09727 

96273 

36 

24 

40 

25 

9.59924 

10.40076 

9.03657 

10.30343 

10.03733 

9.96267 

35 

20 

44 

26 

59954 

40010 

03092 

30308 

03738 

96262 

34 

16 

48 

27 

599S3 

40017 

03726 

30274 

08744 

96256 

33 

12 

52 

28 

60012 

39988 

03701 

36239 

03749 

96251 

32 

8 

56 

29 

60011 

39959 

03796 

36204 

03755 

96245 

31 

4 

34 

30 

9.00070 

10.39930 

9.63830 

10.36170 

10.03760 

9.96240 

30 

3<> 

4 

31 

60099 

39901 

63865 

30135 

03766 

96234 

29 

56 

8 

32 

60128 

39872 

63899 

30101 

03771 

96229 

28 

52 

12 

33 

60157 

39843 

63934 

36066 

03777 

96223 

27 

48 

10 

34 

00186 

39814 

63908 

30032 

03782 

96218 

26 

44 

20 

35 

9.60215 

10.39785 

.9.64003 

10.35997 

10.03788 

9.96212 

25 

40 

24 

30 

00244 

39750 

64037 

35963 

03793 

96207 

24 

36 

28 

37 

60273 

39727 

04072 

35928 

03799 

96201 

23 

32 

32 

38 

60302 

39698 

64100 

35894 

03804 

96196 

22 

28 

36 

39 

60331 

39009 

64140 

35860 

03810 

96190 

21 

24 

40 

40 

9.60359 

10.39041 

9.64175 

10.35825 

10.03815 

9.96185 

20 

20 

44 

41 

00388 

39612 

04209 

35791 

03821 

96179 

19 

16 

48 

42 

00417 

39583 

64243 

35757 

03826 

96174 

18 

12 

52 

43 

00440 

39554 

04278 

35722 

03832 

96] 68 

17 

8 

50 

44 

00474 

39520 

04312 

35688 

03838 

96162 

16 

4 

35 

45 

9.00503 

10.39497 

9.64340 

10.35054 

10.03843 

9.96157 

15 

35 

4 

40 

60532 

39468 

04881 

35019 

03849 

96151 

14 

56 

l 

47 

60501 

39439 

04415 

35585 

03854 

96146 

13 

52 

12 

48 

60589 

39411 

04449 

35551 

03860 

96140 

12 

48 

10 

49 

00618 

39382 

64483 

35517 

03865 

96135 

11 

44 

2C 

50 

9.00040 

10.39354 

9.04517 

10.35483 

10.03871 

9.96129 

10 

40 

24 

51 

60675 

39325 

64552 

35448 

03877 

96123 

9 

36 

28 

52 

00704 

39296 

64586 

35414 

03882 

96118 

8 

32 

32 

53 

60732 

39268 

04620 

35380 

03888 

96112 

7 

28 

36 

54 

60761 

39239 

64054 

35346 

03893 

96107 

6 

24 

40 

55 

9.60789 

10.39211 

9.64088 

10.35312 

10.93899 

9.96101 

6 

20 

44 

56 

60818 

39182 

04722 

35278 

03905 

96095 

4 

16 

48 

57 

60846 

39154 

64756 

35244 

03910 

96090 

3 

12 

f2 

58 

60875 

39125 

04790 

35210 

03916 

96084 

2 

8 

50 

59 

60903 

39097 

64824 

35176 

03921 

96079 

1 

4 

3<> 

00 

60931 

39009 

64858 

35142 

03927 

96073 

0 

34 

M.S. 

?h 

CO 

—H 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

66° 

M.S. 

4 h 















Logarithms Trigonometric. 223 


r - 

p 

24° 


Logarithms. 


155° 

10 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M. S. 

30 

0 

9.60931 

10.39069 

9.64858 

10.35142 

10.03927 

9.96073 

60 

24t 

4 

1 

60960 

39040 

64892 

35108 

03933 

96067 

59 

56 

8 

2 

G0988 

39012 

64926 

35074 

03938 

96062 

58 

52 

12 

3 

61016 

38984 

64960 

35040 

039-44 

96056 

57 

48 

1G 

4 

61045 

38955 

G4994 

35006 

03950 

96050 

56 

44 

20 

5 

9.G1073 

10.38927 

9.65028 

10.34972 

10.03955 

9.96045 

55 

40 

24 

6 

61101 

38899 

65062 

34938 

03961 

96039 

54 

36 

28 

7 

61129 

38871 

65096 

34904 

03966 

96034 

53 

32 

32 

8 

61158 

38842 

65130 

34870 

03972 

96028 

52 

28 

3G 

9 

6118G 

38814 

65164 

34836 

03978 

96022 

51 

24 

40 

10 

9.G1214 

10.38786 

9.65197 

10.34803 

10.03983 

9.90017 

50 

20 

44 

11 

61242 

38758 

65231 

34769 

039S9 

96011 

49 

16 

48 

12 

61270 

38730 

65265 

34735 

03995 

96005 

48 

12 

52 

13 

61298 

38702 

65299 

34701 

04000 

96000 

47 

8 

5G 

14 

61326 

38674 

65333 

31667 

04006 

95994 

46 

4 

37 

15 

9.61354 

10.38646 

9.65366 

10.34634 

10.04012 

9.95988 

45 

23 

4 

1G 

G1382 

38618 

65490 

34600 

04018 

95982 

44 

56 

8 

17 

61411 

38589 

65434 

34566 

04023 

95977 

43 

52 

12 

18 

6143S 

38562 

65467 

34533 

04029 

95971 

42 

48 

1G 

19 

G1466 

38534 

65501 

34499 

04035 

95965 

41 

44 

20 

20 

9.61494 

10.38506 

9.65535 

10.34465 

10.04040 

9.95960 

40 

40 

24 

21 

61522 

38478 

65568 

34432 

04046 

95954 

39 

36 

28 

22 

61550 

38450 

65602 

34398 

04052 

95948 

38 

32 

32 

23 

G1578 

38422 

65636 

34364 

04058 

95942 

37 

28 

36 

24 

61 GOG 

38394 

65669 

34:331 

04063 

95937 

36 

24 

40 

25 

9.61G34 

10.38366 

9.65703 

10.34297 

10.04069 

9.95931 

35 

20 

44 

26 

61G62 

38338 

65736 

34264 

04075 

95925 

34 

16 

48 

27 

61G89 

38311 

65770 

34230 

04080 

95920 

33 

12 

62 

28 

G1717 

38283 

65803 

34197 

04086 

95914 

32 

8 

56 

29 

61745 

38255 

65837 

34163 

04092 

95908 

31 

4 

38 

30 

9.G1773 

10.38227 

9.65870 

10.34130 

10.04098 

9.95902 

30 

22 

4 

31 

61800 

38200 

65904 

34096 

04103 

95897 

29 

56 

8 

32 

61828 

38172 

65937 

34063 

04109 

95891 

28 

52 

12 

33 

6185G 

38144 

65971 

34029 

04115 

95885 

27 

48 

16 

34 

G1883 

38117 

66004 

33996 

04121 

95879 

26 

44 

20 

35 

9.61911 

10.38089 

9.66038 

10.33962 

10.04127 

9.95873 

25 

40 

24 

36 

G1939 

38061 

66071 

33929 

04132 

95868 

24 

36 

28 

37 

61966 

38034 

66104 

33896 

04138 

95862 

23 

32 

32 

38 

61994 

38006 

66138 

338G2 

04144 

95856 

22 

28 

36 

39 

62021 

37979 

66171 

33829 

04150 

95850 

21 

24 

40 

40 

9.62049 

10.37951 

9.66204 

10.33796 

10.04156 

9.95844 

20 

20 

44 

41 

62076 

37924 

66238 

33762 

04161 

95839 

19 

16 

48 

42 

62104 

37896 

66271 

33729 

04167 

95833 

18 

12 

52 

43 

G2131 

37869 

66304 

33696 

04173 

95827 

17 

8 

56 

44 

62159 

37841 

66337 

33663 

04179 

95821 

16 

4 

39 

45 

9.62186 

10.378J 4 

9.66371 

10.33629 

10.04185 

9.95815 

15 

21 

4 

46 

62214 

37786 

66404 

33596 

04190 

95810 

14 

56 

8 

47 

62241 

37759 

66437 

33563 

04196 

95804 

13 

52 

12 

48 

62268 

37732 

66470 

33530 

04202 

95798 

12 

48 

1G 

49 

62296 

37704 

66503 

33497 

04208 

95792 

11 

44 

20 

50 

9.62323 

10.37677 

9.66537 

10.33463 

10.04214 

9.95786 

10 

40 

24 

51 

62350 

37650 

66570 

33430 

04220 

95780 

9 

36 

28 

52 

62377 

37623 

66603 

33397 

04225 

95775 

8 

32 

32 

53 

62405 

37595 

66636 

33364 

04231 

95769 

7 

28 

36 

54 

62432 

37568 

66669 

33331 

04237 

95763 

6 

24 

40 

55 

9.62459 

10.37541 

9.66702 

10.33298 

10 04243 

9.95757 

5 

20 

44 

56 

62486 

37514 

66735 

33265 

04249 

95751 

4 

16 

48 

57 

62513 

37487 

66768 

33232 

04255 

95745 

3 

12 

62 

58 

62541 

37459 

66801 

33199 

04261 

95739 

2 

8 

56 

59 

62568 

37432 

66834 

33166 

04267 

95733 

1 

4 

40 

60 

62595 

37405 

66867 

33133 

04272 

95728 

0 

20 

M.S. 

?b 

M 

114 

Cosine. 

0 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

G5° 

M. S 

4 b 

























224 Logarithms Trigonometric. 


l b 

25 c 



Logarithms. 


154° 

10 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

1 Cosine. 

M 

M. S. 

40 

0 

9.62595 

10.37405 

9.66867 

10.33133 

10.04272 

9.95728 

60 

30 

4 

1 

62622 

37378 

66900 

33100 

04278 

95722 

59 

56 

8 

2 

62649 

37351 

G6933 

33007 

04234 

95716 

58 

52 

12 

3 

62676 

37324 

66966 

33U34 

0-1290 

95710 

57 

48 

10 

4 

62703 

37297 

66999 

33001 

0429G 

95704 

56 

44 

20 

5 

9.62730 

10.37270 

9.67032 

10.32968 

10.04302 

9.95698 

55 

40 

24 

6 

62757 

37243 

67065 

33935 

04308 

95692 

54 

36 

28 

7 

62784 

37216 

67098 

32902 

04314 

95686 

53 

32 

32 

8 

6281L 

37189 

67131 

32869 

04320 

95680 

52 

28 

36 

9 

62838 

37162 

67163 

32837 

04326 

95674 

51 

24 

40 

10 

9.62865 

10.37135 

9.67196 

10.32801 

10.04332 

9.95668 

50 

20 

44 

11 

62892 

37108 

67229 

32771 

04337 

95663 

49 

16 

48 

12 

62918 

37082 

67262 

32738 

04343 

95657 

48 

12 

52 

13 

62945 

37055 

67295 

32705 

04349 

95651 

47 

8 

56 

14 

62972 

37028 

67327 

32673 

047155 

95645 

46 

4 

41 

15 

9.62999 

10.37U01 

9.67360 

10.32610 

10.01361 

9.95639 

45 

19 

4 

16 

63026 

36974 

67393 

32607 

04367 

95633 

44 

56 

8 

17 

63052 

36948 

67426 

32574 

04373 

95627 

43 

52 

12 

18 

63079 

36921 

67458 

32542 

04379 

95621 

42 

48 

16 

19 

63106 

36894 

67491 

32509 

04385 

95615 

41 

44 

20 

20 

9.63133 

10.36867 

9.67524 

10.32476 

10.04391 

9.95609 

40 

40 

24 

21 

63159 

36841 

67556 

22444 

04397 

95003 

39 

36 

28 

22 

63186 

36814 

67689 

32411 

04403 

95597 

38 

32 

32 

23 

63213 

36787 

G7622 

32378 

04409 

95591 

37 

28 

36 

24 

63239 

36761 

67654 

32346 

04115 

95585 

36 

24 

40 

25 

9.63266 

10.36734 

9.67687 

10.32313 

10.04421 

9.95579 

35 

20 

44 

26 

63292 

36708 

67719 

32281 

01427 

95573 

34 

16 

48 

27 

63319 

36681 

67752 

32248 

04433 

95567 

33 

12 

52 

28 

63345 

36655 

67785 

32215 

04139 

95561 

32 

8 

56 

29 

63372 

36628 

67817 

32183 

04145 

95555 

31 

4 

42 

30 

9.63398 

10.36602 

9.67850 

10.32150 

10.04451 

9.95549 

30 

18 

4 

31 

63425 

ikk) 15 

67882 

32118 

04457 

95543 

29 

56 

8 

32 

63451 

36549 

67915 

32085 

04463 

95537 

28 

52 

12 

33 

63478 

36522 

67947 

32053 

04469 

95531 

27 

40 

16 

34 

63504 

36496 

67980 

32020 

04475 

95525 

26 

44 

20 

35 

9.63531 

10.36469 

9.68012 

10.31988 

10.04481 

9.95519 

25 

40 

24 

36 

63557 

36443 

68044 

31956 

044'->7 

95513 

24 

36 

28 

37 

63583 

36117 

68U77 

31923 

04493 

95507 

23 

32 

32 

38 

63610 

36390 

68109 

31891 

04500 

95500 

22 

28 

36 

39 

63636 

36364 

68142 

31858 

04506 

95494 

21 

24 

40 

40 

9.63662 

10.36338 

9.68174 

1031826 

10.01512 

9.9548S 

20 

20 

44 

41 

63689 

36311 

68206 

31794 

04518 

95482 

19 

16 

48 

42 

63715 

36285 

68239 

31761 

04524 

95476 

18 

12 

62 

43 

63741 

36259 

68271 

31729 

04530 

95470 

17 

8 

56 

44 

63767 

36233 

68303 

31697 

045716 

95464 

16 

4 

43 

45 

9.63794 

10.36206 

9.68336 

10.31664 

10.04542 

9.95458 

15 

11 

4 

46 

63820 

36180 

68368 

31632 

04548 

95452 

14 

56 

8 

47 

63846 

36154 

68400 

31600 

04554 

95446 

13 

52 

12 

48 

63872 

36128 

63432 

31568 

(>4560 

954(0 

12 

48 

16 

49 

63898 

36102 

68465 

31535 

04566 

95434 

11 

44 

20 

60 

9.63924 

10.36076 

9.68497 

10.31503 

10.04573 

9.95427 

10 

40 

24 

61 

63950 

36050 

68529 

31471 

04579 

95421 

9 

36 

28 

52 

G397G 

36024 

68561 

31439 

04585 

95415 

8 

32 

32 

53 

64002 

35998 

68593 

31407 

04591 

95409 

7 

28 

36 

54 

64U2S 

35972 

68626 

31374 

04597 

95103 

6 

24 

40 

55 

9.64054 

10.35946 

9.68658 

10.31342 

10.04603 

9.95397 

5 

20 

44 

56 

C4080 

35920 

68690 

31310 

04609 

957291 

4 

16 

48 

67 

64106 

35894 

68722 

31278 

04616 

95384 

3 

12 

62 

58 

64132 

35868 

68754 

31246 

04622 

95378 

2 

8 

56 

59 

64158 

35842 

68786 

31214 

04628 

95372 

1 

4 

44 

60 

64184 

35816 

68818 

31182 

04634 

95366 

0 

1.6 

M.S. 

7 h 

M 

115 

Cosine. 

0 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

64° 

-M. 8. 

4“ 





















Logarithms Trigonometric. 


225 


l h 

26° 



• 

Logarithms. 


153° 

10 h 

M.S 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

44 

0 

9.64184 

10.35816 

9.68818 

10.31182 

10.046:34 

9.95366 

60 

1C> 

4 

1 

64210 

35790 

68850 

31150 

04640 

95360 

59 

56 

8 

2 

64236 

35764 

68882 

31118 

04646 

95354 

58 

52 

12 

3 

64262 

35738 

68914 

31086 

04652 

95348 

57 

48 

1(5 

4 

64288 

35712 

68946 

31054 

04659 

95341 

56 

44 

20 

5 

9.64313 

10.35687 

9.68978 

10.31022 

10.04665 

9.95335 

55 

40 

24 

6 

6433,9 

35661 

69010 

30990 

04671 

95329 

54 

36 

28 

7 

64365 

35635 

69042 

30958 

04677 

95323 

53 

82 

32 

8 

64391 

35609 

69074 

30926 

04683 

95317 

52 

28 

3G 

9 

64417 

35583 

69106 

30894 

04690 

95310 

51 

24 

40 

10 

9.64442 

10.35558 

9.69138 

10.30862 

10.04696 

9.95304 

50 

20 

44 

11 

64468 

35532 

69170 

30830 

04702 

95298 

49 

16 

48 

12 

64494 

35506 

69202 

30798 

04708 

95292 

48 

12 

52 

13 

64519 

35481 

69234 

30766 

04714 

95286 

47 

8 

60 

14 

64545 

35455 

69266 

30734 

04721 

95279 

46 

4 

45 

15 

9.64571 

10.35429 

9.69298 

10.30702 

10.04727 

9.95273 

45 

15 

4 

16 

64596 

35404 

69329 

30671 

04733 

95267 

44 

56 

8 

17 

64622 

35378 

69361 

30639 

04739 

95261 

43 

52 

12 

18 

64647 

35353 

69393 

30607 

04746 

95254 

42 

48 

16 

19 

64073 

35327 

69425 

30575 

04752 

95248 

41 

44 

20 

20 

9.64698 

10.35302 

9.69457 

10.30543 

10.04758 

9.95242 

40 

40 

24 

21 

64724 

35276 

69488 

30512 

04764 

95236 

39 

36 

28 

22 

64749 

35251 

69520 

30480 

04771 

95229 

38 

32 

32 

23 

64775 

35225 

69552 

30448 

04777 

95223 

37 

28 

36 

24 

64800 

35200 

69584 

30416 

04783 

95217 

36 

24 

40 

25 

9.64826 

10.35174 

9.69615 

10.30385 

10.04789 

9.95211 

35 

20 

44 

26 

64851 

35149 

69647 

30353 

04796 

95204 

34 

16 

48 

27 

64877 

35123 

69679 

30321 

04802 

95198 

33 

12 

62 

28 

64902 

35098 

69710 

30290 

04808 

95192 

32 

8 

56 

29 

64927 

35073 

69742 

30258 

04815 

95185 

31 

4 

40 

30 

9.64963 

10.35047 

9.69774 

10.30226 

10.04821 

9.95179 

30 

14 

4 

31 

64978 

35022 

69805 

30195 

04827 

95173 

29 

56 

8 

32 

65003 

34997 

69837 

30163 

04833 

95167 

28 

52 

12 

33 

65029 

3-1971 

69868 

30132 

04840 

95160 

27 

48 

10 

34 

65054 

34946 

69900 

30100 

04846 

95154 

26 

44 

20 

35 

9.65079 

10.34921 

9.69932 

10.30068 

10.04852 

9.95148 

25 

40 

24 

36 

65104 

34896 

69963 

30037 

04859 

95141 

24 

36 

28 

37 

65130 

34870 

69995 

30005 

04865 

95135 

23 

32 

32 

38 

65155 

34845 

70026 

29974 

04871 

95129 

22 

28 

36 

39 

65180 

34820 

70058 

29942 

04878 

95122 

21 

24 

40 

40 

9.65205 

10.34795 

9.70089 

10.29911 

10.04884 

9.95116 

20 

20 

44 

41 

65230 

34770 

70121 

29879 

04890 

95110 

19 

16 

48 

42 

65255 

34745 

70152 

29848 

04897 

95103 

18 

12 

52 

43 

65281 

34719 

70184 

29816 

04903 

95097 

17 

8 

66 

44 

65306 

34694 

70215 

29785 

04910 

95090 

16 

4 

41 

45 

9.65331 

10.34669 

9.70247 

10.29753 

10.04916 

9.95084 

15 

13 

4 

46 

65356 

34644 

70278 

29722 

04922 

95078 

14 

56 

8 

47 

653H1 

34619 

70309 

29691 

04929 

95071 

13 

52 

12 

48 

65400 

34594 

70341 

29659 

04935 

95065 

12 

48 

16 

49 

65431 

34569 

70372 

29628 

04941 

95059 

11 

44 

20 

50 

9.65456 

10.34544 

9.70404 

10.29596 

10.04948 

9.95052 

10 

40 

24 

51 

65481 

34519 

70435 

29565 

04954 

95046 

9 

36 

28 

52 

65506 

34494 

70466 

29534 

04961 

95039 

8 

32 

32 

53 

65531 

34469 

70498 

29502 

04967 

95033 

l 

28 

36 

54 

65556 

34444 

70529 

29471 

04973 

95027 

6 

24 

40 

55 

9.65580 

10.34420 

9.70560 

10.29440 

10.04980 

9.95020 

5 

20 

44 

56 

65605 

34395 

70592 

29408 

04986 

95014 

4 

16 

48 

57 

65630 

34370 

70623 

29377 

04993 

95007 

3 

12 

52 

58 

65655 

34345 

70654 

29346 

04999 

95001 

2 

8 

56 

59 

65680 

34320 

70685 

29315 

05005 

94995 

1 

4 

48 

60 

65705 

34295 

70717 

29283 

05012 

949S8 

0 

13 

M.S. 

M 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

M.S. 

7 h 

116° 







63° 

4 h 


15 



















226 


Logarithms Trigonometric. 


l h 

27° 



Logarithms. 


152° 

10* 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

48 

0 

9.65705 

10.34295 

9.70717 

10.29283 

10.05012 

9.94988 

60 

n 

4 

1 

65729 

34271 

70748 

29252 

05018 

94982 

59 

56 

8 

2 

65754 

34246 

70779 

29221 

05025 

94975 

58 

52 

12 

3 

65779 

34221 

70810 

29190 

05031 

94969 

57 

48 

16 

4 

65804 

34196 

70841 

29159 

05038 

94962 

56 

44 

20 

5 

9.65828 

10.34172 

9.70S78 

10.29127 

10.05044 

9.94956 

55 

40 

24 

6 

65853 

34147 

70904 

24096 

05051 

94949 

54 

36 

28 

7 

65878 

34122 

70935 

29065 

05057 

94943 

53 

32 

32 

8 

65902 

34098 

70966 

29034 

05064 

94936 

52 

28 

36 

9 

65927 

34073 

70997 

29003 

05070 

94930 

51 

24 

4o 

10 

9.65952 

10.34048 

9.71028 

10.28972 

10.05077 

9.94923 

50 

20 

44 

11 

65976 

34024 

71059 

28941 

05083 

94917 

49 

16 

48 

12 

66001 

33999 

71090 

28910 

05089 

94911 

48 

12 

52 

13 

66025 

33975 

71121 

28879 

05096 

94904 

47 

8 

56 

14 

60050 

33950 

71153 

28847 

05102 

94898 

46 

4 

49 

15 

9.66075 

10.33925 

9.71184 

10.28816 

10.05109 

9.94891 

45 

11 

4 

16 

66099 

33901 

71215 

28785 

05115 

9488;5 

4-1 

56 

8 

17 

66124 

33876 

71246 

28754 

05122 

94878 

43 

52 

12 

18 

66148 

33852 

71277 

28723 

05129 

94871 

42 

48 

16 

19 

66173 

33S27 

71308 

28692 

05185 

94865 

41 

44 

20 

20 

9.66197 

10.33803 

9.71339 

10.28661 

10.05142 

9.94853 

40 

40 

24 

21 

66221 

33779 

71370 

2S630 

05148 

94852 

39 

36 

28 

22 

66246 

33754 

71401 

28593 

05155 

94845 

38 

32 

32 

23 

66270 

33730 

71431 

28569 

05161 

94839 

37 

28 

36 

24 

66295 

33705 

71462 

28538 

05168 

94832 

36 

24 

40 

25 

9.66319 

10.33681 

9.71493 

10.28507 

10.05174 

9.94826 

35 

20 

44 

26 

66343 

38647 

71524 

28476 

05181 

94819 

34- 

16 

48 

27 

66368 

33632 

71555 

28445 

05187 

91813 

33 

12 

52 

28 

66392 

33608 

71586 

28414 

05194 

94806 

32 

8 

56 

29 

66416 

33584 

71617 

2S383 

05201 

94799 

31 

4 

50 

30 

9.66441 

10.33559 

9.71648 

10.28352 

10.05207 

9.94793 

30 

10 

4 

31 

66465 

33535 

71679 

28321 

05214 

94786 

29 

56 

8 

32 

66489 

33511 

71709 

28291 

05220 

94780 

28 

52 

12 

33 

66513 

38487 

71740 

28260 

05227 

94773 

27 

48 

16 

34 

66537 

33468 

71771 

28229 

05233 

94767 

26 

41 

20 

35 

9336562 

10.33438 

9.71802 

10.28198 

10.05240 

9.94760 

25 

40 

24 

30 

66586 

33414 

71833 

28167 

05247 

94753 

•■it 

36 

28 

37 

66610 

33390 

71863 

28137 

05253 

94747 

23 

32 

32 

38 

66634 

33866 

71894 

2810) 

05260 

94740 

22 

28 

36 

39 

66658 

33342 

71925 

2S075 

05266 

94734 

21 

24 

40 

40 

9.66682 

10.83318 

9.71955 

10.28045 

10.05273 

9.94727 

20 

20 

44 

4L 

66706 

33294 

71986 

28014 

05280 

94720 

19 

16 

48 

42 

66731 

33264 

72017 

27983 

05286 

94714 

18 

12 

52 

43 

6(37 55 

33245 

72048 

27952 

05293 

94707 

17 

8 

56 

44 

66779 

33221 

72078 

27922 

05300 

94700 

16 

4 

51 

45 

9.66803 

10.33197 

9.72109 

10,27891 

10.05306 

9.94694 

15 

9 

4 

46 

66827 

33173 

72140 

27860 

05313 

946S7 

14 

56 

8 

47 

66851 

38149 

72170 

27830 

05320 

94680 

13 

52 

12 

48 

66875 

33125 

72201 

27799 

05326 

94674 

12 

48 

16 

49 

66899 

33101 

72231 

27769 

05333 

94667 

11 

44 

20 

50 

9.66922 

10.33078 

9.72262 

10.27738 

10.05340 

9.94660 

10 

40 

24 

51 

66946 

33054 

72293 

27707 

05346 

94654 

9 

36 

28 

52 

66970 

33030 

72323 

27677 

05353 

94647 

8 

32 

32 

53 

66994 

33006 

72354 

27646 

05360 

94640 

7 

28 

30 

54 

67018 

32982 

72384 

27616 

05366 

94634 

6 

24 

40 

55 

9.67042 

10.32958 

9.72415 

10.27585 

10.05373 

9.94627 

5 

20 

44 

56 

67066 

32934 

72445 

27555 

05380 

94620 

4 

16 

48 

57 

67090 

32910 

72476 

27524 

05386 

91614 

3 

12 

52 

58 

67113 

32887 

72506 

27494 

05393 

94607 

2 

8 

56 

59 

67137 

32863 

72537 

27463 

05400 

94600 

1 

4 

5 :i 

60 

67161 

32889 

72567 

27433 

05407 

94593 

0 

8 

M.S. 

7 h 

M 

117' 

Cosiuc. 

3 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

62° 

M.S. 

4 h 

























Logarithms Trigonometric. 


227 


l b 


28 c 


Logarithms. 


151 


Sine. 


10 b 


M.S. 

M 

53 

0 

4 

1 

8 

2 

12 

3 

16 

4 

20 

5 

24 

6 

28 

7 

32 

8 

36 

9 

40 

10 

44 

11 

48 

12 

52 

13 

56 

14 

53 

15 

4 

16 

8 

17 

12 

18 

10 

19 

20 

20 

24 

21 

28 

22 

32 

23 

36 

24 

40 

25 

44 

26 

48 

27 

52 

28 

50 

29 

54 

30 

4 

31 

8 

32 

12 

33 

16 

34 

20 

35 

24 

36 

28 

37 

32 

38 

36 

39 

40 

40 

44 

41 

48 

42 

52 

43 

5C 

44 

55 

45 

4 

46 

8 

47 

12 

48 

16 

49 

20 

50 

24 

51 

28 

52 

32 

53 

36 

54 

40 

55 

44 

56 

48 

57 

52 

58 

56 

59 

50 

00 

M.S. 

M 

7 b 

118 


9.07161 
07185 
07208 
67232 
07250 
9.07280 
07303 
07327 
07350 
07374 
9.07398 
67421 
07445 
' 07468 
07492 
9.07515 
07539 
07502 
07586 
07009 
9.07033 
07050 
07080 
07703 
07726 
9.07750 
07773 
07790 
07820 
07843 
9.07866 
67890 
07913 
66936 
07959 
9.G79S2 
08006 
68029 
08052 
68075 
9.08098 
68121 
08144 
6S167 
6S190 
9.08213 
00287 
08260 
68283 
08305 
9.08328 
68361 
68374 
68397 
08420 
9.68443 
68400 
68489 
68512 
68534 
68557 

Cosiue. 


Cosecant. 

10.32839 

32815 

32792 

32768 

32744 

10.32720 

32097 

32073 

82050 

32626 

10.326112 

32579 

32555 

32532 

32508 

10.32485 

32461 

32438 

32414 

32391 

10.32807 

32344 

32320 

32297 

32274 

10.32250 

32227 

32204 

32180 

32157 

10.32134 

32110 

32087 

32004 

32041 

10.32018 

31994 

31971 

31948 

31925 

10.31902 

31879 

31856 

31833 

31810 

10.31787 

31703 

31740 

31717 

31095 

10.31072 

31049 

31020 

31003 

31580 

10.31557 

31534 

31511 

31488 

31466 

31443 

Socant. 


Tangent. 

9.72567 

72598 

72628 

72659 

72689 

9.72720 

72750 

72780 

72811 

72841 

9.72872 

72902 

72932 

72963 

72993 

9.73023 

73054 

73084 

73114 

73144 

9.73175 

78205 

73235 

73265 

73295 

9.73326 

73356 

73386 

73416 

73416 

9.73476 

73507 

73537 

73567 

73597 

9.73627 

73057 

73687 

73717 

73747 

9.73777 

73807 

73837 

73867 

73897 

9.73927 

73957 

73987 

74017 

74047 

9.74077 

74107 

74137 

74166 

74190 

9.74226 

74256 

71286 

74316 

74345 

74375 


Cotangent. 
10.27433 
27402 
27372 
27341 
27311 
10.27280 
27250 
27220 
27189 
27159 
10.27128 
27098 
27008 
27037 
27007 
10.26977 
20946 
20916 
20886 
20856 
10.26825 
20795 
' 20765 
26735 
26705 
10.26674 
20644 
26614 
26584 
26554 
10.26524 
20493 
26463 
26433 
26403 
10.20373 
20343 
26313 
26283 
26253 
10.26223 
20193 
20163 
201S3 
26103 
10.26073 
26043 
26013 
25983 
25953 
10.25923 
25893 

25803 
25834 

25804 
10.25774 

25744 

25714 

25084 

25655 

25025 

Tangent. 


Secant. 

10.05407 
05413 
05420 
05427 
05433 
10.05440 
05447 
05454 
05400 
05407 
10.05474 
05481 
05487 
05494 
05501 
10.05508 
05515 
05521 
05528 
05535 
10.05542 
05549 
05555 
05562 
05569 
10.05576 
05583 
05590 
05590 
05003 
10.05010 
05017 
05624 
05631 
05638 
10.05045 
05051 
05058 
05605 
05072 
10.05079 
05086 
05693 
05700 
05707 
10.05714 
05721 
05727 
05734 
05741 
10.0574S 
05755 
05702 
05769 
03776 
10 05783 
05790 
05797 
05804 
05811 
05818 
Cosecant. 


Cosine. 

9.94593 

94587 

94580 

94573 

94567 

9.94560 

94553 

94546 

94540 

94533 

9.94526 

94519 

94513 

94500 

94499 

9.94492 

94485 

94479 

94472 

94405 

9.94458 

94451 

94445 

94438 

94431 

9.94424 

94417 

94410 

94404 

94397 

9.94390 

94383 

94376 

94369 

94362 

9.94355 

94349 

94342 

94335 

94328 

9.94321 

94314 

94307 

94300 

94293 

9.94280 

94279 

94273 

94260 

94259 

9.94252 

94245 

94238 

94231 

94224 

9.94217 

94210 

94203 

94196 

94189 

94182 

Sine. 


M 

M. S. 

60 

8 

59 

56 

58 

52 

57 

48 

56 

-44 

55 

40 

54 

36 

53 

32 

52 

28 

51 

24 

50 

20 

49 

16 

48 

12 

47 

8 

46 

4 

45 

7 

44 

56 

43 

52 

42 

48 

41 

44 

40 

40 

39 

36 

38 

32 

37 

28 

36 

24 

35 

20 

34 

16 

33 

12 

32 

8 

31 

4 

30 

6 

29 

50 

28 

52 

27 

48 

26 

44 

25 

40 

24 

36 

23 

32 

22 

28 

21 

24 

20 

20 

19 

16 

18 

12 

17 

8 

10 

4 

15 

5 

14 

56 

13 

52 

12 

48 

11 

44 

10 

40 

9 

36 

8 

32 

7 

28 

6 

24 

5 

20 

4 

10 

3 

12 

2 

8 

1 

4 

0 

4: 

M 

M. S. 

Gl° 

4 b 


•Cotangent 

























22S 


Logarithms Trigonometric. 


l b 

29° 



Logarithms. 


150° 

10 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M. S. 

50 

0 

9.68567 

10.31443 

9.74375 

10.25625 

10.05818 

9.94182 

60 

4r 

4 

1 

68580 

31420 

74405 

25595 

05825 

94175 

59 

56 

8 

2 

68603 

31397 

74435 

25565 

05832 

94168 

58 

52 

12 

3 

68625 

31375 

74465 

25535 

05839 

94161 

57 

48 

16 

4 

68648 

31352 

74494 

25506 

05846 

94154 

56 

44 

20 

5 

9.68671 

10.31329 

9.74524 

10.25476 

10.05S53 

9.94147 

55 

40 

24 

6 

6S694 

31306 

74554 

25446 

05860 

94140 

54 

3G 

28 

7 

68716 

31284 

74583 

25417 

05867 

94133 

53 

32 

32 

8 

68739 

3126 L 

74613 

25387 

05874 

94126 

52 

28 

36 

9 

68762 

31238 

74643 

25357 

05881 

94119 

51 

24 

40 

10 

9.68784 

10.31216 

9.74673 

10.25327 

10.05888 

9.94112 

50 

20 

44 

1L 

68807 

31193 

74702 

25298 

05895 

94105 

49 

16 

48 

12 

68829 

31171 

74732 

25268 

05902 

94098 

48 

12 

52 

13 

6S852 

31148 

74762 

25238 

05910 

94090 

47 

8 

56 

14 

68875 

31125 

74791 

25209 

05917 

94083 

46 

4 

57 

15 

9.68S97 

10.31103 

9.74821 

10.25179 

10.05924 

9.94076 

45 

3 

4 

16 

68920 

31080 

74851 

25149 

05931 

94069 

44 

56 

8 

17 

68942 

31058 

74880 

25120 

05938 

94062 

43 

52 

12 

18 

68965 

31035 

74910 

25090 

05945 

94055 

42 

48 

16 

19 

68987 

31013 

74939 

25*'61 

05952 

94048 

41 

44 

20 

20 

9.69010 

10.30990 

9.74969 

10.25031 

10.05959 

9.94041 

40 

40 

24 

21 

69032 

30968 

74998 

25002 

05966 

94034 

39 

36 

28 

22 

69055 

30945 

75028 

24972 

05973 

94027 

38 

32 

32 

23 

69077 

30923 

75058 

24942 

05980 

94020 

37 

28 

36 

24 

69100 

30900 

75087 

24913 

05988 

94012 

36 

24 

40 

25 

9.69122 

10.30878 

9.75117 

10.24883 

10.05995 

9.94005 

35 

20 

44 

26 

69144 

30856 

75146 

24854 

06002 

93998 

34 

16 

48 

27 

69167 

30833 

75176 

24824 

06009 

93991 

33 

12 

52 

28 

691S9 

30811 

75205 

24795 

06016 

93984 

32 

8 

56 

29 

69212 

30788 

75235 

24765 

06023 

93977 

31 

4 

58 

30 

9.69234 

10.30766 

9.75264 

10.24736 

10.06030 

9.93970 

30 

3 

4 

31 

69256 

30744 

75294 

24706 

06037 

93963 

29 

56 

8 

32 

69279 

30721 

75323 

24677 

06045 

93955 

28 

52 

12 

33 

69301 

30699 

75353 

24647 

06052 

93948 

27 

48 

16 

34 

69323 

30677 

75382 

24618 

06059 

93941 

26 

44 

20 

35 

9.69345 

10.30655 

9.75411 

10.24589 

10.06O66 

9.93934 

25 

40 

24 

36 

69368 

30632 

75441 

24559 

06073 

93927 

24 

36 

28 

37 

69390 

30610 

75470 

24530 

06080 

93920 

23 

32 

32 

38 

69412 

30588 

75500 

24500 

06088' 

93912 

22 

28 

36 

39 

69434 

30566 

75529 

24471 

06095 

93905 

21 

24 

40 

40 

9.69456 

10.30544 

9.75558 

10.24442 

10.06102 

9.93898 

20 

20 

44 

41 

69479 

30521 

75588 

24412 

06109 

93891 

19 

16 

48 

42 

69501 

30499 

75617 

24383 

06116 

93884 

18 

12 

62 

43 

69523 

30177 

75647 

24353 

06124 

93876 

17 

8 

66 

44 

69545 

30455 

75676 

24324 

06131 

93869 

16 

4 

51) 

45 

9.69567 

10.30133 

9.75705 

10.24295 

10.06138 

9.93862 

15 

1 

4 

46 

69589 

30411 

75735 

24265 

06145 

93855 

14 

56 

8 

47 

69611 

30389 

75764 

24236 

06153 

93847 

13 

52 

12 

48 

69633 

30367 

75793 

24207 

06160 

93840 

12 

48 

16 

49 

69655 

30345 

75822 

24178 

06167 

93833 

11 

44 

20 

50 

9.69677 

10.30323 

9.75852 

10.24148 

10.06174 

9.93826 

10 

40 

24 

51 

69699 

30301 

75881 

24119 

06181 

93819 

9 

36 

2S 

52 

69721 

30279 

75910 

24090 

06189 

93811 

8 

32 

32 

53 

69743 

30257 

75939 

21061 

06196 

93804 

7 

28 

36 

54 

69765 

30235 

75969 

24031 

062O3 

93797 

6 

24 

40 

65 

9.69787 

10.30213 

9.75998 

10.24002 

10.06211 

9.93789 

5 

20 

44 

56 

69809 

30191 

76027 

23973 

06218 

93782 

4 

16 

48 

67 

69831 

30169 

76066 

23944 

06225 

93775 

3 

12 

52 

58 

69853 

30147 

76086 

23914 

06232 

93768 

2 

8 

56 

59 

6987 5 

30125 

76115 

23885 

06240 

93760 

1 

4 

00 

60 

69897 

30103 

76144 

23866 

06247 

93753 

0 

O 

M.S. 

M 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

M. R. 

7 h 

119 

o 






60° 

4 b 
















Logarithms Trigonometric. 


229 


2 h 

30° 



Logarithms. 


149° 

9 h 

M.S 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

0 

0 

9.69897 

10.30103 

9.76144 

10.23856 

10.00247 

9.93753 

60 

00 

4 

1 

69919 

30081 

76173 

23827 

06254 

93746 

59 

56 

8 

2 

69941 

30059 

76202 

23798 

06262 

93738 

58 

52 

12 

3 

69963 

30037 

76231 

23769 

06269 

93731 

57 

48 

16 

4 

69984 

30016 

76261 

23739 

06276 

93724 

56 

44 

20 

5 

9.70006 

10.29994 

9.76290 

10.23710 

10.06283 

9.93717 

55 

40 

24 

6 

70028 

29972 

76319 

23681 

06291 

93709 

54 

30 

28 

7 

70050 

29950 

76348 

23652 

06298 

93702 

53 

32 

02 

S 

70072 

29928 

76377 

23623 

06305 

93695 

52 

28 

06 

9 

70093 

29907 

76406 

23594 

06313 

93687 

51 

24 

40 

10 

9.70115 

10.29885 

9.76435 

10.23565 

10.06320 

9.93680 

50 

20 

14 

]1 

70137 

29863 

76464 

23536 

00327 

93673 

49 

16 

48 

12 

70159 

29841 

76493 

23507 

06335 

93665 

48 

12 

52 

13 

70180 

29820 

76522 

23478 

06342 

93658 

47 

8 

50 

14 

70202 

29798 

76551 

23449 

06350 

93650 

40 

4 

1 

15 

9.70224 

10.29776 

9.70580 

10.23420 

10.06357 

9.93643 

45 

59 

4 

16 

70245 

29755 

76009 

23391 

06364 

93636 

44 

56 

8 

17 

70267 

29733 

76639 

23361 

06372 

93628 

43 

52 

12 

18 

70288 

29712 

70668 

23332 

06379 

93621 

42 

48 

16 

19 

70310 

29690 

76697 

23303 

06386 

93614 

41 

44 

20 

20 

9.70332 

10.29668 

9.76725 

10.23275 

10.06394 

9.93606 

40 

49 

24 

21 

70353 

22647 

76754 

23246 

06401 

93599 

39 

36 

28 

22 

70375 

29625 

76783 

23217 

06409 

93591 

38 

32 

32 

23 

70396 

29604 

76812 

23188 

06416 

93584 

37 

28 

36 

24 

70418 

295S2 

76841 

23159 

06423 

93577 

36 

24 

40 

25 

9.70439 

10.29561 

9.70S70 

10.23130 

10.06431 

9.93569 

35 

20 

44 

26 

70461 

29539 

76899 

23101 

06438 

93562 

34 

16 

48 

27 

70482 

29518 

76928 

23072 

06446 

93554 

33 

12 

52 

28 

70504 

29496 

76957 

23043 

00453 

93547 

32 

8 

56 

29 

70525 

29475 

76986 

23014 

00461 

93539 

31 

4 

A 

30 

9.70647 

10.29453 

9.77015 

10.22985 

10.06468 

9.93532 

30 

5S 

4 

31 

70568 

29432 

77044 

22956 

06475 

93525 

29 

56 

8 

32 

70590 

29410 

77073 

22927 

06483 

93517 

28 

52 

12 

33 

70611 

29389 

77101 

22899 

06490 

93510 

27 

48 

16 

34 

70633 

29367 

77130 

22870 

00498 

93502 

26 

44 

20 

35 

9.70654 

10.29346 

9.77159 

10.22841 

10.06505 

9.93495 

25 

40 

24 

36 

70675 

29325 

77188 

22812 

06513 

93487 

24 

36 

28 

37 

70697 

29303 

77217 

22783 

06520 

93480 

23 

32 

32 

38 

70718 

29282 

77246 

22754 

06528 

93472 

22 

28 

36 

39 

70739 

29261 

77274 

22726 

06535 

93165 

21 

24 

40 

40 

9.70761 

10.29239 

9.77303 

10.22697 

10.06543 

9.93457 

20 

20 

44 

41 

70782 

29218 

77332 

22668 

06550 

93450 

19 

16 

48 

42 

70803 

29197 

77361 

22639 

06558 

93442 

18 

12 

52 

43 

70824 

29176 

77390 

22610 

06565 

93435 

17 

8 

56 

44 

70846 

29154 

77418 

22582 

06573 

93427 

16 

4 

3 

45 

9.70867 

10.29133 

9.77447 

10.22553 

10.06580 

9.9:4420 

15 

57 

4 

46 

70888 

29112 

77476 

22524 

06588 

93412 

14 

56 

8 

47 

70909 

29091 

77505 

22495 

06595 

93405 

13 

52 

12 

48 

70931 

29069 

77533 

22467 

06603 

93397 

12 

48 

16 

49 

70952 

29048 

77562 

22438 

06610 

93390 

11 

44 

20 

50 

9.70973 

10.29027 

9.77591 

10.22409 

10.00618 

9.93382 

10 

40 

24 

51 

70994 

29006 

77619 

22381 

06625 

93375 

9 

36 

28 

52 

71015 

28985 

77648 

22352 

06633 

93367 

8 

3Z 

32 

53 

71036 

28964 

77677 

22323 

06640 

93360 

7 

28 

36 

51 

71058 

28942 

77706 

22294 

06648 

93352 

6 

24 

40 

55 

9.71079 

10.28921 

9.77734 

10.22266 

10.06656 

9.93344 

5 

20 

44 

56 

71100 

28900 

77763 

22237 

06663 

93337 

4 

16 

48 

57 

71121 

28879 

77791 

22209 

06671 

93329 

3 

12 

52 

58 

71142 

28858 

77820 

22180 

06678 

93322 

2 

8 

56 

59 

71163 

28837 

77849 

22151 

06686 

93314 

1 

4 

4: 

CO 

71184 

28816 

77877 

22123 

06693 

93307 

0 

50 

M.S. 

8 b 

-O 

O 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

59° 

M.S. 

3 b 




















230 Logarithms Trigonometric. 


Oh 

A* 

O 

t—H 

CO 



Lo^nriUimg. 


148° 

9 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

4 

0 

9.71184 

10.28810 

9.77877 

10.22123 

10.00093 

9.93307 

60 

5G 

4 

1 

71205 

287*95 

77906 

22094 

00701 

93299 

59 

56 

8 

2 

71220 

28774 

77935 

22005 

00709 

93291 

58 

52 

12 

3 

71247 

28753 

77903 

22037 

00716 

93284 

57 

48 

10 

4 

71208 

28732 

77992 

22008 

06724 

93276 

56 

44 

20 

5 

9.71289 

10.28711 

9.78020 

10.21980 

10.00731 

9.93209 

55 

40 

24 

0 

71310 

28090 

78049 

21951 

06739 

93201 

54 

36 

28 

7 

71331 

28009 

78077 

21923 

00747 

93253 

53 

32 

32 

8 

71352 

28048 

78106 

21894 

00754 

93240 

52 

28 

3G 

9 

71373 

28027 

78135 

21865 

00702 

93238 

51 

24 

40 

10 

9.71393 

10.28007 

9.78103 

10.21837 

10.00770 

9.93230 

50 

20 

44 

11 

71414 

28580 

78192 

21808 

06777 

93223 

49 

16 

48 

12 

71435 

28505 

78220 

21780 

00785 

93215 

48 

12 

52 

13 

71456 

28544 

78249 

21751 

06793 

93207 

47 

8 

60 

14 

71477 

28523 

78277 

21723 

06800 

93200 

46 

4 

5 

15 

9.71498 

10.28502 

9.78306 

10.21094 

10.06808 

9.93192 

45 

55 

4 

16 

71519 

28481 

78334 

21000 

00810 

93184 

44 

5(5 

8 

17 

71539 

28401 

7S303 

21637 

06823 

93177 

43 

52 

12 

18 

71500 

28440 

78391 

21609 

06831 

93169 

42 

48 

16 

19 

71581 

28419 

78419 

21581 

06S39 

93101 

41 

44 

20 

20 

9.71002 

10.28398 

9.78448 

10.21552 

10.06846 

9.93154 

40 

40 

24 

21 

71622 

23378 

78476 

21524 

00854 

93140 

39 

36 

28 

22 

71043 

28:157 

78503 

21495 

06862 

9313S 

38 

32 

32 

23 

71004 

28336 

785:13 

21467 

06809 

93131 

37 

28 

30 

24 

71085 

28315 

78562 

214438 

06877 

93123 

36 

24 

40 

25 

9.71705 

10.28295 

9.78590 

10.21410 

10.06885 

9.93115 

35 

20 

44 

26 

71720 

28274 

78018 

21382 

00892 

93108 

34 

16 

48 

27 

71747 

28253 

78047 

21353 

06900 

93100 

33 

12 

52 

28 

71707 

28233 

78675 

21325 

00908 

93092 

32 

8 

50 

29 

71788 

28212 

78704 

21296 

00916 

93084 

31 

4 

6 

30 

9.71803 

10.28191 

9.78731 

10.21208 

10.06923 

9.93077 

30 

54 

4 

31 

71829 

28171 

78700 

21240 

06931 

93069 

29 

56 

8 

32 

71850 

28150 

78789 

21211 

06939 

93061 

28 

52 

12 

33 

71870 

28130 

78817 

21183 

06947 

93053 

27 

48 

10 

34 

71891 

28109 

78845 

21155 

06954 

93046 

26 

44 

20 

35 

9.71911 

10.28089 

9.78874 

10.21126 

10.06902 

0.93O3S 

25 

40 

24 

30 

71932 

28008 

78902 

21098 

00970 

93030 

24 

36 

28 

37 

71952 

28048 

78930 

21O70 

00978 

93022 

23 

32 

32 

38 

71973 

28027 

78959 

21041 

00986 

93014 

22 

28 

30 

39 

71994 

28005 

7 8987 

21013 

06993 

93007 

21 

24 

40 

40 

9.72014 

10.27986 

9.79015 

10.20985 

10.07001 

9.92999 

20 

20 

44 

41 

72034 

27900 

79043 

20957 

07009 

92991 

19 

10 

48 

42 

72055 

27945 

79072 

20928 

07017 

92983 

18 

12 

52 

43 

72075 

27925 

79100 

20900 

07024 

92970 

17 

8 

50 

44 

72096 

27904 

79128 

20872 

07032 

92968 

10 

4 

7 

45 

9.72116 

10.27884 

9.79150 

10.20844 

10.07040 

9.929*50 

15 

53 

4 

40 

72137 

27803 

79185 

20815 

07048 

92952 

14 

56 

8 

47 

72157 

27843 

79213 

20787 

07056 

93944 

13 

62 

12 

48 

72177 

27823 

79241 

20759 

07064 

93980 

12 

•J8 

10 

49 

72198 

27802 

79209 

20731 

07071 

92929 

11 

44 

20 

50 

9.72218 

10.27782 

9.79297 

10.20703 

10.07079 

9.92921 

10 

40 

24 

61 

72238 

27702 

79326 

20G74 

O7087 

92913 

9 

36 

28 

52 

72259 

27741 

79354 

20040 

07095 

92905 

8 

32 

32 

53 

72279 

27721 

79382 

20018 

07103 

92897 

7 

28 

30 

54 

72299 

27701 

79410 

20590 

07111 

92889 

6 

24 

40 

55 

9.72320 

10.27080 

9.79438 

10.20562 

10.07119 

9.92881 

6 

20 

44 

60 

72340 

27000 

79406 

20534 

07126 

92874 

4 

16 

48 

57 

72300 

27040 

79495 

20505 

07134 

9280G 

3 

12 

52 

58 

723S1 

27019 

79523 

20477 

07142 

92858 

2 

8 

50 

59 

72401 

27599 

79551 

20449 

07150 

92850 

1 

4 

8 

00 

72421 

27579 

79579 

20421 

07158 

92842 

0 

5^ 

M.S. 

s h 

M 

121 

Cosine. 

0 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

58° 

M.S. 

3 b 

















Logarithms Trigonometric. 231 


2 b 

32° 


Logarithms. 


147 c 

9 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M. S. 

8 

0 

9.72421 

10.27579 

9.79579 

10.20421 

10.07158 

9.92842 

00 

52 

4 

1 

72441 

27559 

79007 

20393 

07100 

92834 

59 

50 

8 

2 

72401 

27539 

79035 

20365 

07174 

92820 

58 

52 

12 

3 

72482 

27518 

79603 

20337 

07182 

92818 

57 

48 

16 

4 

72502 

27498 

79091 

20309 

07190 

92S10 

50 

44 

20 

5 

9.72522 

10.27478 

9.79719 

10.20281 

10.07197 

9.92803 

55 

40 

24 

0 

72542 

27458 

79747 

20253 

07205 

92795 

54 

36 

28 

7 

72502 

27438 

79770 

20224 

07213 

92787 

53 

32 • 

32 

8 

725S2 

27418 

79804 

20196 

07221 

92779 

52 

28 

30 

9 

72002 

27398 

79832 

20168 

07229 

92771 

51 

24 

40 

10 

9.72622 

10.27378 

9.79SOO 

10.20140 

10.07237 

9.927G3 

50 

20 

44 

11 

72643 

27357 

79888 

20112 

07245 

92755 

49 

16 

48 

12 

72603 

27337 

79910 

20084 

07253 

92747 

48 

12 

52 

13 

726S3 

27317 

79944 

20056 

07201 

92739 

47 

8 

50 

14 

. .72703 

27297 

79972 

20028 

07209 

92731 

40 

4 

9 

15 

9.72723 

10.27277 

9.80000 

10.20000 

10.07277 

9.92723 

45 

51 

4 

10 

72743 

27257 

80028 

19972 

07285 

92715 

44 

50 

8 

17 

72703 

27237 

80050 

19944 

07293 

92707 

43 

52 

12 

18 

72783 

27217 

800S4 

19916 

07301 

92699 

42 

48 

10 

19 

72803 

27197 

80112 

19888 

07309 

92691 

41 

44 

20 

20 

9.72823 

10.27177 

9.80140 

10.19860 

10.07317 

9.92083 

40 

40 

24 

21 

72843 

27157 

8010S 

19832 

07325 

92675 

39 

36 

28 

22 

72803 

27137 

80195 

19S05 

07333 

92667 

3S 

32 

32 

23 

72883 

27117 

80223 

19777 

07341 

92659 

37 

28 

30 

24 

72902 

27098 

80251 

19749 

07349 

92651 

36 

24 

40 

25 

9.72922 

10.27078 

9.80279 

10.19721 

10.07357 

9.92043 

35 

20 

44 

20 

72942 

27058 

S0307 

19693 

07305 

92635 

34 

10 

48 

27 

72962 

27038 

80335 

19665 

07373 

92027 

33 

12 

52 

28 

72982 

27018 

80363 

19637 

07381 

92619 

32 

8 

50 

29 

73002 

26998 

80391 

19609 

073S9 

92611 

31 

4 

10 

30 

9.73022 

10.20978 

9.80419 

10.19581 

10.07397 

9.92003 

30 

50 

4 

31 

7304 L 

20959 

80447 

19553 

07405 

92595 

29 

50 

8 

32 

73001 

20939 

80474 

19526 

07413 

92587 

2S 

52 

12 

33 

73081 

20919 

80502 

19498 

07421 

92579 

27 

48 

10 

34 

73101 

20899 

80530 

19470 

07429 

92571 

26 

44 

20 

35 

9.73121 

10.26879 

9.80558 

10.19442 

10.07437 

9.92563 

25 

40 

24 

30 

73140 

20860 

80586 

19414 

07445 

92555 

24 

36 

28 

37 

73160 

20840 

80014 

19386 

07454 

92540 

23 

32 

32 

38 

73180 

20820 

80042 

19358 

07402 

92538 

22 

28 

30 

39 

73200 

20800 

80009 

19331 

07470 

92530 

21 

24 

4u 

40 

9.73219 

10.20781 

9.80097 

10.19303 

10.07478 

9.92522 

20 

20 

44 

41 

73239 

20701 

80725 

19275 

074S0 

92514 

19 

10 

48 

42 

73259 

20741 

807 53 

19247 

07494 

92500 

18 

12 

52 

43 

73278 

20722 

80781 

19219 

07502 

92498 

17 

8 

50 

44 

73298 

20702 

80808 

19192 

07510 

92490 

10 

4 

11 

45 

9.733)8 

10.26682 

9.80836 

10.19164 

10.07518 

9.92482 

15 

40 

4 

40 

73337 

26063 

80804 

19130 

07527 

92473 

14 

50 

8 

47 

73357 

20043 

80892 

19108 

07535 

92465 

13 

52 

12 

48 

73377 

20023 

80919 

19081 

07543 

92457 

12 

48 

10 

49 

73396 

20004 

80947 

19053 

07551 

92449 

11 

44 

20 

50 

9.73410 

10.20584 

9.80975 

10.19025 

10.07559 

9.92441 

10 

40 

24 

51 

73435 

20505 

81003 

18997 

07507 

92433 

9 

36 

28 

52 

73455 

20545 

81030 

18970 

07575 

92425 

8 

32 

32 

53 

73474 

26626 

81058 

18942 

075S4 

92416 

7 

28 

30 

54 

73494 

26506 

81086 

18914 

07592 

92408 

6 

24 

40 

55 

9.73513 

10.26487 

9.81113 

10.18887 

10.07000 

9.92400 

5 

20 

44 

50 

73533 

26467 

81141 

18859 

070.08 

92392 

4 

10 

48 

67 

73552 

20448 

81109 

18831 

07010 

92384 

3 

12 

62 

58 

73572 

26428 

81190 

18804 

07024 

92376 

2 

8 

50 

59 

73591 

26409 

81224 

18776 

07033 

92307 

1 

4 

1* 

00 

73011 

26389 

81252 

18748 

07641 

92359 

0 

48 

M.S. 

8 h 

M 

22' c 

Cosine. 

Secant. 

lotaugent 

Tangent. 

Cosecant. 

Sine. 

M 

37° 

M. S. 

3 h 



























232 Logarithms Trigonometric. 


2 b 

33° 


Logarithms. 


146° 

9 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangeuc. 

Secant. 

Cosine. 

M 

M.S. 

12 

0 

9.73611 

10.26389 

9.81252 

10.18748 

10.07641 

9.92359 

60 

4:8 

4 

1 

73630 

26370 

81279 

18721 

07649 

92351 

59 

56 

8 

2 

73650 

26350 

81307 

18693 

07657 

92343 

58. 

52 

12 

3 

73GG9 

26331 

81335 

18665 

076G5 

92335 

57 

48 

10 

4 

73689 

261111 

81362 

18638 

07674 

923-6 • 

56 

t 1 

20 

5 

9.73708 

10.26292 

9.81390 

10.18610 

10.07682 

9.92318 

55 

40 

24 

6 

73727 

26273 

81418 

18582 

07690 

92310 

54 

36 

.28 

7 

73747 

26253 

81445 

18555 

07698 

92302 

53 

32 

32 

8 

73766 

26234 

81473 

18527 

07707 

92293 

52 

28 

36 

9 

73785 

26215 

81500 

18500 

07715 

92285 

51 

24 

40 

10 

9.73805 

10.26195 

9.81528 

10.18472 

10.07723 

9.92277 

50 

20 

44 

11 

73824 

26176 

81556 

18441 

07731 

92269 

49 

16 

48 

12 

73843 

26157 

81583 

1N4L7 

07740 

92260 

48 

12 

52 

13 

73863 

26137 

81611 

18389 

07748 

92252 

47 

8 

66 

14 

73882 

26118 

81638 

18362 

07756 

92244 

40 

4 

13 

15 

9.73901 

10.26099 

9.81666 

10.18334 

10.07765 

9.92235 

45 

47 

4 

16 

73921 

20079 

81693 

1S307 

07773 

92227 

44 

56 

8 

17 

73940 

26060 

81721 

18279 

07781 

92219 

43 

52 

12 

18 

73959 

20041 

81748 

18252 

07789 

92211 

42 

48 

16 

19 

73978 

26022 

81776 

18224 

07798 

92202 

41 

44 

20 

20 

9.73997 

10.'-'0003 

9.81803 

10.18197 

} 0.07806 

9.92194 

40 

40 

24 

21 

74017 

25983 

81831 

18169 

07814 

92186 

39 

36 

28 

22 

74036 

25964 

81858 

18142 

07823 

92177 

38 

32 

32 

23 

74055 

25945 

81886 

18114 

07831 

92169 

37 

28 

36 

24 

74074 

259-6 

81913 

18087 

07839 

92161 

36 

24 

40 

25 

9.74093 

10.25907 

9.81941 

10.18059 

10.07848 

9.92152 

35 

20 

44 

2G 

74113 

258S7 

81968 

18032 

0785G 

92144 

34 

16 

48 

27 

74132 

25808 

81996 

1.S004 

07864 

92136 

33 

12 

52 

28 

74151 

26849 

82023 

17977 

07873 

92127 

32 

8 

5G 

29 

74170 

25830 

82051 

17949 

07881 

92119 

31 

4 

14 

30 

9.74189 

10.25811 

9.82078 

10.17922 

10.07889 

9.92111 

30 

40 

4 

31 

74208 

25792 

82106 

17894 

07898 

92102 

29 

56 

8 

32 

74227 

25773 

82133 

17867 

07906 

92094 

28 

52 

12 

33 

74246 

25754 

82161 

17839 

07914 

92086 

27 

48 

1G 

34 

74205 

25735 

82188 

17812 

07923 

92077 

26 

44 

20 

35 

9.71284 

10.25716 

9.82215 

10.17785 

10.07931 

9.92069 

25 

40 

24 

36 

74303 

25697 

82213 

17757 

07940 

92060 

24 

36 

28 

37 

74322 

25078 

82270 

17730 

07948 

92052 

23 

32 

32 

38 

74341 

25079 

82298 

17702 

07956 

92044 

22 

28 

36 

39 

74300 

25640 

82325 

17675 

07965 

92035 

21 

24 

40 

40 

9.74379 

10.25621 

9.82352 

10.17648 

10 07973 

9.92027 

20 

20 

14 

41 

74398 

25002 

82380 

1762U 

07982 

9201S, 

19 

16 

48 

42 

74117 

25583 

82407 

17593 

07990 

92010 

18 

12 

52 

43 

74436 

25504 

82435 

175G5 

07998 

92002 

17 

8 

56 

44 

74455 

25545 

82462 

17538 

08007 

91993 

16 

4 

15 

45 

9.74474 

10.255-6 

9.82489 

10.17511 

10.08015 

9 91985 

15 

45 

4 

46 

74493 

25507 

82517 

17483 

08024 

91976 

14 

56 

8 

47 

74512 

25488 

82544 

17456 

08032 

91968 

13 

52 

12 

48 

74531 

25164 

82571 

17429 

0S041 

91959 

12 

48 

16 

49 

74549 

25451 

82599 

17401 

08049 

91951 

11 

44 

20 

50 

9.74568 

10.25432 

9.82626 

10.17374 

10.08058 

9.91942 

10 

40 

24 

51 

74587 

25413 

82653 

17347 

08O06 

91934 

9 

36 

28 

52 

74006 

25394 

82681 

17319 

08075 

91925 

8 

32 

32 

53 

74025 

25375 

82708 

17292 

08083 

91917 

*■» 

1 

28 

36 

64 

74G44 

25856 

82735 

172G5 

08092 

91908 

6 

24 

40 

55 

9.74662 

10.25338 

9.82762 

10.17238 

10 08100 

9.91900 

5 

20 

44 

56 

74081 

25319 

82790 

17210 

08109 

9L891 

4 

16 

48 

57 

74700 

25300 

82817 

17183 

08117 

91883 

3 

12 

52 

58 

74719 

25281 

82844 

17156 

08126 

91874 

2 

8 

56 

59 

74737 

25263 

82>>7 L 

17129 

08134 

91866 

1 

4 

10 

GO 

74766 

25244 

82899 

17101 

OS 143 

'91857 

0 

44 

M.S. 

8 h 

- — 

M 

123 

Cosine. 

0 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

50° 

.\1 S. 

3 h 

















Logarithms Trigonometric. 233 


2 h 

34° 



Logarithms. 


145° 

9' Q 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M. S. 

16 

0 

9.74756 

10.25244 

9.82899 

10.17101 

10.08143 

9.91857 

60 

44 

4 

1 

74775 

25225 

82926 

17074 

08151 

91849 

59 

56 

8 

2 

74794 

25206 

82953 

17047 

08160 

91840 

58 

52 

12 

3 

74812 

25188 

82980 

17020 

08168 

91832 

57 

48 

16 

4 

74831 

25169 

83008 

16992 

08177 

91823 

56 

44 

20 

5 

9.74850 

10.25150 

9.83035 

10.16965 

10.08185 

9.91815 

55 

40 

24 

6 

74868 

25132 

83062 

16938 

08194 

91806 

54 

36 

28 

7 

74887 

25113 

83089 

16911 

08202 

91798 

53 

32 

32 

8 

74906 

25094 

83117 

16883 

08211 

91789 

52 

28 

36 

9 

74924 

25076 

83144 

16856 

08219 

91781 

51 

24 

40 

10 

9.74943 

10.25057 

9.83171 

10.16829 

10.08228 

9.91772 

50 

20 

44 

11 

74961 

25039 

83198 

16802 

08237 

91763 

49 

16 

48 

12 

74980 

25020 

83225 

16775 

08245 

91755 

48 

12 

52 

13 

74999 

25001 

83252 

16748 

08254 

91746 

47 

8 

56 

14 

75017 

24983 

83280 

16720 

08262 

91738 

46 

4 

IT 

15 

9.75030 

10.24964 

9.83307 

10.16693 

10.08271 

9.91729 

45 

4.J 

4 

16 

75054 

24946 

83334 

16666 

08280 

91720 

44 

56 

8 

17 

75073 

24927 

83361 

16639 

08288 

91712 

43 

52 

12 

18 

7509 L 

24909 

83388 

16612 

08297 

91703 

42 

48 

16 

19 

75110 

24896 

83415 

16585 

08305 

91695 

41 

44 

20 

20 

9.75128 

10.24872 

9.83442 

10.16558 

10.08314 

9.91686 

40 

40 

24 

21 

75147 

24853 

83470 

16530 

08323 

91677 

39 

36 

28 

22 

75165 

24835 

83497 

16503 

08:431 

91669 

38 

32 

32 

23 

75184 

24816 

83524 

16476 

08340 

91660 

37 

28 

38 

24 

75202 

24798 

83551 

16449 

08349 

91651 

36 

24 

40 

25 

9.75221 

10.24779 

9.83578 

10.46422 

10.08357 

9.91643 

35 

20 

44 

26 

75239 

24761 

83605 

16395 

08366 

91634 

34 

16 

48 

27 

75258 

24742 

83632 

16368 

08375 

91625 

33 

12 

52 

28 

75276 

21724 

83659 

16:441 

08383 

91617 

32 

8 

56 

29 

75291 

24706 

83686 

16314 

08392 

91608 

31 

4 

18 

30 

9.75313 

10.246S7 

9.83713 

10.16287 

10.08401 

9.91599 

30 

44 

4 

31 

75331 

24669 

83740 

16260 

08409 

91591 

29 

56 

8 

32 

75350 

24650 

83768 

16232 

08418 

91582 

28 

52 

12 

33 

75368 

24632 

83795 

16205 

08427 

91573 

27 

48 

16 

34 

75386 

24614 

83822 

16178 

08435 

91565 

26 

44 

20 

35 

9.75405 

10.24595 

9.83849 

10.16151 

10.08444 

9.91656 

25 

40 

24 

36 

75423 

24577 

8381 0 

16124 

08453 

91547 

24 

36 

28 

37 

75441 

24559 

83903 

16097 

08462 

91538 

23 

32 

32 

38 

75459 

24541 

83930 

16070 

08470 

91530 

2*2 

28 

36 

39 

75478 

24522 

83957 

16043 

08479 

91521 

21 

24 

40 

40 

9.75496 

10.24504 

9.83984 

10.10016 

10.08488 

9.91512 

20 

20 

44 

41 

75514 

244S6 

84011 

15989 

08496 

91504 

19 

16 

48 

42 

75533 

24467 

84038 

15962 

08505 

91495 

18 

12 

52 

43 

75551 

21449 

84065 

15935 

08514 

91486 

17 

8 

56 

44 

75569 

24431 

84092 

15908 

08523 

91477 

16 

4 

11) 

45 

9.75587 

10.24413 

9.84119 

10.15881 

10.08531 

9.91463 

15 

41 

4 

46 

75605 

24395 

84146 

15854 

08540 

91460 

14 

56 

8 

47 

75624 

24376 

84173 

15827 

08549 

91451 

13 

52 

12 

48 

75612 

24358 

84200 

15800 

08558 

91442 

12 

48 

16 

49 

75660 

24340 

84227 

15773 

08567 

91433 

11 

44 

20 

50 

9.75678 

10.24322 

9.84254 

10.15746 

10.08575 

9.91425 

10 

40 

24 

51 

75696 

24304 

84280 

15720 

08584 

91416 

9 

36 

28 

52 

75714 

212S6 

84307 

15693 

08593 

91407 

8 

32 

32 

53 

75733 

24267 

81334 

15666 

08602 

91398 

7 

28 

36 

54 

75751 

24249 

84361 

15639 

08611 

91389 

6 

24 

40 

55 

9.75769 

10.2423 L 

9.84388 

10.15612 

10.08619 

9.91381 

5 

20 

44 

56 

75787 

24213 

84415 

15585 

08628 

91372 

4 

16 

48 

57 

75805 

24195 

84442 

15558 

08637 

91363 

3 

12 

52 

58 

75823 

24177 

S4469 

15531 

08646 

91354 

2 

8 

56 

59 

75841 

24159 

84496 

15504 

08655 

91345 

1 

4 

20 

60 

75859 

24141 

84523 

15477 

08664 

91336 

0 

40 

M. S. 

M 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

M.S. 

8 h 

124° 







55° 

3 h 





















234 Logarithms Trigonometric. 


2“ 

35° 



Logarithms. 


• 144° 

9 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

30 

0 

9.75859 

10.24141 

9.84523 

10.15477 

10.08664 

9.91336 

60 

40 

4 

1 

75877 

24123 

84550 

15450 

08672 

91328 

59 

56 

8 

2 

75895 

24105 

84576 

15424 

0868 L 

91319 

58 

52 

12 

3 

75913 

24087 

84603 

15397 

08690 

91310 

57 

48 

16 

4 

75931 

24069 

84630 

15370 

08699 

91301 

56 

44 

20 

6 

9.75949 

10.24051 

9.84657 

10.15343 

10.08708 

9.91292 

55 

40 

24 

6 

75967 

24033 

84684 

15316 

08717 

91283 

54 

36 

28 

7 

75985 

24015 

84711 

15289 

08726 

91274 

53 

32 

32 

8 

76003 

23997 

84738 

15262 

08734 

91266 

52 

28 

36 

9 

76021 

23979 

84764 

15236 

08743 

91257 

51 

24 

40 

10 

9.76039 

10.23961 

9.84791 

10.15209 

10.08752 

9.91248 

50 

20 

44 

11 

76057 

23943 

84818 

15182 

08761 

91239 

49 

16 

48 

12 

76075 

23925 

84845 

15155 

08770 

91230 

48 

12 

52 

13 

76093 

23907 

84872 

15128 

08779 

91221 

47 

8 

56 

14 

76111 

23889 

84899 

15101 

08788 

91212 

46 

4 

31 

15 

9.76129 

10.23871 

9.84925 

10.15075 

10.08797 

9.91203 

45 

39 

4 

16 

76146 

23854 

84952 

15048 

08806 

91194 

44 

56 

8 

17 

76164 

23836 

84979 

15021 

08815 

91185 

43 

52 

12 

18 

76182 

23818 

85006 

14994 

088.24 

91176 

42 

48 

16 

19 

76200 

23800 

85033 

14967 

08833 

91167 

41 

44 

20 

20 

9.76218 

10.23782 

9.85059 

10.14941 

10.08842 

9.91158 

40 

40 

24 

21 

76236 

23764 

85086 

• 14914 

08851 

91149 

39 

36 

28 

22 

76253 

23747 

85113 

14887 

08859 

91141 

38 

32 

32 

23 

76271 

23729 

85140 

14860 

08868 

91132 

37 

28 

36 

24 

76289 

23711 

85166 

14834 

OS877 

91123 

36 

24 

40 

25 

9.76307 

10.23693 

9.85193 

10.14807 

10.0S886 

9.91114 

35 

20 

44 

26 

76324 

23676 

85220 

14780 

08895 

91105 

34 

16 

48 

27 

76:142 

23658 

85247 

14753 

08904 

91096 

33 

12 

52 

28 

76360 

23640 

85273 

14727 

08913 

91087 

32 

8 

56 

29 

76378 

23622 

85300 

14700 

08922 

91078 

31 

4 

33 

30 

9.76395 

10.23605 

9.85327 

10.14673 

10.08931 

9.91069 

30 

38 

4 

31 

76413 

23587 

85354 

14640 

08940 

91060 

29 

56 

8 

32 

76431 

23569 

85380 

14620 

08949 

91051 

28 

52 

12 

33 

76448 

23552 

85407 

14593 

08958 

91042 

27 

48 

16 

34 

76466 

23534 

85434 

14566 

08967 

91033 

26 

44 

20 

35 

9.76484 

10.23516 

9.85460 

10.14540 

10.08977 

9.91023 

25 

40 

24 

36 

76501 

23499 

85487 

14513 

08986 

91014 

24 

36 

28 

37 

76519 

23481 

85514 

14486 

08995 

91005 

23 

32 

32 

38 

76537 

23463 

85540 

14460 

09004 

90996 

22 

28 

36 

39 

76554 

23446 

85567 

14433 

09013 

90987 

21 

24 

40 

40 

9.76572 

10.23428 

9.85594 

10.14406 

10.09022 

9.90978 

20 

20 

44 

41 

76590 

23410 

85620 

14380 

09031 

90969 

19 

16 

48 

42 

76607 

23393 

85647 

14353 

09040 

90960 

18 

12 

52 

43 

76625 

23375 

85674 

14326 

09049 

90951 

17 

8 

56 

44 

76642 

23358 

85700 

14300 

09058 

90942 

16 

4 

33 

45 

9.76660 

10.23340 

9.S5727 

10.14273 

10.09067 

9.90933 

15 

37 

4 

46 

76677 

23323 

85754 

14246 

09076 

90924 

14 

56 

8 

47 

76695 

23305 

85780 

14220 

09085 

90915 

13 

52 

12 

48 

76712 

23288 

85807 

14193 

09094 

90906 

12 

48 

16 

49 

76730 

23270 

85834 

14166 

09104 

90896 

11 

44 

20 

50 

9.76747 

10.23253 

9.85860 

10.14140 

10.09113 

9.90887 

10 

40 

24 

51 

76765 

23235 

85887 

14113 

(9122 

90878 

9 

36 

28 

52 

76782 

23218 

85913 

14087 

09131 

90869 

8 

32 

32 

53 

76800 

23200 

85940 

140G0 

09140 

90860 

7 

28 

36 

54 

76817 

23183 

85967 

14033 

09149 

90851 

6 

24 

40 

55 

9.76835 

10.23165 

9.85993 

10.14007 

10.09158 

9.90842 

5 

20 

44 

56 

76852 

23148 

86020 

13980 

09168 

90832 

4 

16 

48 

57 

76870 

23130 

86046 

13954 

09177 

90823 

3 

12 

52 

58 

76887 

23113 

86073 

13927 

09186 

90814 

2 

8 

6ij 

59 

76904 

2309G 

8G100 

13900 

09195 

90805 

1 

4 

34 

60 

76922 

23078 

86126 

13874 

09204 

90796 

0 

36 

M.S. 

8 h 

M 

125 c 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

54° 

M.S. 

















Logarithms Trigonometric. 


235 


2 h 

o 

to 

CO 



Logarithms. 


143° 

9 h 

M.S. 

M 

Siuc. 

Cosecaut. 

Tangent. 

Cotaugent. 

Secant. 

Cosine. 

M 

M.S. 

24 

0 

9.76922 

10.23078 

9.86126 

10.13874 

10.09204 

9.90796 

60 

30 

4 

1 

76939 

23061 

86153 

13847 

09213 

90787 

59 

56 

8 

2 

76957 

23043 

86179 

13821 

09223 

90777 

58 

52 

12 

3 

76974 

23026 

86206 

13794 

09232 

90768 

57 

48 

1(3 

4 

76991 

23009 

86232 

13768 

09241 

90759 

56 

44 

20 

5 

9.77009 

10.22991 

9.86259 

10.13741 

10.09250 

9.90750 

55 

40 

24 

6 

77026 

22974 

86285 

13715 

09259 

90741 

54 

36 

2S 

7 

77043 

22957 

86312 

13688 

09269 

9073 L 

53 

32 

32 

8 

77061 

22939 

86338 

13662 

09278 

90722 

52 

28 

3G 

9 

77078 

22922 

86365 

13635 

09287 

90713 

51 

24 

40 

10 

9.77095 

10.22905 

9.86392 

10.13608 

10.09296 

9.90704 

50 

20 

44 

11 

77112 

22888 

86418 

13582 

09306 

90694 

49 

16 

48 

12 

77130 

22S70 

86445 

13555 

09315 

90685 

48 

12 

52 

13 

77147 

22853 

86471 

13529 

09324 

90676 

47 

8 

5G 

14 

* 77164 

22836 

86498 

13502 

09333 

90667 

46 

4 

35 

15 

9.771S1 

10.22819 

9.86524 

10.13476 

10.09343 

9.90657 

45 

35 

4 

1G 

77199 

22801 

86551 

13449 

09352 

90648 

44 

56 

8 

17 

77216 

22784 

86577 

13423 

09361 

90639 

43 

52 

12 

18 

77233 

22767 

86603 

13397 

09370 

90630 

42 

48 

16 

19 

77250 

22750 

86630 

13370 

09380 

90620 

41 

44 

20 

20 

9.77268 

10.22732 

9.86656 

10.13344 

10.09389 

9.9061 L 

40 

40 

24 

21 

77285 

22715 

86683 

13317 

09398 

90602 

39 

36 

28 

22 

77302 

22698 

86709 

13291 

09408 

90592 

38 

32 

32 

23 

77319 

22681 

86736 

13264 

09417 

90583 

37 

28 

36 

24 

77336 

22664 

86762 

13238 

09426 

90574 

36 

24 

40 

25 

9.77353 

10.22647 

9.86789 

10.13211 

10.09435 

9.90565 

35 

20 

44 

20 

77370 

22630 

86815 

13185 

09445 

90555 

34 

16 

48 

27 

77387 

22613 

86812 

13158 

09454 

90546 

33 

12 

52 

28 

77405 

22595 

86868 

13132 

09463 

90537 

32 

8 

56 

29 

77422 

22578 

86894 

13106 

09473 

90527 

31 

4 

30 

30 

9.77439 

10.22561 

9.86921 

10.13079 

10.09482 

9.90518 

30 

34: 

4 

31 

77456 

22544 

86947 

13053 

09491 

90509 

29 

56 

8 

32 

77473 

22527 

86974 

13026 

09501 

90499 

28 

52 

12 

33 

77490 

22510 

87000 

13000 

09510 

90490 

27 

48 

1G 

34 

77507 

22493 

87027 

12973 

09520 

90480 

26 

44 

20 

35 

9.77524 

10.22476 

9.87053 

10.12947 

10.09529 

9.90471 

25 

40 

24 

36 

77541 

22459 

87079 

12921 

09538 

90462 

24 

36 

28 

37 

77558 

22442 

S7I06 

12894 

09548 

90452 

23 

32 

32 

38 

77575 

22425 

87132 

12868 

09557 

90443 

22 

28 

36 

39 

77592 

22408 

87158 

12842 

09566 

90434 

21 

24 

40 

40 

9.77609 

10.22391 

9.87185 

10.12815 

10.09576 

9.90424 

20 

20 

44 

41 

77626 

22374 

87211 

12789 

09585 

90415 

19 

16 

48 

42 

77643 

22357 

87238 

12762 

09595 

90405 

18 

12 

52 

43 

77660 

22340 

87264 

12736 

09604 

90396 

17 

8 

5G 

44 

77677 

22323 

87290 

12710 

09614 

90386 

16 

4 

37 

45 

9.77694 

10.22306 

9.87317 

10.12683 

10.09623 

9.90377 

15 

33 

4 

46 

77711 

22289 

87343 

12657 

09632 

90368 

14 

56 

8 

47 

77728 

22272 

87369 

12631 

09642 

90358 

13 

52 

12 

48 

77744 

22256 

S7396 

12604 

* 09651 

90349 

12 

48 

16 

49 

77761 

22239 

87422 

12578 

09661 

90339 

11 

44 

20 

50 

9.77778 

10.22222 

9.87448 

10.12552 

10.09670 

9.90330 

10 

40 

24 

51 

77795 

22205 

87475 

12525 

09680 

90320 

9 

36 

28 

52 

77812 

22188 

87501 

12499 

09689 

90311 

8 

32 

32 

53 

77829 

22171 

87527 

12473 

09699 

90301 

7 

28 

30 

54 

77846 

22154 

87554 

12446 

09708 

90292 

6 

24 

40 

55 

9.77862 

10.22138 

9.87580 

10.12420 

10.09718 

9.90282 

5 

20 

44 

5G 

77879 

22121 

87606 

12394 

09727 

90273 

4 

16 

48 

57 

77896 

22104 

87633 

12367 

09737 

90263 

3 

12 

52 

58 

77913 

22087 

87659 

12341 

09746 

90254 

2 

8 

56 

59 

77930 

22070 

87685 

12315 

09756 

90244 

1 

4 

28 

GO 

77946 

22054 

87711 

12289 

09765 

90235 

0 

32 

M.S. 

8 h 

M 

126° 

Cosine. 

Secaat. 

Cotaugeut 

Tangent. 

Cosecaut. 

Sine. 

M 

53° 

M.S. 

3> 



















236 Logarithms Trigonometric. 


2 h 

37° 



Logarithms. 


142° 

9 h 

M.S. 

M 

Sine. 

1 Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

28 

0 

9.77946 

10.22054 

9.87711 

10.12289 

10.097G5 

9.90235 

60 

32 

4 

1 

77963 

22037 

87738 

12262 

09775 

90225 

59 

56 

8 

2 

77980 

22020 

87764 

12236 

09784 

90216 

58 

52 

12 

3 

77997 

22003 

87790 

12210 

09794 

90206 

57 

48 

16 

4 

78013 

21987 

87817 

12183 

09803 

90197 

56 

44 

20 

5 

9.78030 

10.21970 

9.87843 

10.12157 

10.09813 

9.90187 

55 

40 

24 

6 

78047 

21953 

87869 

12131 

09822 

90178 

54 

36 

28 

7 

78063 

21937 

87895 

12105 

09832 

90168 

53 

32 

32 

8 

78(180 

21920 

87922 

12078 

09841 

90159 

52 

28 

36 

9 

78097 

21903 

87948 

12052 

09851 

90149 

51 

24 

40 

10 

9.78113 

10.21887 

9.87974 

10.12026 

10.09861 

9.90139 

50 

20 

44 

11 

78130 

21870 

88000 

12000 

09870 

90130 

49 

16 

48 

12 

78147 

21853 

8S027 

11973 

09880 

90120 

48 

12 

52 

13 

78163 

21837 

88053 

11917 

09889 

90111 

47 

8 

56 

14 

78180 

21820 

88079 

11921 

09899 

90101 

46 

4 

39 

15 

9.78197 

10.2181(3 

9.88105 

10.11895 

10.09909 

9.90091 

45 

31 

4 

16 

78213 

21787 

88131 

11869 

09918 

90082 

44 

56 

8 

17 

78230 

21770 

8815S 

J1842 

09928 

90072 

43 

52 

12 

18 

78246 

21754 

88184 

11816 

09937 

90063 

42 

48 

16 

19 

78263 

21737 

88210 

11790 

09947 

91)053 

41 

44 

20 

20 

9.78280 

10.21720 

9.88236 

10.11764 

10.09957 

9.90043 

40 

40 

24 

21 

78296 

21704 

88262 

11738 

09966 

90034 

39 

36 

28 

22 

78313 

21687 

88289 

11711 

09976 

90024 

38 

32 

32 

23 

78329 

21671 

88315 

11685 

09986 

90014 

37 

28 

36 

24 

7S346 

21654 

88341 

11659 

09995 

90005 

36 

24 

40 

25 

9.78362 

10.21638 

9.88367 

10.11633 

10.10005 

9.89995 

35 

20 

44 

20 

78379 

21621 

88393 

11607 

10015 

89985 

34 

16 

48 

27 

78395 

21605 

88420 

11580 

10024 

89976 

33 

12 

52 

28 

78412 

21588 

88446 

11554 

10031 

89966 

32 

8 

56 

29 

78428 

21572 

88472 

11528 

10044 

89956 

31 

4 

30 

30 

9.78445 

10.21555 

9.88498 

10.11502 

10.10053 

9.89947 

30 

30 

4 

31 

78461 

21539 

88524 

11476 

10063 

89937 

29 

56 

8 

32 

78478 

21522 

88550 

11450 

10073 

89927 

28 

52 

12 

33 

78494 

21506 

88577 

11423 

10082 

89918 

27 

48 

16 

34 

78510 

21490 

88603 

11397 

10092 

89908 

26 

44 

20 

35 

9.78527 

10.21473 

9.88629 

10.11371 

10.10102 

9.89898 

25 

40 

24 

36 

78513 

21457 

88655 

11345 

10112 

89888 

24 

36 

28 

37 

78560 

21440 

88681 

11319 

10121 

89879 

23 

32 

32 

38 

78576 

21424 

88707 

11293 

10131 

89869 

22 

28 

86 

39 

78592 

21408 

887:13 

11267 

10141 

89S59 

21 

24 

40 

40 

9.78609 

10.21391 

9.8S759 

10.11241 

10.10151 

9.89849 

20 

20 

44 

41 

78625 

21375 

88780 

11214 

10160 

89840 

19 

16 

48 

42 

78642 

21358 

88812 

11188 

10170 

89830 

IS 

12 

52 

43 

78658 

21342 

88838 

11162 

10180 

89820 

17 

8 

56 

44 

78674 

21326 

888(54 

11136 

10090 

89810 

16 

4 

31 

45 

9.78691 

10.21309 

9.88890 

10.1111O 

10.10199 

9.89801 

15 

29 

4 

46 

787»>7 

21293 

88916 

11084 

10209 

89791 

14 

56 

8 

47 

78723 

21277 

, 88942 

11058 

1(1219 

89781 

13 

52 

12 

48 

78739 

21201 

f 8896S 

11032 

1022) 

89771 

12 

48 

16 

49 

78756 

21244 

88994 

11006 

10239 

89761 

11 

44 

20 

50 

9.78772 

10.21228 

9.89020 

10.10980 

10.10248 

9.89752 

10 

40 

24 

51 

78788 

21212 

89046 

10954 

10258 

89742 

9 

36 

28 

52 

78805 

21195 

89073 

10927 

10268 

89732 

8 

32 

32 

53 

78821 

21179 

89099 

10901 

10278 

89722 

7 

28 

36 

54 

78837 

21163 

89125 

10875 

10288 

89712 

6 

24 

40 

55 

9.78853 

10.21147 

9.89151 

10.10849 

10.10298 

9.89702 

5 

20 

44 

66 

78809 

21131 

89177 

10823 

10307 

89693 

4 

16 

48 

57 

78886 

21114 

89203 

10797 

10317 

89683 

3 

12 

52 

53 

78902 

21098 

89229 

10771 

10327 

89673 

2 

8 

56 

59 

78918 

21082 

89255 

10745 

10337 

89663 

1 

4 

33 

60 

78934 

21006 

89281 

10719 

10347 

89653 

0 

28 

M. S. 

8“ 

M 

127 ( 

Cosine. 

Secant. 

Cotangent 1 

Tangent. 

Cosecant. 

Sine. 

M 

52° 

M.S. 

3 h 






















.Logarithms Trigonometric. 


237 


2' 1 

38° 


Logarithms. 


141° 

9 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

32 

0 

9.78934 

10.21066 

9.89281 

10.10719 

10.10347 

9.89653 

60 

£8 

4 

1 

78950 

21050 

89307 

10693 

10357 

89643 

59 

56 

8 

2 

78967 

21033 

89333 

10667 

10367 

89633 

58 

52 

12 

3 

78983 

21017 

89359 

10641 

10376 

89624 

57 

48 

16 

4 

78999 

21001 

89385 

10615 

10386 

89614 

56 

44 

20 

5 

9.79015 

10.20985 

9.89411 

10.10589 

10.10396 

9.89604 

55 

40 

24 

6 

7903 L 

20969 

89437 

10563 

10406 

89594 

54 

36 

28 

7 

79047 

20953 

89463 

10537 

10416 

89584 

53 

32 

32 

8 

79063 

20937 

89489 

10511 

10426 

89574 

52 

28 

36 

9 

79079 

20921 

89515 

10485 

10436 

89564 

51 

24 

40 

10 

9.79095 

10.20905 

9.89541 

10.10459 

10.10146 

9.89554 

50 

20 

44 

11 

79111 

20889 

89567 

10433 

10456 

89544 

49 

16 

48 

12 

79128 

20872 

89593 

10407 

10466 

89534 

48 

12 

52 

13 

79144 

20856 

89619 

10381 

10476 

89524 

47 

8 

56 

14 

79160 

20840 

89645 

10355 

104S6 

89514 

40 

4 

33 

15 

9.79176 

10.20824 

9.89671 

10.10329 

10.10496 

9.89504 

45 

27 

4 

16 

79192 

26808 

89697 

10303 

105(45 

89 495 

44 

56 

8 

17 

79208 

20792 

89723 

10277 

10515 

89485 

43 

52 

12 

18 

79224 

20776 

89749 

10251 

10525 

89475 

42 

48 

16 

19 

79240 

20760 

89775 

10225 

10535 

89465 

41 

44 

20 

20 

9.79256 

10.29744 

9.89801 

10.10199 

10.10545 

9.89455 

40 

40 

24 

21 

79272 

20728 

89827 

10173 

10555 

89445 

39 

36 

28 

22 

79288 

20712 

89853 

10147 

10565 

89435 

38 

32 

32 

23 

79304 

20696 

89879 

10121 

10575 

89425 

37 

28 

36 

24 

79319 

20681 

89905 

10095 

10585 

89415 

36 

24 

40 

25 

9.79335 

10.20665 

9.89931 

10.10069 

10.10595 

9.89405 

35 

20 

44 

20 

79851 

20649 

89957 

10043 

10605 

89395 

34 

16 

48 

27 

79367 

20633 

89983 

10017 

10615 

89385 

33 

12 

52 

28 

79383 

20617 

90009 

09991 

10625 

89375 

32 

8 

56 

29 

79399 

20601 

90035 

09965 

10636 

89364 

31 

4 

34 

30 

9.79415 

10.20585 

9.90061 

10.09939 

10.10646 

9.89354 

30 

2G 

4 

31 

79431 

20569 

90086 

09914 

10656 

89344 

29 

56 

8 

32 

79447 

20553 

90112 

09888 

10666 

89334 

28 

52 

12 

33 

79463 

20537 

90138 

09862 

10676 

89324 

27 

48 

16 

34 

79478 

20522 

9016 4 

09836 

106S6 

89314 

26 

44 

20 

35 

9.79494 

10.20506 

9.90190 

10.09810 

10.10696 

9.89304 

25 

40 

24 

36 

79510 

20490 

90216 

09784 

10706 

89294 

24 

36 

28 

37 

79526 

20474 

90242 

09758 

10716 

89284 

23 

32 

32 

38 

79542 

20458 

90268 

09732 

10726 

89274 

22 

28 

36 

39 

79558 

20442 

90294 

09706 

10736 

89264 

21 

24 

40 

40 

9.79573 

10.20427 

9.90320 

10.09680 

10 10746 

9.89254 

20 

20 

44 

41 

79589 

20411 

90346 

09654 

10756 

89244 

19 

16 

48 

42 

79605 

20395 

90371 

09029 

10767 

89233 

18 

12 

52 

43 

79621 

20379 

90397 

09603 

10777 

89223 

17 

8 

56 

44 

79636 

20304 

90423 

09577 

10787 

89213 

16 

4 

35 

45 

9.79652 

10.20348 

9.90449 

10.09551 

10.10797 

9 89203 

15 

25 

4 

46 

79668 

20332 

90475 

09525 

10807 

89193 

14 

56 

8 

47 

79684 

20316 

9U501 

09499 

10817 

89183 

13 

52 

12 

48 

79699 

20301 

90527 

09473 

10827 

89173 

12 

48 

16 

49 

79715 

20285 

90553 

09447 

10838 

89162 

11 

44 

20 

50 

9.79731 

10.20269 

9.90578 

10.09422 

10.10S48 

9.89152 

10 

40 

24 

51 

79746 

20254 

90604 

09396 

10858' 

89142 

9 

36 

28 

52 

79762 

20238 

90630 

09:.70 

10868 

89132 

8 

32 

32 

53 

79778 

20222 

90656 

09344 

10878 

89122 

7 

28 

36 

54 

79793 

20207 

90682 

09318 

10888 

89112 

6 

24 

40 

55 

9.79809 

10.20191 

9.90708 

10.09292 

10.10899 

9.89101 

5 

20 

44 

56 

79825 

20175 

90734 

09266 

10909 

89091 

4 

16 

48 

57 

79840 

20160 

90759 

09241 

10919 

89081 

3 

12 

52 

58 

79856 

20144 

90785 

09215 

10929 

89071 

2 

8 

56 

59 

79872 

20128 

90811 

09189 

10940 

89060 

1 

4 

30 

60 

79887 

20113 

90837 

09163 

10950 

89050 

0 


M.S. 

M 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

M.S. 

8 U 

>— > 

K. 

GO 

o 





51° 

3 h 


















238 Logarithms Trigonometric. 


2 b 

39 c 



Logarithms. 


o 

o 

rH 

9 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

ar» 

0 

9.79887 

10.20113 

9.90837 

10.09163 

10.10950 

9.89050 

60 

34 

4 

l 

79903 

20097 

90863 

09137 

10960 

89040 

59 

56 

8 

2 

79918 

20082 

90889 

09111 

10970 

89030 

58 

52 

12 

3 

79934 

20066 

90914 

09086 

10980 

89020 

57 

48 

1G 

4 

79950 

20050 

90940 

09060 

10991 

89009 

56 

44 

20 

5 

9.79965 

10.20035 

9.90966 

10.09034 

10.11001 

9.88990, 

55 

40 

24 

6 

79981 

20019 

90992 

09008 

11011 

88989 

54 

36 

28 

7 

79996 

20004 

91018 

08982 

11022 

88978 

53 

32 

32 

8 

80012 

19988 

91043 

08957 

11032 

88968 

52 

28 

3G 

9 

80027 

19973 

91069 

08931 

11042 

88958 

51 

24 

40 

10 

9.80043 

10.19957 

9.91095 

10.08905 

10.11052 

9.88948 

50 

20 

44 

11 

80058 

19942 

91121 

08S79 

11063 

88937 

49 

16 

48 

12 

80074 

19926 

91147 

08853 

11073 

88927 

48 

12 

52 

13 

80089 

19911 

91172 

08828 

11083 

88917 

47 

8 

5G 

14 

80105 

19895 

91198 

08802 

11094 

88906 

46 

4 

37 

15 

9.80120 

10.19 s 80 

9.91224 

10.08776 

10.11104 

9.88896 

45 

33 

4 

16 

80136 

19864 

91250 

08750 

11114 

88886 

44 

56 

8 

17 

80151 

19849 

91276 

08724 

11125 

88875 

43 

52 

12 

18 

80166 

19834 

91301 

08699 

11135 

88865 

42 

48 

16 

19 

80182 

19818 

91327 

08673 

11145 

88855 

41 

44 

20 

20 

9.80197 

10.19803 

9.91353 

10.08647 

10.11156 

9.88844 

40 

40 

24 

21 

80213 

19787 

91379 

0S621 

11166 

88834 

39 

36 

28 

22 

80228 

19772 

91404 

08596 

11176 

88824 

38 

32 

32 

23 

80244 

19756 

91430 

08570 

11187 

88813 

37 

28 

36 

24 

80259 

19741 

91456 

08544 

11197 

88803 

36 

24 

40 

25 

9.80274 

10.19726 

9.91482 

10.08518 

10.11207 

9.88793 

35 

20 

44 

26 

80290 

19710 

91507 

08493 

11218 

88782 

34 

16 

48 

27 

80305 

19695 

91533 

08467 

11228 

88772 

33 

12 

62 

28 

80320 

19680 

91559 

08411 

11239 

88761 

32 

8 

6G 

29 

80336 

19664 

91585 

08415 

11249 

88751 

31 

4 

38 

30 

9.80351 

10.19649 

9.91610 

10.08390 

10.11259 

9.88741 

30 

33 

4 

31 

80366 

19634 

91636 

08361 

11270 

88730 

29 

56 

8 

32 

80382 

19618 

91662 

08338 

11280 

88720 

28 

52 

12 

33 

80397 

19603 

91688 

08312 

11291 

88709 

27 

48 

16 

34 

80412 

19588 

91713 

08287 

11301 

88699 

26 

44 

20 

35 

9.80428 

10.19572 

9.91739 

10.08261 

10.11312 

9.88688 

25 

40 

24 

36 

80443 

19557 

91765 

08235 

11322 

88678 

24 

36 

28 

37 

8045S 

19542 

91791 

08209 

11332 

88668 

23 

32 

32 

38 

80473 

19527 

9181G 

08181 

11343 

88657 

22 

28 

36 

39 

80489 

19511 

91842 

08158 

11353 

88647 

21 

24 

40 

40 

9.80504 

10.19496 

9.91868 

1008132 

10 11364 

9.88636 

20 

20 

44 

41 

80519 

19481 

91893 

08107 

11374 

88626 

19 

16 

48 

42 

80534 

19466 

91919 

08081 

11385 

88615 

18 

12 

52 

43 

80550 

19450 

91945 

08055 

11395 

88605 

17 

8 

56 

44 

80565 

19435 

91971 

08029 

11406 

88594 

16 

4 

39 

45 

9.80580 

10.19420 

9.91996 

10.08001 

10.11416 

9 88584 

15 

31 

4 

46 

80595 

19405 

92022 

07978 

11427 

88573 

14 

56 

8 

47 

80610 

19390 

92048 

07952 

11437 

88563 

13 

52 

12 

48 

80625 

19375 

92073 

07927 

11448 

88552 

12 

48 

16 

49 

80641 

19359 

92099 

07901 

11458 

885-^2 

11 

44 

20 

50 

9.80656 

10.19344 

9.92125 

10.07875 

10.11469 

9.88531 

10 

40 

24 

51 

80671 

19329 

92150 

07850 

11479 

8852 L 

9 

36 

28 

52 

80686 

19314 

92176 

07824 

11490 

88510 

8 

32 

32 

53 

80701 

19299 

92202 

07798 

11501 

88499 

7 

28 

36 

54 

80716 

19234 

92227 

07773 

11511 

88489 

6 

24 

40 

55 

9.80731 

10.19269 

9.92253 

10.07747 

1011522 

9.88478 

5 

20 

44 

56 

807 46 

19254 

92279 

07721 

11532 

88468 

4 

16 

48 

57 

80762 

19238 

92304 

07696 

11543 

88457 

3 

12 

52 

58 

80777 

19223 

92330 

07670 

11553 

88447 

2 

8 

56 

59 

80792 

19208 

92356 

07644 

11564 

88436 

1 

4 

40 

60 

80807 

19193 

92381 

07619 

11575 

88425 

0 

30 

M.S. 

M 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

M. S. 

8 b 

129 

D 






50° 

3“ 
























Logarithms Trigonometric. 239 


2 h 

o 

o 



Logarithms. 


139° 

9 h 

M.S 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

1 11 

M.S. 

40 

0 

9.80807 

10.19193 

9.92381 

10.07619 

10.11575 

9.88425 

60 

>30 

4 

1 

80822 

19178 

92407 

07593 

11585 

88415 

59 

56 

8 

2 

80837 

19163 

92433 

07567 

11596 

88404 

58 

52 

12 

3 

80852 

19148 

92458 

07542 

11606 

88394 

57 

48 

16 

4 

80867 

19133 

92484 

07516 

11617 

88383 

56 

44 

20 

5 

9.80882 

10.19118 

9.92510 

10.07490 

10.11628 

9.88372 

55 

40 

24 

6 

80897 

19103 

92535 

07465 

11638 

88362 

54 

36 

28 

7 

80912 

19088 

92561 

07439 

11649 

88351 

53 

32 

32 

8 

80927 

19073 

92587 

07413 

11660 

88340 

52 

28 

3G 

9 

80942 

1905S 

92612 

07388 

11670 

88330 

51 

24 

40 

10 

9.80957 

10.19043 

9.92638 

10.07362 

10.11681 

9.88319 

50 

20 

44 

11 

80972 

19028 

92663 

07337 

11692 

88308 

49 

16 

48 

12 

80987 

19013 

92689 

07311 

11702 

88208 

48 

12 

52 

13 

81002 

18998 

92715 

07285 

11713 

88287 

47 

8 

5G 

14 

81017 

18983 

92740 

07260 

11724 

88276 

46 

4 

41 

15 

9.81032 

10.18968 

9.92766 

10.07234 

10.117:44 

9.88266 

45 

19 

4 

16 

81047 

18953 

92792 

07208 

11745 

88255 

44 

56 

8 

17 

81061 

18939 

92817 

07183 

11756 

88244 

43 

52 

12 

18 

81076 

18924 

92843 

07157 

11766 

88234 

42 

48 

1G 

19 

81091 

18909 

92868 

07132 

11777 

88223 

41 

44 

20 

20 

9.81106 

10.18894 

9.92894 

10.07106 

10.11788 

9.88212 

40 

40 

24 

21 

81121 

18879 

92920 

07080 

11799 

88201 

39 

36 

28 

22 

81136 

18864 

92945 

07055 

11809 

88191 

38 

32 

32 

23 

81151 

18849 

92971 

07029 

11820 

88180 

37 

28 

36 

24 

81166 

18S34 

92996 

07004 

11831 

88169 

36 

24 

40 

25 

9.81180 

10.18820 

9.93022 

10.06978 

10.11842 

9.88158 

35 

20 

44 

26 

81195 

18805 

93048 

06952 

11852 

88148 

34 

16 

48 

27 

81210 

18790 

93073 

06927 

11863 

88137 

33 

12 

52 

28 

81225 

18775 

93099 

06901 

11874 

88126 

32 

8 

56 

29 

81240 

18760 

93124 

06876 

11885 

88115 

31 

4 

43 

30 

9.81254 

10.18746 

9.93150 

10.06850 

10.11895 

9.88105 

30 

18 

4 

31 

81269 

18731 

93175 

06825 

11906 

8S094 

29 

56 

8 

32 

81284 

18716 

93201 

06799 

11917 

88083 

28 

52 

12 

33 

81299 

18701 

93227 

06773 

11928 

88072 

27 

48 

16 

34 

81314 

18686 

93252 

06748 

11939 

88061 

26 

44 

20 

35 

9.81328 

10.18672 

9.93278 

10.06722 

10.11949 

9.88051 

25 

40 

24 

36 

81343 

18657 

93303 

06697 

11960 

88040 

24 

36 

28 

37 

81358 

18642 

93329 

06671 

11971 

88029 

23 

32 

32 

38 

81372 

18628 

93354 

06616 

11982 

88018 

22 

28 

36 

39 

81387 

18613 

93380 

06620 

11993 

88007 

21 

24 

40 

40 

9.81402 

10.18598 

9.93406 

10.06594 

10.12004 

9.87996 

20 

20 

4-1 

41 

81417 

18583 

93431 

06569 

12015 

87985 

19 

16 

48 

42 

81431 

18569 

93457 

06543 

12025 

87975 

18 

12 

52 

43 

81446 

18554 

93482 

06518 

12036 

87964 

17 

8 

56 

44 

81461 

18539 

93508 

06492 

12047 

87953 

16 

4 

4.1 

45 

9.81475 

10.18525 

9.93533 

10.06467 

10.12058 

9.87942 

15 

17 

4 

46 

81490 

18510 

93559 

06441 

12069 

87931 

14 

56 

8 

47 

81505 

18495 

93584 

06416 

12080 

87920 

13 

52 

12 

48 

81519 

18481 

93610 

06390 

12091 

87909 

12 

48 

16 

49 

81534 

18466 

93636 

06364 

12102 

87898 

11 

44 

20 

50 

9.81549 

10.18451 

9.93661 

10.06339 

10.12113 

9.87887 

10 

40 

24 

51 

81563 

1S437 

93687 

06313 

12123 

87877 

9 

36 

28 

52 

81578 

18422 

93712 

06288 

12134 

87866 

8 

32 

32 

53 

81592 

18408 

93738 

06262 

12145 

87855 

7 

28 

36 

54 

81607 

18393 

93763 

06237 

12156 

87844 

6 

24 

40 

55 

9.81622 

10.18378 

9.93789 

10.06211 

10.12167 

9.87833 

5 

20 

44 

56 

81636 

18364 

93814 

06186 

12178 

87822 

4 

16 

48 

57 

81651 

18349 

93840 

06160 

12189 

87811 

3 

12 

. 52 

58 

81665 

18335 

93865 

06135 

12200 

87800 

2 

8 

56 

59 

81680 

1§320 

93891 

06109 

12211 

87789 

1 

4 

44 

60 

81694 

18306 

93916 

06084 

12222 

87778 

0 

16 

M.S. 

8 U 

M 

130 c 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

49° 

M.S. 

3 h 




























240 Logarithms Trigonometric. 


2 h 

41° 



Logarithms. 


138° 

9 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

44 

0 

9.81094 

10.18300 

9.93916 

10.00084 

10.12222 

9.87778 

60 

1G 

4 

1 

81709 

18291 

93942 

00058 

12233 

87707 

59 

50 

8 

2 

81723 

18277 

93967 

06033 

12244 

87750 

58 

52 

12 

3 

81738 

18202 

93993 

06007 

12255 

87745 

57 

48 

10 

4 

81752 

18248 

94018 

05982 

12266 

87734 

50 

44 

20 

5 

9.81707 

10.18233 

9.94014 

10.05956 

10.12277 

9.87723 

55 

40 

24 

0 

81781 

18219 

94009 

05931 

12288 

87712 

54 

30 

28 

7 

81790 

18204 

94095 

05905 

12299 

8770 L 

53 

32 

32 

8 

81810 

18190 

94120 

058S0 

12310 

87690 

52 

28 

30 

9 

81825 

18175 

94140 

05854 

1232 L 

87079 

51 

2-1 

40 

10 

9.81839 

10.18101 

9.94171 

10.05829 

10.12332 

9.87008 

50 

20 

44 

11 

81854 

18146 

94197 

05803 

12:143 

87657 

49 

16 

48 

12 

81808 

18132 

94222 

05778 

12354 

87040 

48 

12 

52 

13 

81882 

18118 

94248 

05572 

12365 

87035 

47 

8 

50 

14 

81897 

18103 

94273 

05727 

12376 

87024 

40 

4 

45 

15 

9.81911 

10.18089 

9.94299 

10.05701 

10.12387 

9.87013 

45 

15 

4 

10 

81926 

18074 

94324 

05076 

12399 

87001 

44 

50 

8 

17 

81940 

18000 

94350 

05650 

12410 

87590 

43 

52 

12 

18 

81955 

18045 

94375 

05625 

12421 

87679 

42 

48 

10 

19 

81909 

18031 

94401 

05599 

12432 

87508 

41 

44 

20 

20 

9.81983 

10.18017 

9.94426 

10.05574 

10.12443 

9.87557 

40 

40 

24 

21 

81998 

18002 

94452 

05548 

12454 

87540 

39 

36 

28 

22 

82012 

17988 

94477 

05523 

12465 

87 53«j 

38 

32 

32 

23 

82020 

17974 

9-1503 

05497 

12470 

87524 

37 

28 

36 

24 

82ii41 

17959 

94528 

05472 

12487 

87513 

36 

24 

40 

25 

9.82055 

10.17945 

9.94554 

10.05446 

10.12499 

9.8750L 

35 

20 

44 

20 

82009 

17931 

94579 

05421 

12510 

87490 

34 

16 

48 

27 

82084 

17910 

94604 

05390 

12521 

87 479 

33 

12 

52 

28 

82098 

17902 

94030 

05370 

12532 

87408 

32 

8 

50 

29 

82112 

17888 

94655 

05345 

12543 

87457 

31 

4 

46 

30 

9.82120 

10.17874 

9.9408 L 

10.05319 

10.12554 

9.87440 

30 

14 

4 

31 

82141 

17859 

94706 

05294 

12500 

87434 

29 

50 

8 

32 

82155 

17845 

91732 

05268 

12577 

87423 

28 

52 

12 

33 

82109 

17831 

94757 

05243 

12588 

87412 

27 

48 

10 

34 

82Ln4 

17810 

94783 

05217 

12599- 

87401 

26 

44 

20 

35 

9.82198 

10.17802 

9.9480S 

10.05192 

10.12010 

9.87390 

25 

40 

24 

30 

82212 

177'S 

94834 

05166 

12022 

87378 

24 

30 

28 

37 

82220 

17774 

94859 

05141 

12033 

87307 

23 

32 

32 

38 

82240 

17700 

91884 

05110 

12044 

87350 

22 

28 

36 

39 

82255 

17745 

94910 

05090 

12055 

87345 

21 

24 

40 

40 

9.82209 

10.17731 

9.9493$ 

10.05065 

10.12666 

9.87334 

20 

20 

44 

41 

82283 

17717 

94901 

05039 

12078 

87322 

19 

10 

48 

42 

82297 

17703 

94986 

05014 

12089 

87311 

IS 

12 

52 

43 

82311 

17089 

95012 

04088 

12700 

87300 

17 

8 

50 

4-1 

82320 

17074 

95037 

04903 

12712 

87288 

10 

4 

47 

45 

9.82340 

10.17000 

9.95002 

10.04938 

10.12723 

9.87277 

15 

13 

4 

40 

82354 

17040 

950S8 

04912 

12734 

87206 

14 

56 

8 

47 

82308 

17032 

95113 

04887 

12745 

87255 

13 

52 

12 

48 

82382 

17018 

95139 

04801 

12757 

87243 

12 

48 

10 

49 

82396 

17004 

95104 

04830 

12708 

87232 

11 

44 

20 

50 

9.82410 

10.17540 

9.95190 

10.04810 

10.12779 

9.87221 

10 

40 

24 

51 

82424 

17570 

95215 

04785 

12791 

87209 

9 

36 

28 

52 

82434 

17501 

95240 

04700 

12802 

87198 

8 

32 

32 

53 

82453 

17547 

95206 

04734 

12813 

87187 

7 

28 

30 

54 

82407 

17533 

95291 

04709 

12825 

87175 

6 

24 

40 

55 

9.82481 

10.17519 

9.95317 

10.04083 

10.12836 

9.87104 

5 

20 

44 

50 

82495 

17505 

95342 

04058 

12S47 

87153 

4 

16 

48 

57 

82509 

17491 

95368 

04632 

12859 

87141 

3 

12 

52 

58 

82523 

17477 

95393 

04607 

12870 

87130 

2 

8 

50 

59 

82537 

17403 

95418 

04582 

* 1288 1 

87119 

1 

4 

48 

00 

82551 

17449 

954-14 

04556 

12893 

871C7 

0 

44 

M.S. 

8 h 

M 

131' 

Coaiue. 

3 

Seeaut. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

CO t* 
O 

M.S. 

3 h 























Logarithms Trigonometric. 241 


2 h 

42° 



Logarithms. 


137° 

9 h 

M.S. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

M.S. 

48 

0 

9.82551 

10.17449 

9.95444 

10.04556 

10.12893 

9.87107 

60 

13 

4 

1 

82565 

17435 

95469 

04531 

12904 

87096 

59 

56 

8 

2 

82579 

17421 

95495 

04505 

12915 

87085 

58 

52 

12 

3 

82593 

17407 

95520 

04480 

12927 

87073 

57 

48 

16 

4 

82607 

17393 

95545 

04455 

12938 

87062 

56 

44 

20 

5 

9.82621 

10.17379 

9.95571 

10.04429 

10.12950 

9.87050 

55 

40 

24 

6 

82635 

17365 

95596 

04404 

12961 

87039 

54 

36 

28 

7 

82649 

17351 

95622 

04378 

12972 

87028 

53 

32 

32 

8 

82663 

17337 

95647 

04353 

12984 

87016 

52 

28 

36 

9 

82677 

17323 

95672 

04328 

12995 

87005 

51 

24 

40 

10 

9.82691 

10.17309 

9.95698 

10.04302 

10.13007 

9.86993 

50 

20 

44 

11 

82705 

17295 

95723 

04277 

13018 

86982 

49 

16 

48 

12 

82719 

17281 

9574S 

04252 

13030 

86970 

48 

12 

52 

13 

82733 

17267 

95774 

04226 

13041 

86959 

47 

8 

56 

14 

82747 

17253 

95799 

04201 

13053 

86947 

46 

4 

49 

15 

9.82761 

10.17239 

9.95825 

10.04175 

10.13064 

9.86936 

45 

11 

4 

16 

82775 

17225 

95850 

04150 

13076 

86924 

44 

56 

8 

17 

82788 

17212 

95875 

04125 

13087 

86913 

43 

52 

12 

18 

82802 

17198 

95901 

04099 

13098 

86902 

42 

48 

16 

19 

82816 

171S4 

95926 

04074 

13110 

86890 

41 

44 

20 

20 

9.82830 

10.17170 

9.95952 

10.04048 

10.13121 

9.86879 

40 

40 

24 

21 

82844 

17156 

95977 

04023 

131:33 

86867 

39 

36 

28 

22 

82858 

17142 

96002 

03998 

13145 

86855 

38 

32 

32 

23 

82S72 

17128 

96028 

03972 

13156 

86844 

37 

28 

36 

24 

82885 

17115 

96053 

03947 

13168 

86832 

36 

24 

40 

25 

9.82899 

10.17101 

9.96078 

10.03922 

10.13179 

9.86821 

35 

20 

44 

26 

82913 

17087 

96104 

03896 

13191 

86809 

34 

16 

48 

27 

82927 

17073 

96129 

03871 

13202 

86798 

33 

12 

52 

28 

82941 

17059 

96155 

03845 

13214 

86786 

32 

8 

56 

29 

82955 

17045 

96180 

03820 

13225 

86775 

31 

4 

50 

30 

9.82968 

10.17032 

9.96205 

10.03795 

10.13237 

9.86763 

30 

10 

4 

31 

82982 

17018 

96231 

03769 

13248 

86752 

29 

56 

8 

32 

82996 

17004 

96256 

03744 

13260 

86740 

28 

52 

12 

33 

83010 

16990 

96281 

03719 

13272 

86728 

27 

48 

16 

34 

83023 

16977 

96307 

03693 

13283 

86717 

26 

44 

20 

35 

9.83037 

10.16963 

9.96332 

10.03668 

10.13295 

9.86705 

25 

40 

24 

36 

83051 

16949 

96357 

03643 

13306 

86694 

24 

36 

28 

37 

83065 

16935 

96383 

03617 

13318 

86682 

23 

32 

32 

38 

83078 

16922 

96408 

03592 

13330 

86670 

22 

28 

36 

39 

83032 

16908 

96433 

03567 

13341 

86659 

21 

24 

40 

40 

9.83106 

10.16894 

9.96459 

10.03541 

10.13353 

9.86647 

20 

20 

44 

41 

83120 

16880 

96484 

03516 

13365 

86635 

19 

16 

48 

42 

83133 

16867 

96510 

03490 

13376 

86624 

18 

12 

52 

43 

83147 

16853 

96535 

03465 

13388 

86612 

17 

8 

56 

44 

83161 

16839 

96560 

03440 

13400 

86600 

16 

4 

51 

45 

9.83174 

10.16826 

9.96586 

10.03414 

10.13411 

9.86589 

15 

9 

4 

46 

83188 

16812 

96611 

03389 

13423 

86577 

14 

56 

8 

47 

83202 

16798 

96636 

03364 

13435 

86565 

13 

52 

12 

48 

83215 

16785 

96662 

03338 

13446 

86554 

12 

48 

16 

49 

83229 

16771 

96687 

03313 

13458 

86542 

11 

44 

20 

50 

9.83242 

10.16758 

9.96712 

10.03288 

10.13470 

9.86530 

10 

40 

24 

51 

83256 

16744 

96738 

03262 

13482 

86518 

9 

36 

28 

52 

83270 

16730 

96763 

03237 

13493 

86507 

8 

32 

32 

53 

83283 

16717 

90788 

03212 

13505 

86495 

7 

28 

36 

54 

83297 

16703 

96814 

03186 

13517 

86483 

6 

24 

40 

55 

9.83310 

10.16690 

9.96839 

10.03161 

10.13528 

9.86472. 

5 

20 

44 

56 

83324 

16676 

96864 

03136 

13540 

86460 

4 

16 

48 

57 

83338 

16662 

90890 

03110 

13552 

86448 

3 

12 

52 

58 

83351 

16649 

96915 

03085 

13564 

86436 

2 

8 

56 

59 

83365 

16635 

96940 

03060 

13575 

86425 

1 

4 

53 

60 

83378 

16622 

96966 

03034 

13587 

86413 

0 

8 

MS. 

M 

Cosine. 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

M.S. 

8 h 

132° 






47° 

3 h j 


16 



















242 Logarithms Trigonometric. 


2 h 

43° 



Logarithms. 


136° 

9 h 

M.s. 

M 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

-M.S 

53 

0 

9.83378 

10.16622 

9.96906 

10.03034 

10.13587 

9.86413 

60 

8 

4 

1 

83392 

16608 

90991 

03009 

13599 

86401 

59 

56 

8 

2 

83405 

16595 

97016 

02984 

13611 

86389 

58 

52 

12 

3 

83419 

16581 

97042 

02953 

13623 

86377 

57 

48 

16 

4 

83432 

16568 

97007 

02933 

13634 

Sti366 

56 

44 

20 

5 

9.83446 

10.16554 

9.97092 

10.02908 

10.13046 

9.86354 

55 

40 

24 

6 

83459 

16541 

97118 

02882 

13658 

86342 

54 

36 

28 

7 

83473 

16527 

97143 

02857 

13670 

86330 

53 

32 

32 

8 

83486 

16514 

97168 

02832 

136S2 

86318 

52 

28 

36 

9 

83500 

16500 

97193 

02807 

13694 

86306 

51 

24 

40 

10 

9.83513 

10.10487 

9.97219 

10.02781 

10.13705 

9.86295 

50 

20 

44 

11 

83527 

16473 

97244 

02756 

13717 

86283 

49 

16 

48 

12 

83540 

16400 

97269 

02731 

13729 

86271 

48 

12 

62 

13 

83554 

16446 

97295 

02705 

13741 

86259 

47 

8 

60 

14 

83567 

16433 

97320 

02680 

13763 

86247 

46 

4 

53 

15 

9.83581 

10.16419 

9.97345 

10.02655 

10.13765 

9.86235 

45 

7 

4 

16 

83594 

16406 

97371 

02629 

13777 

86223 

4-1 

56 

8 

17 

83608 

16392 

97396 

02604 

13789 

86211 

43 

52 

12 

18 

83621 

16379 

97421 

02579 

13800 

86200 

42 

48 

16 

19 

83634 

16366 

97447 

02563 

13812 

86188 

41 

44 

20 

20 

9.83648 

10.16352 

9.97472 

10.02528 

10.13824 

9.S6176 

40 

40 

24 

21 

83661 

16339 

97497 

02503 

13836 

86164 

39 

36 

28 

22 

83674 

16326 

97523 

02477 

13848 

86152 

38 

32 

32 

23 

83688 

16312 

97548 

02452 

13860 

86140 

37 

28 

36 

24 

83701 

16299 

97573 

02427 

13872 

86128 

36 

24 

40 

25 

9.83715 

10.16285 

9.97698 

10.02402 

10.13884 

9.86116 

35 

20 

44 

26 

83728 

16272 

97624 

02376 

13896 

86104 

34 

16 

48 

27 

83741 

16259 

97649 

02351 

13908 

86092 

33 

12 

52 

28 

83755 

16245 

97674 

02326 

13920 

86080 

32 

8 

56 

29 

83768 

16232 

97700 

02300 

13932 

86068 

31 

4 

54 

30 

9.83781 

10.16219 

9.97725 

10.02275 

10.13944 

9.86056 

30 

6 

4 

31 

83795 

16205 

97750 

02250 

13956 

86044 

29 

56 

8 

32 

83808 

16192 

97776 

02224 

13968 

86032 

28 

52 

12 

33 

83821 

16179 

97801 

02199 

13980 

86020- 

27 

48 

16 

34 

83834 

16166 

97826 

02174 

13992 

86008 

26 

44 

20 

35 

9.83848 

10.16152 

9.97851 

10.02149 

10.14004 

9.85996 

25 

40 

24 

36 

83861 

16139 

97877 

02123 

14016 

85984 

24 

36 

28 

37 

83874 

16126 

97902 

02098 

14028 

85972 

23 

32 

32 

38 

83887 

16113 

97927 

02073 

14040 

85960 

22 

28 

36 

39 

83901 

16099 

97953 

02047 

14052 

85948 

21 

24 

40 

40 

9.83914 

10.16086 

9.97978 

10.02022 

10.14064 

9.85936 

20 

20 

44 

41 

83927 

16073 

98003 

01997 

14076 

85924 

19 

16 

48 

42 

83940 

16060 

98029 

01971 

14088 

85912 

18 

12 

52 

43 

83954 

16046 

98054 

01946 

14100 

85900 

17 

8 

66 

44 

83967 

16033 

98079 

01921 

14112 

85888 

16 

4 

55 

45 

9.83980 

10.16020 

9.98104 

10.01896 

10.14124 

9.85876 

15 

5 

4 

46 

83993 

16007 

98130 

01870 

14136 

85864 

14 

56 

8 

47 

84006 

15994 

98155 

01845 

14149 

85851 

13 

52 

12 

48 

84020 

15980 

98180 

01820 

14161 

85839 

12 

48 

16 

49 

84033 

15967 

98206 

01794 

14173 

85827 

11 

44 

20 

50 

9.84046 

10.15954 

9.98231 

10.01769 

10.14185 

9.85815 

10 

40 

24 

51 

84059 

15941 

98266 

01744 

14197 

85S03 

9 

36 

28 

52 

84072 

15928 

98281 

01719 

14209 

85791 

8 

32 

32 

63 

84085 

15915 

98307 

01693 

14221 

85779 

7 

28 

36 

54 

84098 

15902 

98332 

01668 

14234 

85766 

6 

24 

40 

55 

9.84112 

10.15888 

9.98357 

10.01643 

10.14246 

9.85754 

5 

20 

44 

56 

84125 

15875 

983S3 

01617 

14258 

85742 

4 

16 

48 

57 

84138 

15862 

98408 

01592 

14270 

85730 

3 

12 

52 

58 

84151 

15849 

98433 

01567 

14282 

85718 

2 

8 

56 

59 

84164 

15836 

98458 

01542 

14294 

85706 

1 

4 

5(> 

60 

84177 

15823 

98484 

01516 

14307 

85693 

0 

4 

M. S. 

M 

133 

Cosine. 

0 

Secant. 

Cotangent 

Tangent. 

Cosecant. 

Sine. 

M 

40° 

M.S. 

3“ 

-s 
















Logarithms Trigonometric, 


243 


Ob 

M 

44° 



Logarithms. 


135° 

M.S. 

M 

Siue. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

M 

56 

0 

9.84177 

10.15823 

9.98484 

10.01516 

10.14307 

9.85093 

GO 

4 

1 

84190 

15810 

98509 

01491 

14319 

85081 

59 

8 

2 

84203 

15797 

98534 

01406 

14331 

86669 

58 

12 

3 

84216 

15784 

98560 

01440 

14343 

85657 

57 

16 

4 

84229 

15771 

98585 

01415 

14355 

85045 

50 

20 

5 

9.84242 

10.15758 

9.98610 

10.01390 

10.14308 

9.85632 

55 

24 

6 

84255 

15745 

98635 

01365 

14380 

85620 

54 

28 

7 

84269 

15731 

98661 

01339 

14392 

85608 

53 

32 

8 

84282 

15718 

98636 

01314 

14404 

85590 

52 

36 

9 

84295 

15705 

93711 

01289 

14417 

85583 

51 

40 

10 

9.84308 

10.15092 

9.98737 

10.01263 

10.14429 

9.85571 

50 

44 

11 

84321 

15679 

93702 

01238 

14441 

85559 

49 

4S 

12 

84334 

15066 

98787 

01213 

14453 

85547 

43 

52 

13 

84347 

15653 

98812 

01188 

14466 

85534 

47 

56 

14 

84360 

15640 

98838 

01162 

14478 

85522 

46 

57 

15 

9.84373 

10.15027 

9.98863 

10.01137 

10.14490 

9.85510 

45 

4 

16 

S4385 

15615 

98388 

01112 

14503 

85497 

44 

8 

17 

84398 

15602 

98913 

01087 

14515 

85485 

43 

12 

18 

84411 

15589 

93939 

01061 

14527 

85473 

42 

16 

19 

84424 

15576 

93964 

01036 

14540 

85400 

41 

20 

20 

9.84437 

10.15563 

9.98989 

10.01011 

10.14552 

9.85448 

40 

24 

21 

84450 

15550 

99015 

00985 

14564 

85436 

39 

28 

22 

84463 

15537 

99040 

00960 

14577 

85423 

38 

32 

23 

84476 

15524 

99065 

00935 

14589 

85411 

37 

36 

24 

84489 

15511 

99090 

00910 

14601 

85399 

36 

40 

25 

9.84502 

10.15498 

9.99116 

10.00884 

10.14614 

9.85380 

35 

44 

26 

84515 

15485 

99141 

00859 

14626 

85374 

34 

48 

27 

84528 

15472 

99166 

00834 

14639 

85301 

33 

52 

28 

84540 

15460 

99191 

00309 

14651 

85349 

32 

56 

29 

84553 

15447 

99217 

00783 

14663 

85337 

31 

58 

30 

9.84506 

10.15434 

9.99242 

10.00758 

10.14676 

9.85324 

30 

4 

31 

84579 

15421 

99267 

00733 

14688 

85312 

29 

8 

32 

84592 

15408 

99293 

00707 

14701 

85299 

28 

12 

33 

84(505 

15395 

99318 

00682 

14713 

86287 

27 

16 

34 

84618 

15382 

99343 

00657 

14726 

85274 

26 

20 

35 

9.84030 

10.15370 

9.99308 

10.00032 

10.14738 

9.85262 

25 

24 

36 

84643 

15357 

99394 

00600 

14750 

85250 

24 

28 

37 

84056 

15344 

99419 

00581 

14763 

85237 

23 

32 

38 

84669 

15331 

99444 

00556 

14775 

85225 

22 

36 

39 

84082 

15318 

99469 

00531 

14788 

85212 

21 

4(> 

40 

9.84694 

10.15306 

9.99495 

10.00505 

10 14800 

9.S5200 

20 

44 

41 

84707 

15293 

99520 

00480 

14813 

85187 

19 

48 

42 

84720 

15280 

99545 

03455 

14825 

85175 

18 

52 

43 

84733 

15267 

99570 

00430 

14838 

85102 

17 

56 

44 

84745 

15255 

99596 

00404 

14850 

85150 

16 

59 

45 

9.84758 

10.15242 

9.99621 

10.00379 

10.14863 

9 85137 

15 

4 

46 

84771 

15229 

99046 

00354 

14S75 

85125 

14 

8 

47 

84784 

15216 

99672 

00328 

14888 

85112 

13 

12 

48 

84796 

15204 

99097 

00303 

14900 

85100 

12 

16 

49 

84809 

15191 

99722 

00278 

14913 

85087 

11 

20 

50 

9.84822 

10.15178 

9.99747 

10.00253 

10.14926 

9.85074 

10 

24 

51 

84835 

15165 

99773 

00227 

14938 

85062 

9 

28 

52 

84847 

15153 

99798 

00202 

14951 

85049 

8 

32 

53 

84S00 

15140 

99823 

00177 

14963 

85037 

7 

36 

54 

84873 

15127 

99S48 

00152 

14976 

85024 

6 

40 

55 

9.84885 

10.15115 

9.99874 

10.00126 

1014988 

9.85012 

5 

44 

56 

84898 

15102 

99899 

00101 

15001 

84999 

4 

48 

57 

84911 

15089 

99924 

00070 

15014 

84986 

3 

52 

58 

84923 

15077 

99949 

00051 

15026 

84974 

2 

56 

59 

84936 

15064 

99975 

00025 

15039 

8496 L 

1 

GO 

60 

84949 

15051 

10.00000 

00000 

15051 

84949 

0 

M.S. 

8*‘| 

M 

134 

Cosine. 

3 

Secant. 

Cotaugent. 

Tangent. 

Cosecant. 

Siue. 

M 

45° 


9 ^ 

M.S. 


0 

M. S. 

3 “ 


l-CK>KCOW*->t-*-Cr>C>'>ii 

#-ooti©c^octvCi c>f“0cK,o^i^00u.0sct^ccivCiOi*»(»[i© <y iH»t0®cif‘00tv0!Ctt‘00b:ci w it‘00»005Oi»-aKioC'i( i '(xiccir 
















214 


Explanation of tile Tables. 


EXPLANATION OF THE TABLES. 

The outer columns in the trigonometrical tables contain the angle in time of 
hours, minutes and seconds, corresponding to the same angle in degrees and min¬ 
utes in the next columns. The hour is noted at the top and bottoirr, the minutes 
in black, and the seconds in ordinary figures. 

To find the Logarithm and Natural Line for Seconds exceeding 

Minutes of a Degree. 

Example 1. Find the logarithm for sin. 38° 47' 55". 


diff. 15. 


flog. sin. 38° 48'= 9.706991 
From table, | .f „ 38° 47'= 9.796841 

Correction, 15 X 55 : 60 = -4-14 nearly. 

The required log. sin. 38° 47' 55" = 9.79698 


In practice, the difference is subtracted direct from the tables. 
Example 2. Find the natural cos. 43° 29' 19". 


From table, cos. 43° 29' = 0.72557 
Correction, 20 X 19 : 60 = — 6 n early. 

The required cos. 43° 29' 19" = 0.72551 

The correction is added when the function is increasing, and subtracted when 
decreasing. 

To find the Angle corresponding to a given Logarithm or Nat¬ 
ural Line. 


Example 3. Log. sin. = 9.56429. Required the angle. 

From table, j lo ~ sin ‘ 2l ° 31 ' = 956440 } diff. 32. 

I « “ 21° 30' = 9.56408 < 

The angle required, “ “ 21° 30'29"= 9.56429} “ 2L 

Correction, 21 X 60 : 32 = 29 seconds nearly. 


Example 4. Cosine = 0.35254. Required the angle. 

... f cos. 69° 22' =0.352391 0 - 

From table, •< \ dm. 2 /. 

<• “ 69° 21' = 0.35266 < 

The required angle, “ 69° 21'27" = 0.35254 j 4 *" 

Correction, 12 X 60 : 27 = 27 seconds, nearly. 


Conversion of Minutes and Seconds into Decimals of a Degree 

or of an Hour. 


M. 

Decimal. 

M. 

Decimal. 

M. 

Decimal. 

S. 

Decimal. 

a. 

Decimal. 

a. 

Decimal. 

1 

.016666 

21 

.350000 

41 

.683333 

1 

.000277 

21 

.005833 

41 

.011388 

2 

.033333 

22 

.366666 

42 

.700000 

2 

.000565 

22 

.006111 

42 

.011666 

3 

.050000 

23 

.383333 

43 

.716666 

3 

.000833 

23 

.006388 

43 

.011944 

4 

.066666 

24 

.400000 

44 

.733333 

4 

.001111 

24 

.006666 

44 

.01222*2 

5 

.083333 

25 

.416666 

45 

.750000 

5 

.001388 

25 

006944 

45 

.012500 

6 

.100000 

26 

.433333 

46 

.766666 

6 

.001666 

26 

.007222 

46 

.012777 

7 

.116666 

27 

.450000 

47 

.783333 

7 

.001944 

27 

.007500 

47 

.013055 

8 

.133333 

28 

.466666 

48 

.800000 

8 

.002222 

28 

.007777 

48 

.013333 

9 

.150000 

29 

.483333 

49 

.816666 

9 

.002500 

29 

.008055 

49 

.013611 

10 

.166666 

30 

.500000 

50 

833333 

10 

.002777 

30 

.008333 

50 

.013888 

11 

.183333 

31 

.516000 

51 

.850000 

11 

.003055 

31 

.008611 

51 

.014166 

12 

.200000 

32 

.533333 

52 

.866666 

12 

.003333 

32 

.008888 

52 

.014444 

13 

.216606 

33 

.550000 

53 

.883333 

13 

.003611 

33 

.009166 

53 

.014722 

14 

.233333 

34 

.566666 

54 

.900000 

14 

.003888 

34 

.009444 

54 

.015000 

15 

.250000 

35 

.583333 

55 

.916666 

15 

.004166 

35 

.009722 

55 

.015277 

16 

.266666 

36 

.600000 

56 

.933333 

16 

.004444 

36 

.010000 

56 

.015555 

17 

.283333 

37 

.616666 

57 

.950000 

17 

.004722 

37 

.010277 

57 

.015833 

18 

.300000 

38 

.633333 

58 

.906666 

18 

.005000 

38 

.010555 

58 

.016111 

19 

.316666 

39 

.650000 

59 

.983333 

19 

.005277 

39 

.010833 

59 

.01G388 

20 

.333333 

40 

.666666 

60 

1.000000 

20 

.005555 

40 

.011111 

60 

.010666 




























Natural Lines 


245 


o h 

0° 

Natural Trigonometrical Functions. 

179° 

ll h 

M.S. 

M. 

Sine. 

Vrs.Cos. 

jCosec’nte 

Tang. 

Cotang. 

Secante. 

Vrs.Sin. 

Cosine. 

M. 

M.S. 

0 

0 

.00000 

1.0000 

Infinite 

.00000 

Infinite 

1.0000 

.00000 

1.0000 

60 

60 

4 

1 

. 0029 

.99971 

3437.7 

. 002!) 

3437.7 

.0000 

. oooo 

.0000 

59 

56 

8 

2 

. 0058 

. 9942 

1718.9 

. 0058 

1718.9 

.0000 

. oooo 

.0000 

58 

52 

12 

3 

. 0087 

. 9913 

1145.9 

. 0087 

1145.9 

.0000 

. oooo 

.0000 

67 

48 

16 

4 

. 0116 

. 9*84 

859.44 

. 0116 

859.44 

.0000 

. oooo 

.0000 

56 

44 

20 

5 

.00145 

.99854 

687.55 

.00145 

687.55 

1.0000 

.00000 

1.0000 

55 

40 

24 

0 

. 0174 

. 9825 

572.96 

. 0174 

572.96 

.0000 

. oooo 

.0000 

54 

36 

2^ 

7 

. 0204 

. 9796 

491.11 

. 0204 

491.11 

.0000 

. oooo 

.0000 

53 

32 

32 

8 

. 0233 

. 9767 

429.72 

. 0233 

429.72 

.0000 

. oooo 

.0000 

52 

28 

36 

9 

. 0262 

. 9738 

3*1.97 

. 0262 

381.97 

.0000 

. oooo 

.0000 

51 

24 

40 

10 

.00291 

.99709 

343 77 

.00291 

313.77 

1.0000 

.ooooo 

.99999 

50 

20 

44 

11 

. 0320 

. 9680 

312.52 

. 0320 

312.52 

.0000 

. oooo 

. 9999 

49 

16 

48 

12 

. 0349 

. 9651 

286.18 

. 0349 

286 48 

.0000 

. 0001 

. 99J9 

48 

12 

52 

13 

. 0378 

. 9622 

64.44 

. 0378 

64.44 

.0000 

. 0001 

. 6939 

47 

8 

56 

14 

. 0407 

. 9593 

45.55 

. 04o7 

45 55 

.0000 

. 0001 

. 9909 

46 

4 

1 

15 

.00436 

.99564 

'229.18 

.00436 

229.18 

1.0000 

.00001 

.99999 

45 

59 

4 

16 

. 0465 

. 9534 

14.86 

. 0465 

14.86 

.0000 

. 0001 

. 9999 

44 

56 

8 

17 

. 0494 

. 9505 

02.22 

. 0494 

02.22 

.0000 

. 0001 

. 9999 

43 

52 

12 

18 

. 0524 

. 9476 

190.99 

. 0524 

190.98 

.0000 

. 0001 

. 9999 

42 

48 

16 

19 

. 0553 

. 9447 

180.93 

. 0553 

180.93 

.0000 

. 0001 

. 9998 

41 

44 

20 

20 

.00582 

.99418 

171.89 

.00582 

171.88 

1.0000 

.00002 

.99998 

40 

40 

24 

21 

. 0611 

. 9389 

63.70 

. 0611 

6(1.70 

.0000 

. 0002 

. 9998 

39 

36 

28 

22 

. 0640 

. 9360 

56.26 

. 0610 

56.26 

.0000 

. 0002 

. 9998 

38 

32 

32 

23 

. 0669 

. 9331 

49.47 

. 0669 

49.46 

.0000 

. 0002 

. 9998 

37 

28 

36 

24 

. 0698 

. 9302 

43.24 

. 0698 

43.24 

.oooo 

. 0002 

. 9997 

36 

24 

40 

25 

.00727 

.99273 

137.51 

.00727 

137.51 

1.0000 

.00003 

.99997 

35 

20 

44 

26 

. 0766 

. 9244 

32.22 

. 0756 

32.22 

.0000 

. 0003 

. 9997. 

34 

16 

48 

27 

. 0785 

. 9215 

27.32 

. 0785 

27.32 

.0000 

. 0003 

. 9997 

33 

12 

52 

28 

. 0814 

. 9185 

22.78 

. 0814 

22.77 

.0000 

. 0003 

. 9997 

32 

8 

56 

29 

. 0843 

. 9156 

18.54 

. 0844 

18.54 

.0000 

. 0003 

. 9996 

31 

4 

2 

30 

.00873 

.99127 

114.59 

.00873 

114.59 

1.0000 

.00004 

.99996 

30 

58 

4 

31 

. 0902 

. 9098 

10.90 

. 0902 

10.89 

.0000 

. 0004 

. 9996 

29 

56 

8 

32 

. 0931 

. 9069 

07.43 

. 0931 

07.43 

.0000 

. 0004 

. 9996 

28 

52 

12 

33 

. 0960 

. 9040 

04.17 

. 0960 

04.17 

.0000 

. 0005 

. 9995 

27 

48 

16 

34 

. 0989 

. 9011 

01.11 

. 0989 

01.11 

.oooo 

. 0005 

. 9995 

26 

44 

20 

35 

.01018 

.98^82 

98.223 

.01018 

98.218 

1.0001) 

.00005 

.99995 

25 

40 

24 

36 

. 1047 

. 8953 

5.495 

. 1047 

5.489 

.0000 

. 0005 

. 9994 

24 

36 

28 

37 

. 1076 

. 8924 

2.914 

. 1076 

2.908 

.0000 

. 0006 

. 9994 

23 

32 

32 

38 

. 1105 

. 8895 

0.169 

. 1105 

0.463 

.0001 

. 0006 

. 9994 

22 

28 

86 

39 

. 1134 

. 8865 

8S.149 

. 1134 

88.143 

.0001 

. 0006 

. 9993 

21 

24 

40 

40 

.01163 

.98836 

85.946 

.01164 

85.940 

1.0001 

.00007 

.79993 

20 

20 

44 

41 

. 1193 

. 8807 

3.849 

. 1193 

3.843 

.0001 

. 0007 

. 9993 

19 

16 

48 

42 

. 1222 

. 8778 

1.853 

. 1222 

1.847 

.0001 

. 0007 

. 9992 

18 

12 

52 

43 

. 1251 

. 8749 

79.950 

. 1251 

79.943 

.0001 

. 0008 

. 9992 

17 

8 

56 

44 

. 1280 

. 8720 

78.133 

. 1280 

78.126 

.0001 

. 0008 

. 9992 

16 

4 

3 

45 

.01309 

.98691 

76.396 

.01309 

76.390 

1.0001 

.00008 

.999!) 1 

15 

57 

4 

46 

. 1338 

. 8662 

4.736 

. 1338 

4.729 

.0001 

. 0009 

. 9991 

14 

56 

8 

47 

. 1307 

. 8633 

3.146 

. 1367 

3.139 

.0001 

. 0009 

. 9991 

13 

52 . 

12 

48 

. 1396 

. 8604 

1.622 

. 1396 

1.615 

.0001 

. 0010 

. 9990 

12 

48 

16 

49 

. 1425 

. 8575 

0160 

. 1425 

0.153 

.0001 

. 0010 

. 9990 

11 

44 

20 

50 

.01454 

.98546 

68.757 

.01454 

68.750 

1.0001 

.00010 

.99989 

10 

40 

24 

51 

. 1483 

. 8516 

7.409 

. 1484 

7.402 

.0001 

. 0011 

. 9989 

9 

36 

28 

52 

. 1512 

. 8487 

6.113 

. 1513 

6.105 

.0001 

. 0011 

. 9988 

8 

32 

32 

53 

. 1542 

. 8458 

4.866 

. 1542 

4.858 

.0001 

. 0012 

. 9988 

7 

28 

36 

54 

. 1571 

. 8429 

3.664 

. 1571 

3.657 

.0001 

. 0012 

. 9988 

6 

24 

40 

55 

.01600 

.98400 

62.507 

.01600 

62.499 

1.0001 

.00013 

.99987 

5 

20 

44 

56 

. 1029 

. 8371 

1.391 

. 1629 

1.383 

.0001 

. 0013 

. 9987 

4 

16 

48 

57 

. 1658 

. 8342 

0.314 

. 1658 

0.306 

.0001 

. 0014 

. 9987 

3 

12 

52 

58 

. 1687 

. 8313 

59.274 

. 1687 

59.266 

.0001 

. 0014 

. 9986 

2 

8 

56 

59 

. 1716 

. 8284 

8.270 

. 1716 

8.261 

.0001 

. 0015 

. 9985 

1 

4 

4 

60 

. 1745 

. 8255 

7.299 

. 1745 

7.290 

.0001 

. 0015 

. 9985 

0 

56 

MS. 

6 h 

M. 

90° 

Cosine. 

Vrs.Sin. 

Secante. 

Cotang. (Tangent. 

Natural. 

Cosec’nt 

Vrs.Cos. 

Sine. 

M. 

89° 

M.S. 

5 b 


























246 


Natural Lines, 


o h 

1° 

Natural Trigonometrical Functions. 

o 

CO 

ll h 

M.S. 

M 

Sine. 

Yrs.Gos. 

Cosec’nte 

Tang. 

Cotang. 

Secante. 

Vrs.Sin 

Cosine. 

M 

M.S. 

A 

0 

.01745 

.98255 

57.299 

.01745 

57.290 

1.0001 

.00015 

.99985 

60 

5G 

4 

1 

. 1774 

. 8226 

50.359 

. 1775 

56.350 

.0001 

. 0016 

7 9984 

59 

56 

8 

2 

. 1803 

. S196 

55.450 

. 1804 

55.441 

.0002 

. 0016 

. 9984 

58 

52 

12 

3 

. 1832 

. 8167 

54.570 

. 1833 

54.561 

.0002 

. 0017 

. 9983 

67 

48 

16 

4 

. 1861 

. 8138 

53.718 

. 1862 

63.708 

.0002 

. 0017 

. 9983 

50 

44 

20 

5 

.01891 

.98109 

52891 

.01891 

52.882 

1.0002 

.00018 

.99982 

55 

40 

24 

6 

. 1920 

. 8080 

2.090 

. 1920 

2.081 

.0002 

. 0018 

. 9981 

54 

36 

28 

7 

. 1949 

. 8051 

1.313 

. 1949 

1.303 

.0002 

. 0019 

. 99S1 

53 

32 

32 

8 

. 1978 

. 8022 

0.558 

. 1978 

0.548 

.0002 

. 0019 

. 9980 

52 

28 

36 

9 

. 2007 

. 7993 

49.826 

. 2007 

49.816 

.0002 

. 0020 

. 9980 

51 

24 

40 

10 

.02036 

.97964 

49.114 

.02036 

40.104 

1.0002 

.00021 

.99979 

50 

20 

41 

11 

. 2065 

. 7935 

8.422 

. 2066 

8.412 

.0002 

. 0021 

. 9979 

49 

16 

48 

12 

. 2094 

. 7906 

7.750 

. 2095 

7.739 

.0002 

. 0022 

. 9978 

48 

12 

-62 

13 

. 2123 

. 7877 

7.096 

. 2124 

7.085 

.0002 

. 0022 

. 9977 

47 

8 

56 

14 

. 2152 

. 7847 

6.460 

. 2153 

6.449 

.0002 

. 0023 

. 9977 

46 

4 

5 

15 

.02181. 

.97818 

45>40 

.02182 

45.829 

1.0002 

.00024 

.99976 

45 

55 

4 

16 

. 2210 

. 7789 

5.237 

. 2211 

5.226 

.0002 

. 0024 

. 9975 

44 

56 

s 

17 

. 2240 

. 7760 

4.650 

. 2240 

4.638 

.0002 

. 0025 

. 9975 

43 

52 

12 

18 

. 2269 

. 7731 

4.077 

. 2269 

4.0G6 

.0002 

. 0026 

. 9974 

42 

48 

16 

19 

. 2298 

. 7702 

3.520 

. 2298 

3.508 

.0003 

. 0026 

. 9974 

41 

44 

20 

20 

.02327 

.97673 

42.976 

.02327 

42.964 

1.0003 

.00027 

.99973 

40 

40 

24 

21 

. 2356 

. 7644 

2.445 

. 2357 

2.433 

.0003 

. 0028 

. 9972 

39 

36 

28 

22 

. 2385 

. 7615 

1.92S 

. 2386 

1.916 

.0003 

. 0028 

. 9971 

38 

32 

32 

23 

. 2414 

. 7586 

1.423 

. 2415 

1.410 

.0003 

. 0029 

. 9971 

37 

28 

3( 

24 

. 2443 

. 7557 

0.930 

. 2444 

0.917 

.0003 

. 0030 

. 9970 

36 

24 

40 

25 

.02 472 

.97528 

40.448 

.02173 

40.436 

1.0003 

.00030 

.99969 

35 

20 

44 

26 

, . 2501 

. 7499 

39.978 

. 2502 

39.965 

.0003 

. 0031 

. 9969 

34 

16 

48 

27 

. 2530 

. 7469 

9.518 

. 2531 

9.506 

.0003 

. 0032 

. 996S 

33 

12 

52 

28 

. 2559 

. 7440 

9.069 

. 2560 

9.057 

.0003 

. 0033 

. 9967 

32 

8 

56 

29 

. 2589 

. 7411 

8.631 

. 2589 

8.618 

.0003 

. 0033 

. 9966 

31 

4 

0 

30 

.02618 

.97382 

38.201 

.02618 

38.188 

1.0003 

.00034 

.99966 

30 

5F 

4 

31 

. 2647 

. 7353 

7.782 

. 2648 

7.769 

.0003 

. 0035 

. 9965 

29 

56 

8 

32 

. 2676 

. 7324 

7.371 

. 2677 

7.358 

.0003 

. 0036 

. 9964 

28 

52 

12 

33 

. 2705 

. 7295 

6.969 

. 2706 

6.956 

.0004 

. 0036 

. 9963 

27 

48 

10 

34 

. 2734 

. 7266 

6.576 

. 2735 

6.663 

.0004 

. 0037 

. 9963 

26 

44 

20 

35 

.02763 

.97237 

36.191 

.02764 

36.177 

1.0004 

.00038 

.99962 

25 

40 

24 

36 

. 2792 

. 7208 

5.814 

. 2793 

5.800 

.0004 

. 0039 

. 9961 

24 

36 

28 

37 

. 2821 

. 7179 

5.445 

. 2822 

5.431 

.0004 

. 0040 

. 9960 

23 

32 

32 

38 

. 2850 

. 7150 

5.084 

. 2851 

6.069 

.0004 

. 0041 

. 9959 

22 

28 

36 

39 

. 2879 

. 7121 

4.729 

. 2880 

4.715 

.0004 

. 0041 

. 9958 

21 

24 

40 

40 

.02908 

.97091 

31.382 

.02910 

34.368 

1.0004 

.00042 

.99958 

20 

20 

44 

41 

. 2937 

. 7062 

4.042 

. 2939 

4.027 

.0004 

. 0043 

. 9957 

19 

16 

48 

42 

. 2967 

. 7033 

3.708 

. 2968 

3X93 

.0004 

. 0044 

. 9956 

18 

12 

52 

43 

. 2996 

. 7004 

3.381 

. 2997 

3.366 

.0004 

. 0045 

. 99551 

17 

8 

56 

44 

. 3025 

. 6975 

3.060 

. 3026 

3.045 

.0004 

. 0046 

. 9954 

16 

4 

7 

45 

.03054 

.96946 

32.745 

.03055 

32.730 

1.0005 

.00046 

.99953 

15 

53 

4 

46 

. 3083 

. 9692 

2.437 

. 3084 

2.421 

.0005 

. 0047 

. 9952 

14 

56 

8 

47 

. 3112 

. 6888 

2.134 

. 3113 

2.118 

.0005 

. 0048 

. 9951 

13 

52 

12 

48 

. 3141 

. 6859 

1.836 

. 3143 

1.820 

.0005 

. 0049 

. 9951 

12 

48 

16 

49 

. 3170 

. 6830 

1.544 

. 3172 

1.528 

.0005 

. 0050 

. 9950 

11 

44 

20 

60 

.03199 

.96801 

31.257 

.03201 

31.241 

1.0005 

.00051 

.99949 

10 

40 

24 

51 

. 8228 

. 6772 

0.976 

. 3230 

0.960 

.0005 

. 0052 

. 9948 

9 

36 

28 

52 

. 3257 

. 6743 

0.699 

. 3259 

0.683 

.0005 

. 0053 

. 9947 

8 

32 

32 

53 

. 3286 

. 6713 

0.428 

. 3288 

0.411 

.0006 

. 0054 

. 9946 

7 

28 

36 

54 

. 3315 

. 6684 

0.161 

. 3317 

0.145 

.0005 

. 0055 

. 9045 

6 

24 

40 

55 

.03344 

.96655 

29.899 

.03346 

29.882 

1.0005 

.00056 

.99944 

5 

20 

41 

56 

. 3374 

. 6626 

9.641 

• 3»i7 o 

9.624 

.0006 

. 0057 

. 9943 

4 

16 

48 

67 

. 3403 

. 6597 

9.38S 

. 3405 

9.371 

.0006 

. 0058 

. 9942 

3 

12 

52 

58 

. 3432 

• 6568 

9.139 

. 3434 

9.122 

.0006 

. 0059 

. 9941 

2 

8 

66 

59 

. 3461 

. 6539 

8.894 

. 3463 

8.877 

.0006 

. 0060 

. 9940 

1 

4 

8 

60 

. 3490 

. 6510 

8.654 

. 3492 

8.636 

.0006 

. 0061 

. 9939 

0 

53 

M.S. 

M 

Cosine. 

Vrs.Sin! 

Secante. 

Cotaug.iTnngent. 

Cosec’ut 

Vrs.Cos 

Sine. 

M 

M.S. 





Natural. 




88° 

5 h 























Natural Lines. 


247 


o h 

2° 

Natural Trigonometrical Functions. 

177° 

ll h 

M.S. 

M 

Sine. 

Vrs.Cos. 

Cosec'nte 

Tang. 

Cotang. 

Secante. 

Vrs.Sin 

Cosine. 

M 

M.S. 

8 

0 

.03490 

.90510 

28.054 

.03492 

28 030 

1.0006 

.00001 

.99939 

00 

52 

4 

1 

. 3519 

. 6481 

8.417 

. 3521 

8.399 

.0000 

. 0062 

. 9938 

69 

50 

8 

2 

. 3548 

. 6452 

8.184 

. 3550 

s.yio 

7.937 

.0006 

. 0003 

. 9937 

58 

52 

12 

3 

4 

. 3577 

. 6423 

7.955 

. 3579 

.0006 

. 0064 

. 9930 

57 

48 

10 

. 3000 

. 6394 

7.730 

. 3608 

7.712 

.oooo 

. 0065 

. 9935 

50 

44 

2*> 

5 

.03035 

.90305 

27.508 

.03638 

27.490 

1.0007 

.00000 

.99934 

55 

40 

24 

6 

. 3001 

. 6330 

7.290 

. 3007 

7.271 

.0007 

. 0007 

. 9933 

54 

30 

28 

7 

. 3093 

. 0300 

7.075 

. 3696 

7.056 

.0007 

. 0008 

. 9932 

53 

32 

32 

33 

8 

. 3722 

. 6277 

0.864 

. 3725 

6.845 

.0007 

. 0069 

. 9931 

52 

28 

9 

. 3751 

. 0248 

6.655 

. 3764 

6.037 

.0007 

. 0070 

. 9930 

51 

24 

40 

10 

.03781 

.90219 

20.150 

.03783 

26.432 

1.0007 

.00071 

.99928 

50 

20 

41 

11 

. 3810 

. 0190 

6 249 

. 3812 

6.230 

.0007 

. 0073 

. 9927 

49 

10 

48 

12 

. 3839 

. 6101 

6.050 

. 3812 

0.031 

.0007 

. 0074 

. 9926 

48 

12 

52 

13 

. 3808 

. 0132 

5.854 

. 3871 

5.835 

.0007 

. 0075 

. 9925 

47 

8 

50 

14 

. 3897 

. 6103 

5.661 

. 3900 

5.042 

.0008 

. 0076 

. 9924 

46 

4 

9 

15 

.03920 

.90074 

25.471 

.03929 

25.452 

1.0008 

.00077 

.99923 

45 

51 

4 

10 

. 3955 

. 0045 

5.284 

. 3958 

5.264 

.0008 

. 0078 

. 9922 

44 

50 

8 

12 

17 

. 3984 

. 6010 

5.100 

. 3987 

5.080 

.0008 

. 0079 

. 9921 

43 

52 

18 

. 4013 

. 5987 

4.918 

. 4010 

4.898 

.0008 

. 0080 

. 9919 

42 

48 

10 

19 

. 4042 

. 5958 

4.739 

. 4045 

4.718 

.000S 

. 0082 

. 9918 

41 

44 

20 

20 

.04071 

.95929 

24.502 

.04075 

24.542 

1.0008 

.00083 

.99917 

40 

40 

24 

21 

. 4100 

. 5900 

4.388 

. 4104 

4.367 

.0008 

. 0084 

. 9910 

39 

30 

28 

22 

. 4129 

. 5870 

4.210 

. 4133 

4.196 

.0008 

. 0085 

. 9915 

38 

32 

32 

23 

. 4158 

. 5841 

4.047 

. 4102 

4.026 

.0009 

. 0080 

. 9913 

37 

28 

30 

24 

. 4187 

. 5812 

3 880 

. 4191 

3.859 

.0009 

. 0088 

. 9912 

30 

24 

40 

25 

.04217 

.95783 

23.710 

.04220 

23.094 

1.0009 

.00089 

.99911 

35 

20 

44 

20 

. 4240 

. 5754 

3.653 

. 4249 

3.532 

.0009 

. 0090 

. 9910 

34 

16 

48 

27 

. 4275 

. 5725 

3.393 

. 4279 

3.372 

.0009 

. 0091 

. 9908 

33 

12 

52 

28 

. 4304 

. 5690 

3.235 

. 4308 

3.214 

.0009 

. 0093 

. 9907 

32 

8 

50 

29 

. 4333 

. 5007 

3.079 

. 4337 

3.058 

.0009 

. 0094 

. 9900 

31 

4 

10 

80 

.04302 

.95038 

22.925 

.04306 

22.904 

1.0009 

.00095 

.99905 

30 

50 

4 

31 

. 4391 

. 5009 

2.774 

. 4395 

2.752 

.0010 

. 0090 

. 9903 

29 

56 

8 

32 

. 4420 

. 5580 

2.624 

. 4424 

2.602 

.0010 

. 0098 

. 9902 

28 

52 

12 

33 

. 4449 

. 5551 

2.476 

. 4453 

2.454 

.0010 

. 0099 

. 9901 

27 

48 

10 

34 

. 4478 

. 5522 

2.330 

. 4483 

2.308 

.0010 

. 0100 

. 9900 

26 

44 

20 

35 

.04507 

.95493 

22.186 

.04512 

22.164 

1.0010 

.00102 

.99898 

25 

40 

24 

30 

. 4530 

. 5404 

2.044 

. 4541 

2.022 

.0010 

. 0103 

. 9897 

24 

36 

28 

37 

. 4565 

. 5435 

1.904 

. 4570 

1.881 

.0010 

. 0104 

. 9890 

23 

32 

32 

38 

. 4594 

. 5405 

1.705 

. 4599 

1.742 

.0010 

. 0100 

. 9894 

22 

28 

30 

39 

. 4023 

. 5376 

1.029 

. 4628 

1.600 

.0011 

. 0107 

. 9893 

21 

24 

40 

40 

.04652 

.95317 

21.494 

.04057 

21.470 

1.0011 

.00108 

.99892 

20 

20 

44 

41 

. 408 L 

. 5318 

1.300 

. 4687 

1.337 

.0011 

. 0110 

. 9890 

19 

16 

48 

42 

. 4711 

. 5289 

1.228 

. 4716 

1.205 

.0011 

. 0111 

. 9889 

18 

12 

52 

43 

. 4710 

. 5200 

1.098 

. 4745 

1.075 

.0011 

. 0112 

. 9888 

17 

8 

50 

44 

. 4769 

. 6231 

0.970 

. 4774 

0.946 

.0011 

. 0114 

. 9886 

10 

4 

11 

45 

.04798 

.95202 

20.843 

.04803 

20.819 

1.0011 

.00115 

.99885 

15 

40 

4 

40 

. 4827 

. 5173 

0.717 

. 4832 

0.093 

.0012 

. 0110 

.9883 

14 

66 

8 

47 

. 4856 

. 5144 

0.593 

. 4802 

0.509 

.0012 

. 0118 

. 9882 

13 

52 

12 

48 

. 4885 

. 5115 

0.471 

. 4891 

0.446 

.0012 

. 0119 

. 9881 

12 

48 

10 

49 

. 4914 

. 5080 

0.350 

. 4920 

0.325 

.0012 

. 0121 

. 9879 

11 

44 

20 

50 

.01943 

.95057 

20.230 

.04949 

20.205 

1.0012 

.00122 

.99878 

10 

40 

24 

51 

. 4972 

. 5028 

0.112 

. 4978 

0.087 

.0012 

. 0124 

. 9876 

9 

36 

28 

52 

. 5001 

. 4999 

19.995 

. 5007 

19.970 

.0012 

. 0125 

. 9875 

8 

32 

32 

53 

. 5030 

. 4970 

9.880 

. 5037 

9.854 

.0013 

. 0127 

. 9873 

7 

28 

3G 

54 

. 5059 

. 4941 

9.706 

. 5000 

9.740 

.0013 

. 0128 

. 9872 

6 

24 

40 

55 

.05088 

.94912 

19.653 

.05095 

19.627 

1.0013 

.00129 

.99870 

5 

20 

41 

50 

. 5117 

. 4883 

9.541 

. 5124 

9.515 

.0013 

. 0131 

. 9869 

4 

16 

48 

57 

. 5140 

. 4853 

9.431 

. 5153 

9.405 

.0013 

. 0132 

. 9867 

3 

12 

52 

58 

. 5175 

. 4824 

9.322 

. 5182 

9.296 

.0013 

. 0134 

. 9866 

2 

8 

50 

69 

. 5204 

. 4795 

9.214 

. 5212 

9.188 

.0013 

. 0135 

. 9804 

1 

4 

12 

60 

. 5234 

. 4700 

9.107 

. 5241 

9.081 

.0014 

. 0137 

. 9803 

0 

48 

M.S. 

t> h 

M 

92° 

Cosine. 

Ws.Sin. 

Secaute. 

Cotaug.! 

Na 

Tangent. I 

tural. 

Cosec'nt 

Vrs.Cos 

Sine. 

O 

< I"* 
GO 

M.S. 

5 h 

























248 Natural Lines. 


O b 

CO 

O 

Natural Trigonometrical 

Functions. 

176° 

11“ 

M.S. 

M 

Sine. 

Yrs.Cos. 

Coscc'nte 

Tang. 

Cotang. 

Secaute. 

Vrs.Sin 

Cosine. 

M 

M.S. 

12 

0 

.05234 

.94766 

19.107 

.05241 

19.081 

1.0014 

.0013T 

.99863 

60 

48 

4 

1 

. 5263 

. 4737 

9.002 

. 5270 

8.975 

.0014 

. 0138 

. 9861 

59 

56 

8 

2 

. 5292 

. 4708 

8.897 

. 5299 

8.871 

.0014 

. 0140 

. 9800 

58 

52 

12 

3 

. 5321 

. 4679 

8.794 

. 5328 

8.768 

.0014 

. 0142 

. 9858 

57 

48 

16 

4 

. 5350 

. 4650 

8.692 

. 5357 

8.665 

.0014 

. 0143 

. 9857 

56 

44 

20 

5 

.05379 

.94621 

18.591 

.05387 

18.564 

1.0014 

.00145 

.99855 

55 

40 

24 

6 

. 5408 

. 4592 

8.491 

. 5416 

8.464 

.0015 

. 0146 

. 9854 

54 

36 

28 

7 

. 5437 

. 4563 

8.393 

. 5445 

8.365 

.0015 

. 0148 

. 9852 

53 

32 

82 

8 

. 5466 

. 4534 

8.295 

. 5474 

8.268 

.0015 

. 0149 

. 9850 

52 

28 

86 

9 

. 5495 

. 4505 

8.198 

. 5503 

8.171 

.0015 

. 0151 

. 9S49 

51 

24 

40 

10 

.05524 

.94476 

18.103 

.05532 

18.075 

1.0015 

.00153 

.99847 

50 

20 

41 

11 

. 5553 

. 4447 

8.008 

. 6562 

7.980 

.0015 

. 0154 

. 9846 

49 

16 

48 

12 

. 5582 

. 4418 

7.914 

. 5591 

7.886 

.0016 

. 0156 

. 9844 

48 

12 

52 

13 

. 5611 

. 4389 

7.821 

. 5620 

7.793 

.0016 

. 0157 

. 9842 

47 

8 

66 

14 

. 5640 

. 4360 

7.780 

. 5649 

7.701 

.0016 

. 0159 

. 9841 

46 

4 

13 

15 

.05669 

.94331 

17.639 

.05678 

17.610 

1.0016 

.00161 

.99839 

45 

47 

4 

16 

. 5698 

. 4302 

7.549 

. 5707 

7.520 

.0016 

. 0162 

. 9837 

44 

56 

8 

17 

. 5727 

. 4273 

7.460 

. 5737 

7.431 

.0016 

. 0164 

. 9836 

43 

52 

12 

18 

. 5756 

. 4244 

7.372 

. 5766 

7.343 

.0017 

. 0166 

. 9834 

42 

48 

16 

19 

. 5785 

. 4214 

7.2S5 

. 5795 

7.256 

.0017 

. 0167 

. 9832 

41 

44 

20 

20 

.05814 

.94185 

17.198 

.05824 

17.169 

1.0017 

.00169 

.99831 

40 

40 

24 

21 

. 5843 

. 4156 

7.113 

. 5853 

7.084 

.0017 

. 0171 

. 9829 

39 

3G 

28 

22 

. 5872 

. 4127 

7.028 

. 5883 

6.999 

.0017 

. 0172 

. 9827 

38 

32 

32 

23 

. 5902 

. 4098 

6.944 

. 5912 

6.915 

.0017 

. 0174 

. 9826 

37 

28 

36 

24 

. 5931 

. 4069 

6.861 

. 5941 

6.832 

.0018 

. 0176 

. 9824 

36 

24 

40 

25 

.05960 

.94040 

16.779 

.05970 

16.750 

1.0018 

.00178 

.99822 

35 

20 

44 

26 

. 5989 

. 4011 

6.698 

. 5999 

6.668 

.0018 

. 0179 

. 9820 

34 

16 

48 

27 

. 6018 

. 3982 

6.617 

. 6029 

6.587 

.0018 

. 0181 

. 9819 

33 

12 

52 

28 

. 6047 

. 3953 

6.538 

. 6058 

6.507 

.0018 

. 0183 

. 9817 

32 

8 

56 

29 

. 6076 

. 3924 

6.459 

. 6087 

6.428 

.0018 

. 0185 

. 9815 

31 

4 

14 

30 

.06105 

.93895 

16.3S0 

.06116 

16.350 

1.0019 

.00186 

.99813 

36 

40 

4 

31 

. 6134 

. 3866 

6.303 

. 6145 

6.272 

.0019 

. 0188 

. 9812 

29 

56 

8 

32 

. 6163 

. 3837 

6.226 

. 6175 

6.195 

.0019 

. 0190 

. 9810 

28 

52 

12 

33 

. 6192 

. 3808 

6.150 

. 6204 

6.119 

.0019 

. 0192 

. 9808 

27 

48 

16 

34 

. 6221 

. 3777 

6.075 

. 6233 

6.043 

.0019 

. 0194 

. 9806 

26 

44 

20 

35 

.06250 

.93750 

16.000 

.06262 

45.969 

1.0019 

.00195 

.99804 

25 

40 

24 

36 

. 6279 

. 3721 

5.926 

. 6291 

5.894 

.0020 

. 0197 

. 9803 

24 

36 

28 

37 

. 6308 

. 3692 

5.853 

. 6321 

5.821 

.0020 

. 0199 

. 9801 

23 

32 

32 

38 

. 6337 

. 3663 

5.780 

. 6330 

5.748 

.0020 

. 0201 

. 9799 

22 

28 

86 

39 

. 6366 

. 3634 

5.708 

. 6379 

5.676 

.0020 

. 0203 

. 9797 

21 

24 

40 

40 

.06395 

.93605 

15.637 

.06408 

15.605 

1.0020 

.00205 

.99795 

20 

20 

44 

41 

. 6424 

. 3576 

5.566 

. 6437 

5.534 

.0021 

. 0206 

. 9793 

19 

16 

48 

42 

. 6453 

. 3547 

5.496 

. 6467 

5.464 

.0021 

. 0208 

. 9791 

18 

12 

52 

43 

. C4S2 

. 351S 

5.427 

. 6496 

5-394 

.0021 

. 0210 

. 9790 

17 

8 

56 

44 

. 6511 

. 3489 

5.358 

. 6525 

5.325 

.0021 

. 0212 

. 9788 

16 

4 

15 

45 

.06540 

.93460 

15.290 

.06554 

15.257 

1.0021 

.00214 

.99786 

15 

45 

4 

46 

. 6569 

. 3431 

4.222 

. 6583 

5.189 

.0022 

.0216 

. 9784 

14 

56 

8 

47 

. 6598 

. 3402 

5.155 

. 6613 

5.122 

.0022 

. 0218 

. 9782 

13 

52 

12 

48 

. 6627 

. 3373 

5.089 

. 6642 

5.056 

.0022 

. 0220 

. 9780 

12 

48 

16 

49 

. 6656 

. 3343 

5.023 

. 6671 

4.990 

.0022 

. 0222 

. 9778 

11 

4-4 

20 

50 

.06685 

.93314 

14.958 

.06700 

14.924 

1.0022 

.00224 

.99776 

10 

40 

24 

51 

. 6714 

. 3285 

4.893 

. 6730 

4.860 

.0023 

. 0226 

. 9774 

9 

36 

28 

52 

. 6743 

. 3256 

4.829 

. 6759 

4.795 

.0023 

. 0228 

. 9772 

8 

32 

32 

53 

. 6772 

. 3227 

4.765 

. 6788 

4.732 

.0023 

. 0230 

. 9770 

7 

28 

86 

54 

. 6801 

. 3198 

4.702 

. 6817 

4.668 

.0023 

. 0231 

. 9768 

6 

24 

40 

55 

.06830 

.93169 

14.640 

.06846 

14.606 

1.0023 

.00233 

.99766 

5 

20 

*44 

56 

. 6859 

. 3140 

4.578 

. 6876 

4.544 

.0024 

. 0235 

. 9764 

4 

16 

48 

57 

. 6888 

. 3111 

4.517 

. 6905 

4.482 

.0024 

. 0237 

. 9762 

3 

12 

62 

58 

. 6918 

. 3082 

4.456 

. 6934 

4.4‘Jl 

.0024 

. 0239 

. 9760 

2 

8 

66 

59 

. 6947 

. 3053 

4.395 

. 6963 

4.361 

.0024 

. 0241 

. 9758 

1 

4 

10 

60 

. 6976 

. 3024 

4.335 

. 6993 

4.301 

.0024 

. 0243 

. 9756 

0 

44 

MS. 

M 

Cosiue. 

Vrs.Sin. 

Seeuuic. 

Cotang.;Tangent. 

Cosee'nt 

Vrs. Cos 

Sine. 

M 

M.S. 

6 h 

93° 




Natural. 




86° 

5 h 



















Natural Lines. 


249 


o h 

4° 

Natural Trigonometrical Functions. 

175° 

ll h 

M.S. 

M 

Sine. 

Vrs.Cos 

Coseo'nte 

Tang. 

Cotang. 

Sccante 

[Vrs.Sin 

Cosine. 

M 

M.S. 

10 

0 

.06976 

.93024 

14.335 

.06993 

14.301 

1.0024 

.00243 

.99756 

CO 

44 

4 

1 

. 7005 

. 2995 

4.276 

. 7022 

4.241 

.0025 

. 0246 

. 9754 

59 

56 

8 

2 

. 7034 

. 2966 

4.217 

. 7051 

4.182 

.0025 

. 024S 

. 9752 

58 

52 

12 

3 

. 7063 

. 2937 

4.159 

. 7080 

4.123 

.0025 

. 0250 

. 9750 

57 

48 

10 

4 

. 7092 

. 2908 

4.101 

. 7110 

4.065 

.0025 

. 0252 

. 9748 

56 

44 

20 

5 

.07121 

.92879 

14.043 

.07139 

14.008 

1.0025 

.00254 

.99746 

55 

40 

24 

6 

. 7150 

. 2850 

3.986 

. 7168 

3.951 

.0026 

. 0256 

. 9744 

54 

36 

28 

7 

. 7179 

. 2821 

3.930 

. 7197 

3.894 

.0026 

. 0258 

. 9742 

53 

32 

82 

8 

. 7208 

. 2792 

3.874 

. 7226 

3.838 

.0026 

. 0260 

. 9740 

52 

28 

36 

9 

. 7237 

. 2763 

3.818 

. 7256 

3.782 

.0026 

. 0262 

. 9738 

51 

24 

40 

10 

.07-66 

.92734 

13.763 

.07285 

13.727 

1.0026 

.00264 

.99736 

50 

20 

44 

11 

. 7295 

. 2705 

3.708 

. 7314 

3.672 

.()(27 

. 0266 

. 9733 

49 

16 

48 

12 

. 7324 

. 2676 

3.654 

. 7343 

3.617 

.0027 

. 0268 

. 9731 

48 

12 

52 

13 

. 7353 

. 2647 

3.600 

. 7373 

3.563 

.0027 

. 0271 

. 9729 

47 

8 

56 

14 

. 7382 

. 2618 

3.547 

. 7402 

3.510 

.0027 

. 0273 

. 9727 

46 

4 

1? 

15 

.07411 

.925^9 

13.494 

.07431 

13.457 

1.0027 

.00275 

.99725 

45 

43 

4 

1G 

. 7440 

. 2560 

3.441 

. 7460 

3.404 

.0028 

. 0277 

. 9723 

44 

56 

8 

17 

. 7469 

. 2531 

3.389 

. 7490 

3.351 

.002S 

. 0279 

. 9721 

43 

52 

12 

18 

. 7498 

. 2502 

3.337 

. 7519 

3.299 

.0028 

. 0281 

. 9718 

42 

48 

1G 

19 

. 7527 

. 2473 

3.286 

. 7548 

3.248 

.0028 

. 0284 

. 9716 

41 

44 

20 

20 

.07556 

.92441 

13.235 

.07677 

13.197 

1.0029 

.00286 

.99714 

40 

40 

24 

21 

. 7585 

. 2415 

3.184 

. 7607 

3.146 

.0029 

. 0288 

. 9712 

39 

36 

28 

22 

. 7614 

. 2386 

3.134 

. 7636 

3.096 

.0029 

. 0290 

. 9710 

38 

32 

32 

23 

. 7643 

. 2357 

3.084 

. 7665 

3.046 

.0029 

. 0292 

. 9707 

37 

28 

36 

24 

. 7672 

. 2328 

3.034 

. 7694 

2.996 

.0029 

. 0295 

. 9705 

36 

24 

40 

25 

.07701 

.92293 

12.985 

.07724 

12.947 

1.0030 

.00297 

.99703 

35 

20 

44 

26 

. 7730 

. 2270 

2.937 

. 7753 

2.898 

.0030 

. 0299 

. 9701 

34 

16 

48 

27 

. 77o9 

. 2241 

2.888 

. 7782 

2.849 

.0030 

. 0301 

. 9698 

33 

12 

52 

28 

. 7788 

. 2212 

2.840 

. 7812 

2.801 

.0030 

. 0304 

. 9696 

32 

8 

5G 

29 

. 7817 

. 21S3 

2.793 

. 7841 

2.754 

.0031 

. 0306 

. 9694 

31 

4 

IK 

30 

.07846 

.92154 

12.745 

.07870 

12.706 

1.0031 

.00308 

.99692 

30 

43 

4 

31 

. 7875 

. 2125 

2.698 

. 7899 

2.659 

.0031 

. 0310 

. 9689 

29 

56 

8 

32 

. 7904 

. 2096 

2.652 

. 7929 

2.612 

.0031 

. 0313 

. 9687 

28 

52 

12 

33 

. 7933 

. 2067 

2.606 

. 7958 

2.566 

.0032 

. 0315 

. 9685 

27 

48 

16 

34 

. 7962 

. 2038 

2.560 

. 7987 

2.520 

.0032 

. 0317 

. 9CS2 

26 

44 

20 

35 

.07991 

.92009 

12.514 

.08016 

12.474 

1.0032 

.00320 

.99680 

25 

40 

24 

36 

. 8020 

. 1980 

2.469 

. 8046 

2.429 

.0032 

. 0322 

. 9678 

24 

36 

28 

37 

. 8049 

. 1951 

2.424 

. 8075 

2.384 

.0032 

. 0324 

. 9675 

23 

32 

32 

38 

. 8078 

. 1922 

2.379 

. 8104 

2.339 

.0033 

. 0327 

. 9673 

22 

28 

36 

39 

. 8107 

. 1893 

2.335 

. 8134 

2.295 

.0033 

. 0329 

. 9671 

21 

24 

40 

40 

.08136 

.91864 

12.291 

.08163 

12.250 

1.0033 

.00331 

.99668 

20 

20 

44 

41 

. 8165 

. 1835 

2.248 

. 8192 

2.207 

.0033 

. 0334 

. 9666 

19 

16 

48 

42 

. 8194 

. 1806 

2.204 

. 8221 

2.163 

.0034 

. 0336 

. 9664 

18 

12 

52 

43 

. 8223 

. 1777 

2.161 

. 8251 

2.120 

.0034 

. 0339 

. 9661 

17 

8 

56 

44 

. 8252 

. 1748 

2.118 

. 8280 

2.077 

.0034 

. 0341 

. 9659 

16 

4 

19 

45 

.08281 

.91719 

12.076 

.08309 

12.035 

1.0034 

.00343 

.99656 

15 

41 

4 

46 

. 8310 

. 1690 

2.034 

. 8339 

1.992 

.0035 

. 0346 

. 9654 

14 

56 

8 

47 

. 8339 

. 1GG1 

1.992 

. 8368 

1.950 

.0035 

. 0348 

. 9652 

13 

52 

12 

48 

. 8368 

. 1632 

1.950 

. 8397 

1.909 

.0035 

. 0351 

. 9649 

12 

48 

1G 

49 

. 8397 

. 1603 

1.909 

. 8426 

1.867 

.0035 

. 0353 

. 9647 

11 

44 

20 

50 

.08426 

.91574 

11.868 

.08456 

11.826 

1.0036 

.00356 

.99644 

10 

40 

24 

51 

. 8455 

. 1545 

1.828 

. 8485 

1.785 

.0036 

. 0358 

. 9642 

9 

36 

28 

52 

. 8484 

. 1516 

1.787 

. 8514 

1.745 

.0036 

. 0360 

. 9639 

8 

32 

32 

53 

. 8513 

. 1487 

1.747 

. 8544 

1.704 

.0036 

. 0363 

. 9637 

7 

28 

36 

54 

. 8542 

. 1458 

1.707 

• 85 i 3 

1.664 

.0037 

. 0365 

. 9634 

6 

24 

40 

55 

.08571 

.91429 

11.668 

.08602 

11.625 

1.0037 

.00368 

.99632 

5 

20 

44 

56 

. 8600 

. 14(H) 

1.628 

. 8632 

1.585 

.0037 

. 0370 

. 9629 

4 

16 

48 

57 

. 8629 

. 1371 

1.589 

. 8661 

1.546 

.0037 

0373 

. 9627 

3 

12 

52 

58 

. 8658 

. 1342 

1.550 

. 8690 

1.507 

.0038 

. 0375 

. 9624 

2 

8 

56 

59 

. 8687 

. 1313 

1.612 

. 8719 

1.4G8 

.0038 

. 0378 

. 9622 

1 

4 

20 

60 

. 8715 

. 1284 

1.474 

. 8749 

1.430 

.0038 

Cosee'ntl 

. 0380 

Vrs.Cos 

. 9619 

0 

40 

M. S. 

G h 

M 

Cosine. 

Vrs.Sin. 

Secame. 

Co tang.1 

Natn 

Tangent. 

ral. 

Sine. 

M 

85° 

M.S. 

5 fa 

































250 


Natural Lines. 


o h 

5° 

Natural Trigonometrical 

Functions. 

1 

74° 

ll h 

M. S. 

M 

Sine. 

Vrs.Cos. 

Coscc'ute 

Tang. 

Cotang. 

Secante. 

Vrs. Sin 

Cosine. 

M 

M.S. 

ao 

0 

.08715 

.91284 

11.474 

.08749 

11.430 

1.0038 

.00380 

.99619 

CO 

40 

4 

1 

. 8744 

. 1255 

1.430 

. 8778 

1.392 

.0038 

. 0383 

. 9617 

59 

56 

8 

2 

. 8773 

. 1220 

1.398 

. 8807 

1.354 

.0039 

. 0386 

. 9(514 

58 

52 

12 

3 

. 8802 

. 1197 

1.300 

. 8837 

1.316 

.0039 

. 0388 

. 9612 

57 

48 

1(5 

4 

. 8831 

. 1108 

] .323 

. 8806 

1.279 

.0039 

. 0391 

. 9609 

56 

44 

20 

5 

.08800 

.91139 

11.280 

.08895 

11.242 

1.0039 

.00393 

.99607 

55 

40 

24 

0 

. 8889 

. mo 

1.249 

. 8925 

1.2*6 

.0040 

. 0396 

. 9004 

54 

36 

’ 28 

7 

. 8918 

. 1082 

1.213 

. 8954 

1.168 

.0040 

. 0398 

. 9601 

53 

32 

32 

8 

. 8947 

. 1053 

1.170 

. 8983 

1.132 

.0040 

. 0401 

. 9599 

52 

28 

3(5 

9 

. 8970 

. 1024 

1.140 

. 9013 

1.095 

.0040 

. 0404 

. 9590 

51 

24 

40 

10 

.09005 

.90995 

11.101 

.09042 

11.059 

1.0041 

.00406 

.99594 

50 

20 

44 

14 

. 9034 

. 09(56 

1.009 

. 9071 

1.024 

.0041 

. 0409 

. 9591 

49 

16 

48 

12 

. 9003 

. 0937 

1.033 

. 9101 

0.988 

.0041 

. 0411 

. 95S8 

48 

12 

52 

13 

. 9092 

. 0908 

0.998 

. 9130 

0.953 

.0041 

. 0414 

. 9586 

47 

8 

66 

14 

. 9121 

. 0879 

0.963 

. 9159 

0.918 

.0042 

. 0417 

. 9583 

46 

4 

£1 

15 

.09150 

.90850 

10.929 

.09189 

10.883 

1.0042 

.00419 

.99580 

45 

30 

4 

10 

. 9179 

. 0821 

0.894 

. 9218 

0.848 

.0042 

. 0422 

. 9578 

44 

56 

s 

17 

. 92US 

. ('792 

0.860 

. 9247 

0.814 

.0043 

. 0425 

. 9575 

43 

52 

12 

18 

. 9237 

. 0703 

0.820 

. 9277 

0.780 

.0043 

. 0427 

. 9572 

42 

48 

1G 

19 

. 9200 

. 0734 

0.792 

. 9306 

0.746 

.0043 

. 0430 

. 9570 

41 

44 

20 

20 

.09295 

.90705 

10.758 

.09335 

10.712 

1.0043 

.00433 

.99567 

40 

40 

24 

21 

. 9324 

. 0670 

0.725 

. 9305 

0.678 

.0044 

. 0436 

. 9564 

39 

36 

28 

22 

. 9353 

. 0647 

0.092 

. 9394 

0.645 

.0044 

. 0138 

. 9562 

38 

32 

32 

23 

. 9382 

. 0018 

0.059 

. 9423 

0.612 

.0044 

. 0441 

. 9569 

37 

28 

30 

24 

. 9411 

. 0589 

0 020 

. 9453 

0.579 

.0044 

. 0444 

. 9556 

36 

24 

40 

25 

.09440 

.90500 

10.593 

.( 9482 

10.546 

1.0045 

.00446 

.99553 

35 

20 

44 

26 

. 9409 

. 0531 

0.501 

. 9511 

0.514 

.0045 

. 0449 

. 9551 

34 

16 

48 

27 

. 9498 

. U5(>2 

0 529 

. 9541 

0.481 

.0045 

. 0452 

. 9548 

33 

12 

52 

28 

. 9527 

. 0473 

0.497 

. 9570 

0.449 

.0046 

. 0455 

. 9545 

32 

8 

50 

29 

. 955(5 

. 0441 

0.465 

. 9599 

0.417 

.0016 

. (1458 

. 9542 

3L 

4 

22 

30 

.09584 

.90415 

10.433 

.09029 

10.385 

1.0046 

.00460 

.99540 

30 

38 

4 

31 

. 9013 

. 0380 

0.402 

. 9658 

0.354 

.0046 

. 0463 

. 9537 

29 

56 

8 

32 

. 9042 

. 0357 

0.371 

. 9088 

0.322 

.0047 

. 0466 

. 9534 

28 

52 

12 

33 

. 9071 

. 0328 

0.310 

. 9717 

0.291 

.0047 

. 0469 

. 9531 

27 

48 

16 

34 

. 97(K) 

. 0300 

0.809 

. 9740 

0.260 

.0047 

. 0472 

. 9528 

26 

44 

20 

35 

.09729 

.90271 

10.278 

.09770 

10.229 

1.0048 

.00474 

.99525 

25 

40 

24 

30 

. 9758 

. 0212 

0.248 

. 9805 

0.199 

.0048 

. 0477 

. 9523 

21 

36 

28 

37 

. 9787 

. 0213 

0.217 

. 9834 

0.168 

.0048 

. 0480 

. 9520 

23 

32 

32 

38 

. 9816 

. 0181 

0.187 

. 9804 

0.138 

.0048 

. 0483 

. 9517 

22 

28 

30 

39 

. 9815 

. 0156 

0.157 

. 9893 

0.108 

.0049 

. 0486 

. 9514 

21 

24 

40 

40 

.09874 

.90120 

10.127 

.09922 

10.078 

1.0049 

.00489 

.99511 

20 

20 

44 

41 

. 9903 

. 0097 

0.098 

. 9952 

0.048 

.0049 

. 0491 

. 9508 

19 

16 

48 

42 

. 9932 

. 0008 

0.008 

. 9981 

0.019 

.0050 

. 0494 

. 9505 

18 

12 

52 

43 

. 9901 

. 0039 

0.039 

.10011 

9.9893 

.0050 

. 0497 

. 9503 

17 

8 

66 

44 

. 9990 

. 0010 

0.010 

.10040 

.9(01 

.0050 

. 0500 

. 9500 

16 

4 

2.4 

45 

110019 

.89981 

9.9812 

.10069 

.9310 

1.0050 

.00503 

.99497 

15 

37 

4 

40 

. 0018 

. 9952 

.9525 

. 0099 

.9021 

.0051 

. 0506 

. 9494 

14 

56 

8 

47 

. 0077 

. 9925 

.9239 

. 0128 

.8734 

.0051 

. 0609 

. 9491 

13 

52 

12 

48 

. 0106 

. 9894 

.8955 

. 0158 

.8448 

.0051 

. 0512 

. 9488 

12 

48 

10 

49 

. 0134 

. 9865 

.8072 

. 0187 

.8164 

.0052 

. 0515 

. 9485 

11 

44 

20 

50 

.10103 

.89830 

9.8391 

.10216 

9.7882 

1.0052 

.00518 

.99482 

10 

40 

24 

51 

. 0192 

. 9807 

.8112 

. 0246 

.7601 

.0052 

. 05_'l 

. 9479 

9 

36 

28 

52 

. 0221 

. 9779 

.7834 

. 0275 

.7322 

.0053 

. 0524 

. 9476 

8 

32 

32 

53 

. 0250 

. 9750 

.7558 

. 0305 

.7044 

.0053 

. 0527 

. 9473 

7 

28 

36 

54 

. 0279 

. 9721 

.7283 

. 0334 

.6768 

.0053 

. 0530 

. 9470 

G 

24 

40 

65 

.10308 

.89(592 

9.7010 

.10363 

9.6493 

1.0053 

.00533 

.99467 

5 

20 

44 

56 

. 0337 

. 9(503 

.0739 

. 0393 

.6220 

.0054 

. 0536 

. 9464 

4 

16 

48 

57 

. 0366 

. 9034 

.0409 

. 0422 

.6949 

.0054 

. 0539 

. 9461 

3 

12 

52 

58 

. 0395 

. 9605 

.0200 

. 0452 

.5679 

.0054 

. 0542 

. 9458 

2 

8 

56 

59 

. 0424 

. 9570 

.5933 

. 0481 

.5411 

.0055 

. 0545 

. 9455 

1 

4 

21 

60 

. 0453 

. 9547 

.5608 

. 0510 

.5144 

.0055 

. 0548 

. 9452 

0 

30 

MS. 

M 

Cosine. 

Vrs.Sin. 

Secame. 

Cotang. [Tangent. 

Cosec'nt 

Vrs. Cos 

Sine. 

M 

M.S. 

6 h 

95° 




Natural. 




84° 

6 h 

































Natural Lines. 


251 


O h 

6° 

Natural Trigonometrical Functions. 

173° 

ll h 

M.S 

M 

Sine. 

Via.Cos. 

Coseo’nto 

Tang. 

Cotang. 

Secante. 

' Vrs. Sin 

Cosine. 

M 

M.S. 

24: 

0 

.10453 

.89547 

9.5668 

.10510 

9.5144 

1.0055 

.00548 

.99452 

60 

36 

4 

1 

. 0482 

. 9518 

.5404 

. 0540 

.4S78 

.0055 

. 0551 

. 9449 

59 

56 

s 

2 

. 0511 

. 94.89 

.5141 

. 0569 

.4614 

.0056 

. 0554 

. 9446 

58 

52 

12 

3 

. 0540 

. 9460 

.4880 

. 0599 

.4351 

.0056 

. 0557 

. 9443 

57 

48 

1G 

4 

. 0568 

. 9431 

.4620 

. 0628 

.4090 

.00-6 

. 0560 

. 9410 

56 

44 

20 

5 

.10597 

.89402 

9.4362 

.10657 

9.3831 

1.0057 

.00563 

.99437 

55 

40 

24 

6 

. 0026 

. 9373 

.4105 

. 0687 

.3572 

.0057 

. 0566 

. 9434 

54 

36 

28 

7 

. 0655 

. 9345 

.3850 

. 0716 

.3315 

.0057 

. 0569 

. 9431 

53 

32 

32 

8 

. 0084 

. 9310 

.3596 

. 0746 

.3060 

.0057 

. 0572 

. 9428 

52 

28 

36 

9 

. 0713 

. 9287 

.3343 

. 0775 

.2806 

.0058 

. 0575 

. 9424 

51 

24 

40 

10 

.10742 

.89258 

9.3092 

.10805 

9.2553 

1.0058 

.00579 

.99421 

50 

20 

44 

11 

. 0771 

. 9229 

.2842 

. 0834 

.2302 

.0058 

. 0582 

. 9418 

49 

16 

48 

12 

. 0800 

. 9200 

.2593 

. 0863 

.2051 

.0059 

. 0585 

. 9415 

48 

12 

52 

13 

. 0829 

. 9171 

.2346 

. 0893 

.1803 

.0059 

. 0588 

. 9412 

47 

8 

5G 

14 

. 0858 

. 9142 

.2100 

. 0922 

.1555 

.0059 

. 0591 

. 9409 

46 

4 

535 

15 

.10887 

.89113 

9.1855 

.10952 

9.1309 

1.0060 

.00594 

.99406 

45 

35 

4 

16 

. 0916 

. 9084 

.1612 

. 0981 

.1064 

.0060 

. 0597 

. 9402 

44 

56 

8 

17 

. 0944 

. 9055 

.1370 

. 1011 

.0821 

.0060 

. 0601 

. 9399 

43 

52 

12 

18 

. 0973 

. 9026 

.1129 

. 1040 

.0579 

.0061 

. 0604 

. 9396 

42 

48 

16 

19 

. 1002 

. 8908 

.0890 

. 1069 

.0338 

.0061 

. 0607 

. 9393 

41 

44 

20 

20 

.11031 

.88969 

9.0651 

.11099 

9.0098 

1.0061 

.00110 

.99390 

40 

40 

24 

21 

. 1060 

. 8940 

.0414 

. 1128 

8.9860 

.0062 

. 0013 

. 9386 

39 

36 

28 

22 

. 1089 

. 8911 

.0179 

. 115S 

.9623 

.0062 

. 0617 

. 9383 

38 

32 

32 

23 

. 1118 

. 8882 

.9944 

. 1187 

.9387 

.0062 

. 0620 

. 9380 

37 

28 

36 

24 

. 1147 

. 8853 

8.9711 

. 1217 

.9152 

.0003 

. 0623 

. 9377 

36 

24 

40 

25 

.11170 

.88824 

8.9479 

.11246 

8.8918 

1.0063 

.00626 

.99373 

35 

20 

44 

26 

. 1205 

. 8795 

.9248 

. 1276 

.8686 

.0063 

. 0630 

. 9370 

34 

16 

48 

27 

. 1234 

. 8766 

.9018 

. 1305 

.8455 

.0064 

. 0633 

. 9367 

33 

12 

52 

28 

. 1262 

. 8737 

.8790 

. 1335 

.8225 

.0064 

. 0636 

. 9364 

32 

8 

56 

29 

. 1291 

. 8708 

.8563 

. 1364 

.7996 

.0064 

. 0039 

. 9360 

31 

4 

26 

30 

.11320 

.88680 

8.8837 

.11393 

8.7769 

1.0065 

.00643 

.99357 

30 

34 

4 

31 

. 1349 

. 8651 

.8112 

. 1423 

.7542 

.0065 

. 0646 

. 9354 

29 

56 

8 

32 

. 1378 

. 8622 

.7888 

. 1452 

.7317 

.0065 

. 0649 

. 9350 

28 

52 

12 

33 

. 1407 

. 8593 

.7665 

. 1482 

.7093 

.0066 

. 0653 

. 9347 

27 

48 

16 

34 

. 1436 

. 8564 

.7444 

. 1511 

.6870 

.0066 

. 0656 

. 9344 

26 

44 

20 

35 

.11465 

.88535 

8.7223 

.11541 

8.6648 

1.0066 

.00659 

.99341 

25 

40 

24 

36 

. 1494 

. 8506 

.7004 

. 1570 

.6427 

.0067 

. 0663 

. 9337 

24 

36 

2S 

37 

. 1523 

. 8477 

.6786 

. 1600 

.6208 

.0067 

. 0666 

. 9334 

23 

32 

32 

38 

. 1551 

. 8448 

.6569 

. 1629 

.5989 

.0067 

. 0669 

. 9330 

22 

28 

36 

39 

. 1580 

. 8420 

.6353 

. 1659 

.5772 

.0068 

. 0673 

. 9327 

21 

24 

40 

40 

.11609 

.88391 

8.6138 

.11688 

8.5555 

1.0068 

.00676 

.99324 

20 

20 

44 

41 

. 1638 

. 8362 

.5924 

. 1718 

.5340 

.0068 

. 0679 

. 9320 

19 

16 

48 

42 

. 1667 

. 8333 

.5711 

. 1747 

.5126 

.0069 

. 0683 

. 9317 

18 

12 

52 

43 

. 1696 

. 8304 

.5499 

. 1777 

.4913 

.0069 

. 0686 

. 9314 

17 

8 

56 

44 

. 1725 

. 8272 

.5289 

. 1806 

.4701 

.01’69 

. 0690 

. 9310 

16 

4 

27 

45 

.11754 

.88246 

8.5079 

.11836 

8.4489 

1.0070 

.00693 

99307 

15 

33 

4 

46 

. 1783 

. 8217 

.4871 

. 1865 

.4279 

.0070 

. 0696 

. 9303 

14 

56 

8 

47 

. 1811 

. 8188 

.4663 

. 1895 

.4070 

.0070 

. 0700 

. 9300 

13 

52 

12 

48 

. 1840 

. 8160 

.4457 

. 1924 

.3862 

.0071 

. 0703 

. 9296 

12 

48 

16 

49 

. 1809 

. 8131 

.4251 

. 1954 

.3655 

.0071 

. 0707 

. 9293 

11 

44 

20 

50 

.11898 

.88102 

8.4046 

.11983 

8.3449 

1.0071 

.00710 

.99290 

10 

40 

24 

51 

. 1927 

. 8073 

.3843 

. 2013 

.3244 

.0072 

. 0714 

. 9286 

9 

36 

28 

52 

. 1956 

. 8044 

.3640 

. 2042 

.3040 

.0072 

. 0717 

. 9283 

8 

32 

32 

53 

. 1985 

. 8015 

.3439 

. 2072 

.2837 

.0073 

. 0721 

. 9279 

7 

28 

36 

54 

. 2014. 

. 7986 

.3238 

. 2101 

.2635 

.0073 

. 0724 

. 9276 

G 

24 

40 

55 

.12042 

.87957 

8.3039 

.12131 

8.2434 

1.0073 

.00728 

.99272 

5 

20 

44 

56 

. 2071 

. 7928 

.2840 

. 2160 

.2234 

.0074 

. 0731 

. 9269 

4 

16 

48 

57 

. 2100 

. 7900 

.2642 

. 2190 

.2035 

.0074 

. 0735 

. 9265 

3 

12 

52 

58 

. 2129 

. 7871 

.2446 

. 2219 

.1837 

.0074 

. 0738 

. 9262 

2 

8 

56 

59 

. 2158 

. 7842 

.2250 

. 2249 

.1640 

.0075 

. 0742 

. 9258 

1 

4 

28 

60 

. 2187 

. 7813 

.2055 

. 2278 

.1443 

.0075 

. 0745 

. 9255 

0 

32 

M.S. 

M 

Cosine. 

Vrs.Sin. 

Secaute. 

Cotang, i 

Tangent. 

Cosec'ut 

Vrs. Cos 

Sine. 

M 

M.S. 

6 h 

96° 




Natural. 




83° 

5 h 


































252 


Natural Lines. 


o h 

7° 

\ 

Natural Trigonometrical Functions 

172° 

ll h 

M.S. 

M 

Sine. 

Vrs. Cos. 

Cosec’nte 

Tang. 

Cotang. 

Secante. 

Vrs. Sin 

Cosine. 

M 

M.S. 

as 

0 

.12187 

.87813 

8.2055 

.12278 

8.1443 

1.0075 

.00745 

.99255 

60 

3 a 

4 

1 

. 2216 

. 7787 

.186 L 

. 2308 

.1248 

.0075 

. 074* 

. 9251 

59 

56 

8 

2 

. 2245 

. 7755 

.1668 

. 2337 

.1053 

.0076 

. 0752 

. 9247 

58 

52 

12 

3 

. 2273 

. 7726 

.1476 

. 2367 

.0860 

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57 

48 

lb 

4 

. 2302 

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44 

20 

5 

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8.0476 

1.0077 

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55 

40 

24 

6 

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36 

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7 

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32 

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7.9906 

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7.9971 

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15 

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7.9240 

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45 

31 

4 

16 

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56 

8 

17 

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12 

18 

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7.8344 

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24 

40 

25 

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7.7469 

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7.6821 

1.0084 

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35 

20 

44 

26 

. 2937 

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34 

16 

48 

27 

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33 

12 

52 

28 

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8 

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29 

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30 

30 

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7.6613 

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7.5957 

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30 

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26 

44 

20 

35 

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7.5776 

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7.5113 

1.00S8 

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25 

40 

24 

36 

. 8226 

. 6774 

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3G 

28 

37 

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. 6745 

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23 

32 

32 

38 

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. 6717 

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36 

39 

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. 60'S 

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. 3432 

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24 

40 

40 

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7.4957 

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7.4287 

1.0090 

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20 

20 

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18 

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8 

56 

44 

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4 

31 

45 

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7.4156 

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7.3479 

1.0092 

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ao 

4 

46 

. 3514 

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14 

56 

8 

47 

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52 

12 

48 

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. 6428 

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12 

48 

16 

49 

. 3600 

. 6400 

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. 3728 

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11 

44 

20 

50 

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7.3372 

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7.2687 

1.0094 

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10 

40 

24 

51 

. 3658 

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9 

36 

28 

52 

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8 

32 

32 

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7 

28 


54 

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6 

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40 

55 

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7.1912 

1.0096 

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5 

20 

44 

56 

. 38'2 

. 6198 

.2453 

. 3935 

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4 

16 

48 

57 

. 3831 

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3 

12 

52 

58 

. 3860 

. 6110 

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2 

8 

56 

59 

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. 4024 

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1 

4 

3 a 

60 

. 3917 

, 6083 

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. 4054 

.1154 

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. 0973 

. 9027 

0 

as 

M.S. 

M 

Cosine. 

Vrs. Sin. 

Secante. 

Cotang.| 

Tangent. 

Cosec’nt 

Vrs. Cos 

Sine. 

M 

M.S. 

O' 1 

97° 




Natural. 




82° 

5 h 

























253 


Natural Lines. 


o h 

GO 

O 


Natural Trigonometrical Functions. 

171° 

ll h 

M. S 

M 

Sine. 

Vrs.Cos. 

Cosec'nte 

Tang. 

Cotaug. 

1 Secante 

Vrs.Sin 

Cosine. 

M 

M.S 

*£.'4 

0 

.13917 

.86083 

7.1853 

.14054 

7.1154 

1.0098 

.00973 

.99027 

60 

28 

4 

1 

. 3946 

. 6054 

.1704 

. 4084 

.1004 

.0099 

. 0977 

. 9023 

59 

56 

8 

2 

. 3975 

. 6025 

.1557 

. 4113 

.0854 

.0099 

. 0981 

. 9019 

58 

52 

12 

3 

. 4004 

. 5996 

.1403 

. 4143 

.07(16 

.0099 

. 0985 

. 9015 

57 

48 

re 

4 

. 4032 

. 5967 

.1263 

. 4173 

.0558 

.0100 

. 0989 

. 9010 

56 

41 

20 

5 

.14061 

.85939 

7.1117 

.14202 

7.0410 

1.0100 

.00993 

.99006 

55 

40 

24 

G 

. 4090 

. 5910 

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54 

36 

28 

7 

. 4119 

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53 

32 

32 

8 

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6.9972 

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52 

28 

3b 

9 

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6.9827 

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. 1010 

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51 

24 

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10 

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7.0396 

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6.9682 

1.0102 

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50 

20 

44 

11 

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. 5766 

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. 1018 

. 89S2 

49 

16 

48 

12 

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. 8978 

48 

12 

52 

13 

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6.9971 

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. 8973 

47 

8 

55 

14 

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6.9830 

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46 

4 

3*£ 

15 

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6.9690 

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6.8969 

1.0104 

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27 

4 

16 

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44 

56 

8 

17 

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43 

52 

12 

18 

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42 

48 

1G 

19 

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41 

44 

20 

20 

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6.8998 

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6.8269 

1.0107 

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40 

40 

24 

21 

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39 

36 

28 

22 

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38 

32 

32 

23 

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37 

28 

30 

24 

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. 8927 

36 

24 

40 

25 

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6.8320 

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6.7584 

1.0109 

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35 

20 

44 

26 

. 4666 

. 5334 

.8185 

. 4826 

.7448 

.0109 

. 10S1 

. 8919 

34 

16 

48 

27 

. 4695 

. 6305 

.8052 

. 4856 

.7313 

.0110 

. 1085 

. 8914 

33 

12 

52 

28 

. 4723 

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. 8910 

32 

8 

5G 

29 

. 4752 

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. 4915 

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.0111 

. 1094 

. 8906 

31 

4 

34 

30 

.14781 

.85219 

6.7655 

.14945 

6.6911 

1.0111 

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30 

20 

4 

31 

. 4810 

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.6779 

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. 1103 

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29 

56 

8 

32 

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28 

52 

12 

33 

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27 

48 

16 

34 

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.0113 

. 1116 

. 8884 

26 

44 

20 

35 

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6.7003 

.15094 

6.6252 

1.0113 

.01120 

.98880 

25 

40 

24 

36 

. 4953 

. 5046 

.6874 

. 5123 

.6122 

.0114 

. 1124 

. 8876 

24 

36 

28 

37 

. 4982 

. 5018 

.6745 

. 5153 

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.0114 

. 1129 

. 8871 

23 

32 

32 

38 

. 5011 

. 4989 

.6617 

. 5183 

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. 1133 

. 8867 

22 

28 

36 

39 

. 5040 

. 4960 

.6490 

. 5213 

.5734 

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. 1137 

. 8862 

21 

24 

40 

40 

.15068 

.84931 

6.6363 

.15243 

6.5605 

1.0115 

.01142 

.98858 

20 

20 

44 

41 

. 5097 

. 4903 

.6237 

. 5272 

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.0116 

. 1146 

. 8854 

19 

16 

48 

42 

. 5126 

. 4874 

.6111 

. 5302 

.5350 

.0116 

. 1151 

. 8849 

18 

12 

52 

43 

. 6155 

. 4845 

.5985 

. 5332 

.5223 

.0117 

. 1155 

. 8845 

17 

8 

56 

44 

. 5183 

. 4816 

.6860 

. 5362 

.5097 

.0117 

. 1159 

. 8840 

16 

4 

33 

45 

.15212 

.84788 

6.5736 

.15391 

6.4971 

1.0118 

.01164 

.98836 

15 

25 

4 

46 

. 5241 

. 4759 

.5612 

. 5421 

.4845 

.0118 

. 1168 

. 8832 

14 

56 

8 

47 

. 5270 

. 4730 

.5488 

. 5451 

.4720 

.0119 

. 1173 

. 8827 

13 

52 

12 

48 

. 5298 

. 4701 

.5365 

. 5481 

.4596 

.0119 

. 1177 

. 8823 

12 

48 

16 

49 

. 5328 

. 4672 

.5243 

. 5511 

.4472 

.0119 

. 1182 

. 8818 

11 

44 

20 

50 

.15356 

.84614 

6.5121 

.15540 

6.4348 

1.0120 

.01186 

.98814 

10 

40 

24 

51 

. 5385 

. 4615 

.4999 

. 5570 

.4225 

.0120 

. 1190 

. 8809 

9 

36 

28 

52 

. 5413 

. 4586 

.4878 

. 5600 

.4103 

.0121 

. 1195 

. 8805 

8 

32 

32 

53 

. 5442 

. 4558 

.4757 

. 5(530 

.3980 

.0121 

. 1199 

. 8800 

7 

28 

36 

54 

. 5471 

. 4529 

.4637 

. 5659 

.3S59 

.0122 

. 1204 

. 8796 

6 

24 

40 

55 

.15500 

.84500 

6.4517 

.15689 

6.3737 

1.0122 

.01208 

.98791 

5 

20 

44 

56 

. 5528 

. 4471 

.4398 

. 5719 

.3616 

.0123 

. 1213 

. 8787 

4 

16 

48 

57 

. 5557 

. 4413 

.4279 

. 5749 

.3496 

.0123 

. 1217 

. 8782 

3 

12 

52 

58 

. 6586 

. 4414 

.4160 

. 5779 

.3376 

.0124 

. 1222 

. 8778 

2 

8 

56 

59 

. 5615 

. 43S5 

.4042 

. 5809 

.3257 

.0124 

. 1227 

. 8773 

1 

4 

3G 

60 

. 5643 

. 4356 

.3924 

. 5838 

.3137 

.0125 

. 1231 

. 8769 

0 

241 

M.S. 

M 

Cosine. 

Vrs.Sin. 

Secnute. 

Cotang. 

raugeut. 

Cosec’nt 

Sine. 

Vrs.Cos 

M 

M.S. 

6 h 

98° 




Natural. 



81° 

5 h 






















254 


Natural Lines. 


o h 

9° 

Natural Trigonometrical 

Functions. 

1 

70° 

ll h 

MS. 

M 

Sine. 

Vrs.Cos. 

Cosec'nte 

Tang. 

Cotang. 

Sccantc. 

Vrs.Sin 

Cosine. 

M 

MS. 

30 

0 

.15643 

.84356 

6.3924 

.15838 

6.3137 

1.0125 

.01231 

.98769 

60 

34 

4 

1 

. 5672 

. 4328 

.3807 

. 5868 

.3019 

.0125 

. 1236 

. 8764 

59 

56 

8 

2 

. 5701 

. 4299 

.3690 

. 5898 

.2901 

.0125 

. 1240 

. 8760 

58 

52 

12 

3 

. 5730 

. 4270 

.3574 

. 5928 

.2783 

.0126 

. 1245 

. 8755 

57 

48 

16 

4 

. 5758 

. 4242 

.3458 

. 5958 

.2665 

.0126 

. 1249 

. 8750 

56 

44 

20 

5 

.15787 

.84213 

6.3343 

.15987 

6.2548 

1.0127 

.01254 

.98746 

55 

40 

24 

G 

. 5816 

. 4184 

.3228 

. 6017 

.2432 

.0127 

. 1259 

. 8741 

54 

36 

28 

7 

. 5844 

. 4155 

.3113 

. 6047 

.2316 

.0128 

. 1263 

. 8737 

53 

32 

32 

8 

. 5873 

. 4127 

.2999 

. 6077 

.2200 

.0128 

. 1268 

. 8732 

52 

28 

36 

9 

. 5902 

. 4098 

.2885 

. 6107 

.2085 

.0129 

. 1272 

. 8727 

51 

24 

40 

10 

.15931 

.84069 

6.2772 

.16137 

6.1970 

1.0129 

.01277 

.98723 

50 

20 

44 

11 

. 5959 

. 4041 

.2659 

. 6167 

.1856 

.0130 

. 1282 

. 8718 

49 

16 

48 

12 

. 59*8 

. 4012 

.2546 

. 6196 

.1742 

.0130 

. 1286 

. 8714 

48 

12 

52 

13 

. 6017 

. 3983 

.2434 

. 6226 

.1628 

.0131 

. 1291 

. 8709 

47 

8 

56 

14 

. 6045 

. 3954 

.2322 

. 6256 

.1515 

.0131 

. 1296 

. 8704 

46 

4 

37 

15 

.16074 

.83926 

6.2211 

.16286 

6.1402 

1.0132 

.01300 

.98700 

45 

33 

4 

16 

. 6103 

. 3897 

.2100 

. 6316 

.1290 

.0132 

. 1305 

. 8695 

44 

56 

8 

17 

. 6132 

. 3868 

.1990 

. 6346 

.1178 

.0133 

. 1310 

. 8690 

43 

52 

12 

IS 

. 6160 

. 3840 

.1880 

. 6376 

.1066 

.0133 

. 1314 

. 8685 

42 

48 

16 

19 

. 6189 

. 3811 

.1770 

. 6405 

.0955 

.0134 

. 1319 

. 8681 

41 

44 

20 

20 

.16218 

.83782 

6.1661 

.16435 

6.0844 

1.0134 

.01324 

.98676 

40 

40 

24 

21 

. 6246 

. 3753 

.1552 

. 6465 

.0734 

.0135 

. 1328 

. 8671 

39 

36 

28 

22 

. 6275 

. 3725 

.1443 

. 6495 

.0624 

.0135 

. 1333 

. 8667 

38 

32 

32 

23 

. 6304 

. 3696 

.1335 

. 6525 

.0514 

.0136 

. 1338 

. 8662 

37 

28 

36 

24 

. 6333 

. 3667 

.1227 

. 6555 

.0405 

.0136 

. 1343 

. 8657 

36 

24 

40 

25 

.16361 

.83639 

6.1120 

.16585 

6.0296 

1.0136 

.01317 

.98662 

35 

20 

44 

26 

. 6390 

. 3610 

.1013 

. 6615 

.0188 

.0137 

. 1352 

. 8648 

34 

16 

48 

27 

. 6419 

. 3581 

.0906 

. 6644 

.0080 

.0137 

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33 

12 

52 

28 

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5.9972 

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32 

8 

56 

29 

. 6476 

. 3524 

.0694 

. 6704 

5.9865 

.0138 

. 1367 

. 8633 

31 

4 

38 

30 

.16505 

.83495 

6.0588 

.16734 

5.9758 

1.0139 

.01371 

.98628 

30 

33 

4 

31 

. 6533 

. 3466 

.0483 

. 6764 

.9651 

.0139 

. 1376 

. 8624 

29 

56 

8 

32 

. 6562 

. 3438 

.0379 

. 6794 

.9545 

.0140 

. 1381 

. 8619 

28 

52 

12 

33 

. 6591 

. 3409 

.0274 

. 6824 

.9439 

.0140 

. 1386 

. 8614 

27 

48 

16 

34 

. 6619 

. 3380 

.0170 

. 6854 

.9333 

.0141 

. 1391 

. 8609 

26 

44 

20 

35 

.16648 

.83352 

6.0066 

.16884 

5.9228 

1.0141 

.01395 

.98604 

25 

40 

24 

36 

. 6677 

. 3323 

5.9963 

. 6914 

.9123 

.0142 

. 1400 

. 8600 

24 

36 

28 

37 

. 6705 

. 3294 

.9860 

. 6944 

.9019 

.0142 

. 1405 

. 8595 

23 

32 

32 

38 

. 6734 

. 3266 

.9758 

. 6973 

.8915 

.0143 

. 1410 

. 8590 

22 

28 

36 

39 

. 6763 

. 3237 

.9655 

. 7003 

.8811 

.0143 

. 1411 

. 8585 

21 

24 

40 

40 

.16791 

.83208 

5.9554 

.17033 

5.8708 

1.0144 

.01420 

.98580 

20 

20 

44 

41 

. 6820 

. 3180 

.0452 

. 7063 

.8605 

.0144 

. 1425 

. 8575 

19 

16 

48 

42 

. 6849 

. 3151 

.9351 

. 7093 

.8502 

.0145 

. 1430 

. 8570 

18 

12 

52 

43 

. 6878 

. 3122 

.9250 

. 7123 

.8400 

.0145 

. 1434 

. 8565 

17 

8 

56 

44 

. 6906 

. 3094 

.9150 

. 7153 

.8298 

.0146 

. 1439 

. 8560 

16 

4 

30 

45 

.16935 

.83065 

5.9049 

.17183 

5.8196 

1.0146 

.01444 

.98556 

15 

31 

4 

46 

. 6964 

. 3036 

.8950 

. 7213 

.8095 

.0147 

. 1449 

. 8551 

14 

56 

8 

47 

. 6992 

. 3008 

.8850 

. 7243 

.7994 

.0147 

. 1454 

. 8546 

13 

52 

12 

48 

. 7021 

. 2979 

.8751 

. 7273 

.7894 

.0148 

. 1459 

. 8541 

12 

48 

16 

49 

. 7050 

. 2950 

.8652 

. 7303 

.7793 

.0148 

. 1464 

. 8536 

11 

44 

20 

50 

.17078 

.82922 

5.8554 

.17333 

5.7694 

1.0149 

.01469 

.98531 

10 

40 

24 

51 

. 7107 

. 2893 

.8456 

. 7363 

.7594 

.0150 

. 1474 

. 8526 


36 

28 

52 

. 7136 

. 2864 

.8358 

. 7393 

.7495 

.0150 

. 1479 

. 8521 

8 

32 

32 

53 

. 7164 

. 2836 

.8261 

. 7423 

.7396 

.0151 

. 1484 

. 8516 

7 

28 

36 

54 

. 7193 

. 2807 

.8163 

. 7453 

.7297 

.0151 

. 1489 

. 8511 

6 

24 

40 

55 

.17221 

.82778 

5.8067 

.17483 

5.7199 

1.0152 

.01494 

.98506 

5 

20 

44 

56 

. 7250 

. 2750 

.7970 

. 7513 

.7101 

.0152 

. 1499 

. 8501 

4 

16 

48 

57 

. 7279 

. 2721 

.7874 

. 7543 

.7004 

.0153 

. 1504 

. 8496 

3 

12 

52 

58 

. 7307 

. 2692 

.7778 

. 7573 

.6906 

.0153 

. 1509 

. 8491 

2 

8 

56 

59 

. 7336 

. 2664 

.7683 

. 7603 

.6809 

.0154 

. 1514 

. 8486 

1 

4 

40 

60 

. 7365 

. 2635 

.7588 

. 7633 

.6713 

.0154 

. 1519 

. 8481 

0 

30 

MS. 

M 

Cosine. 

Vrs.Sin. 

Secant e. 

Cotang. 

Tangent. 

Coscc nt I Vrs.Cos 

Sine. 

M 

MS. 

G h 

99° 




Natural. 




80° 

5 h 

















Natural Lines. 


255 


o h 

10° 

Natural Trigonometrical Functions 

169° 

n h 

M.S 

M 

Sine. 

Vrs. Cos 

Cosec'nte 

Tang. 

Co tang. 

Secante 

Vrs. Sin 

Cosine. 

M 

M.S. 

40 

0 

.17365 

.32635 

5.7588 

.17633 

5.6713 

1.0154 

.01519 

.98481 

60 

30 

4 

1 

. 7393 

. 2606 

.7493 

. 7663 

.6616 

.0155 

. 1524 

. 8476 

59 

56 

8 

2 

. 7422 

. 2578 

.7398 

. 7693 

.6520 

.0155 

. 1529 

. 8471 

58 

52 

12 

3 

. 7451 

. 2549 

.7304 

. 7723 

.6425 

.0150 

. 1534 

. 8*165 

57 

48 

16 

4 

. 7479 

. 2521 

.7210 

. 7753 

.6329 

.0156 

. 1539 

. 8460 

5G 

44 

20 

5 

.17508 

.82492 

5.7117 

.17783 

5.62:34 

10157 

.01544 

.98*155 

55 

40 

! 24 

6 

. 7537 

. 2463 

.7023 

. 7813 

.6140 

.0157 

. 1550 


54 

36 

28 

7 

. 7563 

. 2436 

.6930 

. 7843 

.6045 

.0158 

. 1555 

. 8445 

53 

32 

82 

8 

. 7594 

. 2406 

.6838 

. 7873 

.5951 

.0158 

. 1560 

. 8440 

52 

28 

36 

9 

. 7622 

. 2377 

.6745 

. 7903 

.5857 

.0159 

. 1565 

. 8435 

51 

24 

40 

10 

.17651 

.82349 

5.6653 

.17933 

5.5764 

1.0159 

.01570 

.98430 

50 

20 

44 

11 

. 7680 

. 2320 

.6561 

. 7963 

.5670 

.0160 

. 1575 

. 8425 

49 

16 

48 

12 

. 7708 

. 2291 

.6470 

. 7993 

.5578 

.0160 

. 1580 

. 8419 

48 

12 

52 

13 

. 7737 

. 2263 

.6379 

. 8023 

.5485 

.6161 

. 1585 

. 8414 

47 

8 

56 

14 

. 7766 

. 2234 

.6288 

. 8053 

.5393 

.0162 

. 1591 

. 8409 

46 

4 

41 

15 

.17794 

.82206 

5.6197 

.18083 

5.5301 

1.0162 

.01596 

.98404 

45 

19 

4 

16 

. 7823 

. 2177 

.6107 

. 8113 

.5209 

.0163 

. 1601 

. 8399 

44 

56 

8 

17 

. 7852 

. 2148 

.6017 

. 8143 

.5117 

.0163 

. 1606 

. 8394 

43 

52 

12 

18 

. 7880 

. 2120 

.5928 

. 8173 

.5026 

.0164 

. 1611 

. 8388 

42 

48 

16 

19 

. 7909 

. 2091 

.5838 

. 8203 

.4936 

.0104 

. 1617 

. 8383 

41 

44 

20 

20 

.17937 

.82062 

5.5749 

.18233 

5.4815 

1.0165 

.01622 

.98378 

40 

40 

24 

21 

. 7966 

. 2034 

.5660 

. 8263 

.4755 

.0165 

. 1627 

. 8373 

39 

36 

28 

22 

. 7995 

. 2005 

.5572 

. 8293 

.4665 

.0160 

. 1632 

. 8308 

38 

32 

32 

23 

. 8023 

. 1977 

.5484 

. 8323 

.4575 

.0166 

. 1638 

. 8362 

37 

28 

36 

24 

. 8052 

. 1948 

.5396 

. 8353 

.4486 

.0167 

. 1643 

. 8357 

36 

24 

40 

25 

.18080 

.81919 

5.5308 

.18383 

5.4396 

1.0167 

.01648 

.98352 

35 

20 

4*4 

26 

. 8109 

. 1891 

.5221 

. 8413 

.4308 

.0168 

. 1653 

. 83*17 

34 

16 

48 

27 

. 8138 

. 1862 

.5134 

. 8444 

.4219 

.0169 

. 1659 

. 8:341 

33 

12 

52 

28 

. 8166 

. 1834 

.5047 

. 8474 

.4131 

.0169 

. 1664 

. 8336 

32 

8 

56 

29 

. 8195 

. 1805 

.4960 

. 8504 

.4043 

.0170 

. 1669 

. 8331 

31 

4 

4>i 

30 

.18223 

.81776 

5.4874 

.18534 

5.3955 

1.0170 

.01674 

.98325 

30 

18 

4 

31 

. 8252 

. 1748 

.4788 

. 8564 

.3868 

.0171 

. 1680 

. 8320 

29 

56 

8 

32 

. 8281 

. 1719 

.4702 

. 8594 

.3780 

.0171 

. 1685 

. 8315 

28 

52 

12 

33 

. 8309 

. 1691 

.4617 

. 8624 

.3694 

.0172 

. 1690 

. 8309 

27 

48 

16 

34 

. 8338 

. 1662 

.4532 

. 8654 

.3607 

.0172 

. 1696 

. 8304 

26 

44 

20 

35 

.18366 

.81633 

5.4447 

.18684 

5.3521 

1.0173 

.01701 

.98299 

25 

40 

24 

36 

. 8395 

. 1605 

.4362 

. 8714 

.3434 

.0174 

. 1706 

. 8293 

24 

36 

28 

37 

. 8424 

. 1576 

.4278 

. 8745 

.3349 

.0174 

. 1712 

. 8288 

23 

32 

32 

38 

. 8452 

. 1548 

.4194 

. 8775 

.3263 

.0175 

. 1717 

. 8283 

22 

28 

36 

39 

. 8481 

. 1519 

.4110 

. 8805 

.3178 

.0175 

. 1722 

. 8277 

21 

24 

40 

40 

.18503 

.81490 

5.4026 

.18835 

5.3093 

1.0176 

.01728 

.98272 

20 

20 

44 

41 

. 8538 

. 1462 

.3943 

. 8865 

.3008 

.0176 

. 1733 

. 8267 

19 

16 

48 

42 

. 8567 

. 1433 

.3860 

. 8895 

.2923 

.0177 

. 1739 

. 8261 

18 

12 

52 

43 

. 8595 

. 1405 

.3777 

. 8925 

.2839 

.01*77 

. 1744 

. 8256 

17 

8 

56 

44 

. 8624 

. 1376 

.3695 

. 8955 

.2755 

.0178 

. 1749 

. 8250 

16 

4 

43 

45 

.18652 

.81348 

5.3612 

.18985 

5.2671 

1.0179 

.01755 

.9S245 

15 

17 

4 

46 

. 8681 

. 1319 

.3530 

. 9016 

.2588 

.0179 

. 1760 

. 8240 

14 

56 

8 

47 

. 8709 

. 1290 

.3449 

. 9046 

.2505 

.0180 

. 1766 

. 8234 

13 

52 

12 

48 

. 8738 

. 1262 

.3367 

. 9076 

.2422 

.0180 

. 1771 

. 8229 

12 

48 

16 

49 

. 8767 

. 1233 

.3286 

. 9106 

.2339 

.0181 

. 1777 

. 8223 

11 

44 

20 

50 

.18795 

.81205 

5.3205 

.19136 

5.2257 

1.0181 

.01782 

.98218 

10 

40 

24 

51 

. 8824 

. 1176 

.3124 

. 9166 

.2174 

.0182 

. 1788 

. 8212 

9 

36 

28 

52 

. 8852 

. 1147 

.3044 

. 9197 

.2092 

.0182 

. 1793 

. 8207 

8 

32 

32 

53 

. 8881 

. 1119 

.2963 

. 9227 

.2011 

.0183 

. 1799 

. 8201 

7 

28 

36 

54 

. 8909 

. 1090 

.2883 

. 9257 

.1929 

.0184 

. 1804 

. 8196 

6 

24 

40 

55 

.18938 

.81062 

5.2803 

.19287 

5.1848 

1.0184 

.01810 

.98190 

5 

20 

*44 

56 

. 8967 

. 1033 

.2724 

. 9317 

.1767 

.0185 

. 1815 

. 8185 

4 

16 

48 

57 

. 8995 

. 1005 

.2645 

. 9347 

.1686 

.0185 

. 1821 

. 8179 

3 

12 

52 

58 

. 9024 

. 0976 

.2566 

. 9378 

.1606 

.0180 

. 1826 

. 8174 

2 

8 

56 

59 

. 9052 

. 0948 

.2487 

. 9403 

.1525 

.0186 

. 1832 

. 8168 

1 

4 

44 

60 

. 9081 

. 0919 

.2408 

. 9438 

.1445 

.0187 

. 1837 

. 8163 

0 

10 

M.S. 

M 

Cosine. 

Vrs. Sin. 

Seen me. 

Cotang. 

rangent. 

losec’ut i Vrs.Cos 

Sine. I 

M 

M.S. 

6" 

o 

c 

0 




Natural. 




7i) J 

5 h 










































256 


Natural Likes, 


o h 

11° 

Natural Trig 

onometrical 

Functions. 

168° 

— 

ll b 

M.S. 

M 

Sine. 

Vrs.Cos. 

Cosec’nte 

Tang. 

Cotang. 

Secante.'Vrs.Sin 

Cosine. 

14 

M.S. 

44 

0 

.19081 

.80919 

6.2408 

.19438 

5.1445 

1.0187 

.01837 

.98163 

60 

Iti 

4 

1 

. 9109 

. 0890 

.2330 

. 9468 

.1366 

.0188 

. 1843 

. 8157 

59 

56 

8 

2 

. 9138 

. 0862 

.2252 

. 9498 

.1286 

.0188 

. 1848 

. 8152 

58 

52 

12 

3 

. 9166 

. 0833 

.2174 

. 9529 

.1207 

.0189 

. 1854 

. 8146 

57 

48 

16 

4 

. 9195 

. 0805 

.2097 

. 9559 

.1128 

.0189 

. 1859 

. 8140 

56 

44 

2(» 

5 

.19224 

.80776 

5.2019 

.19589 

5.1- 49 

1.0190 

.01865 

.98135 

55 

40 

24 

6 

. 9252 

. 0748 

.1942 

. 9619 

.0970 

.0191 

. 1871 

. 8129 

54 

36 

28 

7 

. 9281 

. 0719 

.1865 

. 9649 

.0892 

.0191 

. 1876 

. 8124 

53 

32 

32 

8 

. 9309 

. 0691 

• .1788 

. 9680 

.0814 

.0192 

. 1882 

. 8118 

52 

2S 

36 

9 

. 9338 

. 0662 

.1712 

. 9710 

.0736 

.01)2 

. 1887 

. 8112 

51 

24 

40 

10 

.19366 

.80634 

6.1636 

.19740 

5.0658 

1.0193 

.018573 

.98107 

50 

20 

44 

11 

. 9395 

. 0605 

.1560 

. 9770 

.0581 

.0193 

. 1899 

. 8101 

49 

16 

48 

12 

. 9423 

. 0576 

.1484 

. 9800 

.0504 

.0194 

. 1904 

. 8095 

48 

12 

52 

13 

. 9452 

. 0548 

.1409 

. 9831 

.0427 

.0195 

. 1910 

. 8090 

47 

8 

56 

14 

. 9480 

. 0519 

.1333 

. 9861 

.0350 

.019.7 

. 1916 

. 8084 

46 

4 

45 

15 

.19509 

.80491 

5.1258 

.19891 

5.0273 

1.0196 

.01921 

.98078 

45 

15 

4 

16 

. 9537 

. 0462 

.1183 

. 9921 

.0197 

.0196 

. 1927 

. 8073 

44 

56 

8 

17 

. 9566 

. 0434 

.1109 

. 9952 

.0121 

.0197 

. 1933 

. 8067 

43 

52 

12 

18 

. 9595 

. 0405 

.1034 

. 9982 

.0015 

.0198 

. 1938 

. 8061 

42 

48 

16 

1!) 

. 9623 

. 0377 

.0960 

.20012 

4.9969 

.0198 

. 1944 

. 8066 

41 

44 

20 

20 

.19652 

.80348 

5.0886 

.20042 

4.9894 

1.0199 

.01950 

.98050 

40 

40 

24 

21 

. 9680 

. 0320 

.0812 

. 0073 

.9819 

.0199 

. 1956 

. 8044 

39 

36 

28 

22 

. 9709 

. 0291 

.0739 

. 0103 

.9744 

.0200 

. 1961 

. 8039 

38 

32 

32 

23 

. 9737 

. 0263 

.0666 

. 0133 

.9669 

.0201 

. 1967 

. 80-33 

37 

28 

36 

24 

. 9766 

. 0234 

.0693 

. 0163 

.9594 

.0201 

. 1973 

. 8027 

36 

24 

40 

25 

.19794 

.80206 

6.0520 

.20194 

4.9520 

1.0202 

.01979 

.98021 

35 

20 

44 

26 

. 9823 

. 0177 

.0447 

. 0224 

.9446 

.0202 

. 1984 

. 8016 

34 

16 

48 

27 

. 9851 

. 0149 

.0375 

. 0254 

.9372 

.0203 

. 1990 

. 8010 

33 

12 

52 

28 

. 9880 

. 0120 

.0302 

. 0285 

.9298 

.0204 

. 1996 

. 8004 

32 

8 

56 

29 

. 9908 

. 0092 

.0230 

. 0315 

.9225 

.0204 

. 2002 

. 7998 

31 

4 

40 

30 

.19937 

.80063 

5.0158 

.20345 

4.9151 

1.0206 

.02007 

.97992 

30 

14 

4 

31 

. 9965 

. 0035 

.0087 

. 0375 

.9078 

.0205 

. 2013 

. 7987 

29 

56 

8 

32 

. 9994 

. 0006 

.0015 

. 0406 

.9006 

.0206 

. 2019 

. 7981 

28 

52 

12 

33 

.20022 

.79978 

4.9944 

. 0436 

.8933 

.0207 

. 2025 

. 7975 

27 

48 

16 

34 

.20051 

.79949 

4.9873 

. 0466 

.8860 

.0207 

. 2031 

. 7969 

26 

44 

20 

35 

.20079 

.79921 

4.9802 

.20497 

4.8788 

1.0208 

.02037 

.97963 

25 

40 

24 

36 

. 0108 

. 0892 

.9732 

. 0527 

.8716 

.0208 

. 2042 

. 7957 

24 

36 

28 

37 

. 0136 

. 9863 

.9661 

. 0557 

.8644 

.0209 

. 2048 

. 7952 

23 

32 

32 

38 

. 0165 

. 9835 

.9591 

. 0588 

.8573 

.0210 

. 2054 

. 7946 

22 

28 

36 

39 

. 0193 

. 9807 

.9521 

. 0618 

.8501 

.0210 

. 2060 

. 7940 

21 

24 

40 

40 

.20222 

.79778 

4.9452 

.20648 

4.8430 

1.0211 

.02066 

.97934 

20 

20 

44 

41 

. 0250 

. 9750 

.9382 

. 0679 

.8359 

.0211 

. 2072 

. 7928 

19 

16 

48 

42 

. 0279 

. 9721 

.9313 

. 0709 

.8288 

.0212 

. 2078 

. 7922 

18 

12 

52 

43 

. 0307 

. 9693 

.9243 

. 0739 

.8217 

.0213 

. 2084 

. 7916 

17 

8 

56 

44 

. 0336 

. 9664 

.9175 

. 0770 

.8147 

.0213 

. 2089 

. 7910 

16 

4 

47 

45 

.20564 

.79636 

4.9106 

.20800 

4.8077 

1.0214 

.02095 

.97904 

15 

13 

4 

46 

. 0393 

. 9607 

.9037 

. 0830 

.8007 

.0215 

. 2101 

. 7899 

14 

56 

8 

47 

. 0421 

. 9579 

.8969 

. 0S61 

.7937 

.0215 

. 2107 

. 7893 

13 

52 

12 

48 

. 0450 

. 9550 

.8901 

. 0891 

.7867 

.0216 

. 2113 

. 7887 

12 

48 

16 

49 

. 0478 

. 9522 

.8833 

. 0921 

.7798 

.0216 

. 2119 

. 7S81 

11 

44 

20 

50 

.20506 

.79493 

4.8765 

.20952 

4.7728 

1.0217 

.02125 

,9787«y 

10 

40 

24 

51 

. 0535 

. 9465 

.8697 

. 0982 

.7659 

.0218 

. 2131 

. 7869 

9 

36 

28 

52 

. 0563 

. 9436 

.8630 

. 1012 

.7591 

.0218 

. 2137 

. 7S63 

8 

32 

32 

53 

. 0592 

. 9408 

.8563 

. 1043 

.7522 

.0219 

. 2143 

. 7857 

7 

28 

36 

54 

. 0620 

. 9379 

.8496 

. 1073 

.7453 

.0220 

. 2149 

. 7851 

6 

24 

40 

55 

.20649 

.79351 

4.8429 

.21104 

4.7385 

1.0220 

.02155 

.97845 

5 

20 

44 

56 

. 0677 

. 9323 

.8362 

. 1134 

.7317 

.0221 

. 2161 

. 7839 

4 

16 

48 

57 

. 0706 

. 9294 

.8296 

. 1104 

.7249 

.0221 

. 2167 

. 7833 

3 

12 

52 

58 

. 0734 

. 9266 

.8229 

. 1195 

.7181 

.0222 

. 2173 

. 7827 

2 

8 

56 

59 

. 0763 

. 9237 

.8163 

. 1225 

.7114 

.0223 

. 2179 

. 7821 

1 

4 

48 

60 

. 0791 

. 9209 

.8097 

. 1256 

.7046 

.0223 

. 2185 

. 7815 

0 

13 

M.S. 

6 h 

M 

ior 

Cosine. 

5 

Vrs.Sin. 

Seoante. 

Cotaug.'Tangeuc. 

Natural. 

Cosec'nt 

Vrs.Cos 

Sine. 

M 

78° 

M.S. 

5 h 
























Natural Lines. 


257 


0 ” 


oo 


12 


Natural Trigonometrical Functions. 


167 


IP 


M.S 

M 

Sine. 

Vrs.Cos 

. Cosec'ntt 

Tang. 

Cotang 

. Secante.’Vrs.Sii 

Cosine 

M 

M.S. 


0 

.20791 

.79209 

4.8097 

.21256 

4.7046 

1.0223 

.02185 

.97815 

60 

12 


1 

. 0820 

. 9180 

.8032 

. 1286 

.6979 

.0224 

. 2191 

. 7809 

59 

56 

o 

12 

10 

z 

. 084S 

. 9152 

.7966 

. 1316 

.6912 

.0225 

. 2197 

. 7803 

58 

52 

o 

. 0870 

. 9123 

.7901 

. 1347 

.6845 

.0225 

. 2203 

. 7806 

57 

48 

4: 

. 0905 

. 9105 

.7835 

. 1377 

.6778 

.0226 

. 2209 

. 7790 

56 

44 

z 0 

O 

.20933 

.79000 

4.7770 

.21408 

4.6712 

1.0226 

.02215 

.97784 

55 

40 

24 

O 

. 0902 

. 9038 

.7700 

. 1438 

.6646 

.0227 

. 2222 

. 7778 

54 

36 

Zo 

< 

. 0990 

. 9010 

.7641 

. 1468 

.6580 

.0228 

. 2228 

. 7772 

53 

32 

oZ 

36 

o 

. 1019 

. 8981 

.7570 

. 1499 

.6514 

.0228 

. 2234 

. 7766 

52 

28 

y 

10 

. 1047 

. 8953 

.7512 

. 1529 

.6448 

.0229 

. 2240 

. 7760 

51 

24 

JO 

.21070 

.78924 

4.7448 

.21560 

4.0382 

1.0230 

.02246 

.97754 

50 

20 

44 

11 

. 1104 

. 8896 

.7384 

. 1590 

.0317 

.0230 

. 2252 

. 7748 

49 

16 

48 

12 

. 1132 

. 8807 

.7320 

. 1621 

.6252 

.0231 

. 2258 

. 7741 

48 

12 

52 

13 

. 11G1 

. 8839 

.7257 

. 1651 

.6187 

.0232 

. 2264 

. 7735 

47 

8 

50 

14 

. 1189 

. 8811 

.7193 

. 1682 

.6122 

.0232 

. 2271 

. 7729 

46 

4 

49 

15 

.21218 

.78782 

4.7130 

.21712 

4.6057 

1.0233 

.02277 

.97723 

45 

11 

4 

16 

. 1240 

. 8754 

.7007 

. 1742 

.5993 

.0234 

. 2283 

. 7717 

44 

56 

8 

17 

. 1275 

. 8725 

.70n4 

. 1773 

.5928 

.0234 

. 2289 

. 7711 

43 

52 

12 

18 

. 1303 

. 8697 

.0942 

. 1803 

.5864 

.0235 

. 2295 

. 7704 

42 

48 

10 

19 

. 1331 

. 8008 

.0879 

. 1834 

.5800 

.0235 

. 2302 

. 7698 

41 

44 

20 

20 

.21360 

.78040 

4.0817 

.23864 

4.5736 

1.0236 

.02308 

.97692 

40 

40 

24 

21 

. 1388 

. S012 

.0764 

. 1895 

.5673 

.0237 

. 2314 

. 7086 

39 

36 

28 

22 

. 1417 

. 8583 

.6092 

. 1925 

.5609 

.0237 

. 2320 

. 7680 

38 

32 

32 

23 

. 1445 

. 8555 

.0631 

. 1956 

.5546 

.0238 

. 2326 

. 7673 

37 

28 

30 

24 

. 1473 

. 8520 

.6569 

. 1986 

.5483 

.0239 

. 2333 

. 7667 

36 

24 

40 

25 

.21502 

.78508 

4.6507 

.22017 

4.5420 

1.0239 

.02339 

.97061 

35 

20 

44 

20 

. 1530 

. 8470 

.6446 

. 2047 

.5357 

.0240 

. 2345 

. 7656 

34 

16 

48 

27 

. 1559 

. 8441 

.6385 

. 2078 

.5294 

.0241 

. 2351 

. 7648 

33 

12 

52 

2S 

. 1587 

. 8413 

.6324 

. 2108 

.5232 

.0241 

. 2358 

. 7642 

32 

8 

50 

29 

. 1015 

. 8384 

.6263 

. 2139 

.5109 

.0242 

. 2364 

. 7036 

31 

4 

50 

30 

.21G44 

.7S356 

4.6202 

.22169 

4.5107 

1.0243 

.02370 

.97630 

30 

10 

4 

31 

. 1072 

. 8328 

.6142 

. 2200 

.5015 

.0243 

. 2377 

. 7623 

29 

56 

8 

32 

. 1701 

. 8299 

.6081 

. 2230 

.4983 

.0244 

. 2383 

. 7617 

28 

52 

12 

33 

. 1729 

. 8-71 

.6021 

. 2261 

.4921 

.0245 

. 2389 

. 7611 

27 

48 

10 

34 

. 1757 

. 8242 

.5961 

. 2291 

.4860 

.0245 

. 2396 

. 7604 

26 

44 

20 

3.\ 

.21780 

.78214 

4.5901 

.22322 

4.4799 

1.0246 

.02402 

.97598 

25 

40 

24 

30 

. 1814 

. 8180 

.5841 

. 2353 

.4737. 

.0217 

. 2408 

. 7592 

24 

36 

28 

37 

. 1813 

. 8154 

.5782 

. 2383 

.4676 

.0247 

. 2415 

. 7585 

23 

32 

32 

38 

. 1871 

. 8129 

.5722 

. 2414 

.4615 

.0248 

. 2421 

. 7579 

22 

28 

30 

39 

. 1899 

. 8100 

.6663 

. 2444 

.4555 

.0249 

. 2427 

. 7573 

21 

24 

40 

40 

.21928 

.78072 

4.5604 

.22475 

4.4494 

1.0249 

.02434 

.97566 

20 

20 

14 

41 

. 1956 

. 8043 

.5515 

. 2505 

.4434 

.0250 

. 2440 

. 7560 

19 

16 

48 

42 

. 1985 

. 8015 

.5486 

. 2536 

.4373 

.0251 

. 2446 

. 7553 

18 

12 

52 

43 

. 2013 

. 7987 

.5428 

. 2506 

.4313 

.0251 

. 2453 

. 7547 

17 

8 

50 

44 

. 2041 

. 7959 

.5369 

. 2597 

.4253 

.0252 

. 2459 

. 7541 

1G 

4 

51 

45 

.22070 

.77930 

4.5311 

.22628 

4.4194 

1.0253 

.02466 

.97534 

15 

9 

4 

40 

. 2098 

. 7902 

.5253 

. 2658 

.4134 

.0253 

. 2472 

. 7528 

14 

56 

8 

47 

. 2120 

. 7873 

.5195 

. 2689 

.4074 

.0254 

. 2479 

. 7521 

13 

52 

12 

48 

. 2155 

. 7845 

.5137 

. 2719 

.4015 

.0255 

. 2185 

. 7515 

12 

48 

10 

49 

. 2183 

. 7S17 

.6079 

. 2750 

.3956 

.0255 

. 2491 

. 7508 

11 

44 

20 

50 

.22211 

.77788 

4.5021 

.22781 

4.3897 

1.0256 

.02498 

.97502 

10 

40 

24 

51 

. 2240 

. 7700 

.4964 

. 2811 

.3833 

.0257 

. 2504 

. 7495 

9 

36 

28 

52 

. 2208 

. 7732 

.4907 

. 2842 

.3779 

.0257 

. 2511 

. 7489 

8 

32 

32 

53 

. 2297 

. 7703 

.4850 

. 2872 

.3721 

.0258 

. 2517 

. 7483 

7 

28 

30 

54 

. 2325 

. 7075 

.4793 

. 2903 

.3662 

.0259 

. 2524 

. 7476 

6 

24 

40 

55 

.22353 

.77047 

4.4736 

.22934 

4.3604 

1.0260 

.02530 

.97470 

5 

20 

44 

56 

. 2382 

. 7018 

.4679 

. 2904 

.3546 

.0260 

. 2537 

. 7463 

4 

16 

48 

57 

. 2410 

. 7590 

.4623 

. 2995 

.3488 

.0261 

. 2543 

. 7457 

3 

12 

62 

58 

. 2438 

. 7501 

.4560 

. 3025 

.3430 

.0262 

. 2550 

. 7450 

2 

8 

50 

59 

. 2407 

. 7533 

.4510 

. 3056 

.3372 

.0262 

2556 

. 7443 

1 

4 

52 

60 

. 2495 

. 7505 

.4454 

. 3087 

.3315 

.0263 

2563 

. 7437 

0 

8 

M. 8. 
6 h 

M 

02° 

Cosine. 

Vrs.Sin. 

Seoaute. 

Cotang. Tangent. 

Natural. 

Cosec’nt I Vrs.Cos 

Sine. 

r 

M 

’7 o 

M.S 

5 b 


17 






































258 


Natural Lines. 


o h 

13° 

Natural Trigonometrical Functions 

166° 

M.S. 

M 

Sine. 

Vrs.Cos. 

Coscc'nte 

Tang. 

Cotang. 

Secan te. 

Vrs. Sin 

Cosine. 

M 

5 2 

0 

.22495 

.77505 

4.4454 

.23087 

4.3315 

1.0263 

.02563 

.97437 

60 

4 

1 

. 2523 

. 7476 

.4398 

. 3117 

.3257 

.0204 

. 2569 

. 7430 

59 

8 

2 

. 2552 

. 7448 

.4342 

. 3148 

.3200 

.0264 

. 2576 

. 7424 

58 

12 

3 

. 2580 

. 7420 

.4287 

. 3179 

.3143 

.0265 

. 2583 

. 7417 

57 

16 

4 

. 2608 

. 7391 

.4231 

. 3209 

.3086 

.0266 

. 2589 

. 7411 

56 

20 

5 

.22637 

.77363 

4.4176 

.23240 

4.3029 

1.0266 

.02596 

.97404 

55 

24 

6 

. 2665 

. 7335 

.4121 

. 3270 

.2972 

.0267 

. 2002 

. 7898 

54 

28 

7 

. 2693 

. 7306 

.4065 

. 3301 

.2916 

.0268 

. 2609 

. 7391 

58 

32 

8 

. 2722 

. 7278 

.4011 

. 3332 

.2859 

.0268 

. 2616 

. 7384 

52 

30 

9 

. 2750 

. 7250 

.3956 

. 3363 

.2803 

.0269 

. 2622 

. 7378 

51 

40 

10 

.22778 

.77221 

4.3901 

.23393 

4.2747 

1.0270 

.02629 

.97371 

50 

44 

11 

. 2807 

. 7193 

.3847 

. 3124 

.2691 

.0271 

. 2635 

. 7364 

49 

48 

12 

. 2835 

. 7165 

.3792 

. 34- 5 

.2035 

.0271 

. 2642 

. 735S 

48 

52 

13 

. 2863 

. 7136 

.3738 

. 3485 

.2579 

.0272 

. 2049 

. 7351 

47 

56 

14 

. 2892 

. 7108 

.3084 

. 3516 

.2524 

.0273 

. 2655 

. 7344 

4(5 

53 

15 

.22920 

.77080 

4.3630 

.23547 

4.2468 

1.0273 

.02662 

.97338 

45 

4 

16 

. 2948 

. 7052 

.3576 

. 3577 

.2413 

.0274 

. 2669 

. 7331 

44 

8 

17 

. 2977 

. 7023 

.3522 

. 3608 

.2358 

.0275 

. 2675 

. 7324 

48 

12 

18 

. 3005 

. 6995 

.3469 

. 3639 

.2303 

.0276 

. 2682 

. 7318 

42 

16 

19 

. 3033 

. 6967 

.3415 

. 3670 

.2248 

.0276 

. 2089 

. 7311 

41 

20 

20 

.23061 

.76938 

4.33G2 

.23700 

4.2193 

1.0277 

.02695 

.97304 

40 

24 

21 

. 3090 

. 6910 

.3309 

. 3731 

.2139 

.0278 

. 2702 

. 7298 

39 

28 

22 

. 3118 

. 6882 

.3256 

. 3762 

.2084 

.0278 

. 2709 

. 7291 

38 

32 

23 

. 3146 

. 6853 

.3203 

. 3793 

.2030 

.0279 

. 2716 

. 7284 

37 

36 

24 

. 3175 

. 6S25 

.3150 

. 3823 

.1976 

.0280 

. 2722 

. 7277 

36 

40 

25 

.23203 

.76797 

4.3098 

.23854 

4.1921 

1.0280 

.02729 

.97271 

35 

44 

26 

. 3231 

. 6769 

.3045 

. 3885 

.1867 

.0281 

. ^736 

. 7264 

34 

48 

27 

. 3260 

. 6740 

.2993 

. 3916 

.1814 

.0282 

. 2743 

. 7257 

33 

52 

28 

. 3288 

. 6712 

.2941 

. 3946 

.1760 

.0283 

. 2749 

. 7250 

32 

56 

29 

. 3316 

. 6684 

.2888 

. 3977 

.1706 

.0283 

. 2756 

. 7244 

31 

54 

30 

.23344 

.76655 

4.2836 

.24008 

4.1653 

1.0284 

.02763 

.97237 

30 

4 

31 

. 3373 

. 6627 

.2785 

. 4039 

.1600 

.0285 

. 2770 

. 7230 

29 

8 

32 

. 3401 

. 6599 

.2733 

. 4069 

.1546 

.0285 

. 2777 

. 7223 

28 

12 

33 

. 3429 

. 6571 

.2681 

. 4100 

.1493 

.0286 

. 2783 

. 7216 

27 

16 

34 

. 3458 

. 6542 

.2630 

. 4131 

.1440 

J1287 

. 2790 

. 7210 

26 

20 

35 

.23486 

.76514 

4.2579 

.24162 

4.1388 

1.0288 

.02797 

.97203 

25 

24 

30 

. 3514 

. 6486 

.2527 

. 4192 

.1335 

.0281$ 

. 2804 

. 7196 

'24 

28 

37 

. 3542 

. 6457 

.2476 

. 4223 

.1282 

.0289 

. 2811 

. 7189 

23 

32 

38 

. 3571 

. 6429 

.2425 

. 4254 

.1230 

.0290 

. 2818 

. 7182 

22 

36 

39 

. 3599 

. 6401 

.2375 

. 4285 

.1178 

.0291 

. 2824 

. 7175 

21 

40 

40 

.23627 

.76373 

4.2324 

.24316 

4.1126 

1.0291 

.02831 

.97169 

20 

44 

41 

. 3655 

. G344 

.2273 

. 4346 

.1073 

.0292 

. 2838 

. 7162 

19 

48 

42 

. 3084 

. 63 L6 

.2223 

. 4377 

.1022 

.0293 

. 2845 

. 7155 

18 

52 

43 

. 3712 

. 6288 

.2173 

. 4408 

.0970 

.0293 

. 2852 

. 7148 

17 

56 

44 

. 3740 

. 6260 

.2122 

. 4439 

.0918 

.0294 

. 2859 

. 7141 

16 

55 

45 

.23768 

.76231 

4.2072 

.24470 

4.0867 

1.0295 

.02866 

.97134 

15 

4 

46 

. 3797 

. 6203 

.2022 

. 4501 

.0S15 

.0296 

. 2873 

. 7127 

14 

8 

47 

. 3825 

. 6175 

.1972 

. 4531 

.0764 

.0296 

. 2880 

. 7120 

13 

12 

48 

. 3653 

. 6147 

.1923 

. 4562 

.0713 

.0297 

. 2886 

. 7113 
. 7L)0 

12 

16 

49 

. 3881 

. 6118 

.1873 

. 4593 

.0662 

.0298 

. 2893 

11 

20 

50 

.23910 

.76090 

4.1824 

.24624 

4.0611 

1.0299 

.02900 

.97099 

10 

24 

61 

. 3938 

. 6062 

.1774 

. 4655 

.0560 

.0299 

. 2907 

. 7092 

9 

28 

52 

. 3966 

. 6031 

.1725 

. 4686 

.0509 

.0300 

. 2914 

. 7086 

8 

32 

53 

. 3994 

. 6005 

.1676 

. 4717 

.0458 

.0301 

. 2921 

. 7079 

7 

30 

54 

. 4023 

. 5977 

.1627 

. 4747 

.0408 

.0302 

■. 2928 

. 7072 

6 

40 

65 

.21051 

.75949 

4.1578 

.24778 

4.0358 

1.0302 

.02935 

.97065 

5 

44 

56 

. 4079 

. 5921 

.1529 

. 4809 

.0307 

.0303 

. 2912 

. 7058 

4 

48 

57 

. 4107 

. 5892 

.1481 

. 4840 

.0257 

.0304 

. 2949 

. 7051 

3 

52 

58 

. 4136 

. 5864 

.1432 

. 4871 

.0207 

.0805 

. 2956 

. 7044 

2 

56 

59 

. 4164 

. 5836 

.1384 

. 4902 

.0157 

.0305 

. 2963 

. 7087 

1 

5<> 

60 

. 4192 

. 5808 

.1330 

. 4933 

.0108 

.0306 

. 2970 

. 7029 

0 

M.S. 

6 U 

— 

M 

103 

Cosine. 

Vrs.Siu.i 

Secan le. 

Colaug.[Tangent. 

Natural. 

Coseo'ut 1 

Vrs.Cos 

Sine. 

M 

76° 





































Natural Lines. 


259 


O h 


8 

12 

1G 

20 

24 

28 

32 

3G 

40 

44 

48 

52 

56 

58 
4 
8 

12 

16 

20 

24 

28 

32 

36 

40 

44 

48 

52 

56 

59 
4 
8 

12 

16 

20 

24 

28 

32 

36 

40 

41 
48 
52 
66 
«M> 
MS. 

G h 


14° 

Natural Trigonometrical Functions. 

165° 

ll h 

3. M 

Sine. 

Vrs. Cos 

Cosee'nte 

Tang. 

Cotang. 

Secante 

' Vrs. Sin 

Cosine. 

M 

M.S. 

» 0 

.24192 

.75808 

4.1336 

.24933 

4.0108 

1:0306 

.02970 

.97029 

60 

41 

1 

. 4220 

. 6779 

.1287 

. 4964 

.0058 

.0307 

. 2977 

. 7022 

59 

56 

2 

. 4249 

. 6751 

.1239 

. 4995 

.0009 

.0308 

.. 2984 

. 7015 

58 

52 

3 

. 4277 

. 6723 

.1191 

. 5025 

.9959 

.0308 

. 2991 

. 7008 

57 

48 

4 

. 4305 

. 5695 

.114-1 

. 6056 

3.9910 

.0309 

. 2999 

. 7001 

56 

44 

5 

.24333 

.75667 

4.1096 

.25087 

3.9861 

1.0310 

.03006 

.96994 

55 

40 

G 

. 4361 

. 5638 

.1048 

. 6118 

.9812 

.0311 

. 3013 

. 6987 

54 

36 

7 

. 4300 

. 6610 

.1001 

. 5149 

.9763 

.0311 

. 3020 

. 6980 

53 

32 

8 

. 4418 

. 6582 

.0953 

. 5180 

.9714 

.0312 

. 3027 

. 6973 

52 

28 

9 

. 4446 

. 5554 

.0906 

. 6211 

.9665 

.0313 

. 3034 

. 6966 

51 

2-4 

10 

.24474 

.75526 

4.0859 

.25242 

3.9616 

1.0314 

.03041 

.96959 

50 

20 

11 

. 4502 

. 5497 

.0812 

. 5273 

.9568 

.0314 

. 3048 

. 6952 

49 

16 

12 

. 4531 

. 5469 

.0765 

. 5304 

.9520 

.0315 

. 3055 

. 6944 

48 

12 

13 

. 4559 

. 5441 

.0718 

. 5335 

.9471 

.0316 

. 3063 

. 6937 

47 

8 

14 

. 4587 

. 5413 

.0672 

. 5366 

.9423 

.0317 

. 3070 

. 6930 

46 

4 

15 

.24615 

.75385 

4.0625 

.26397 

3.9375 

1.0317 

.03077 

.96923 

45 

3 

16 

. 4643 

. 6356 

.0679 

. 5428 

.9327 

.0318 

. 3084 

. 6916 

44 

50 

17 

. 4672 

. 5328 

.0532 

. 5459 

.9279 

.0319 

. 3091 

. 6909 

43 

52 

18 

. 4700 

. 5300 

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.9231 

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. 3098 

. 6901 

42 

48 

19 

. 4728 

. 5272 

.0440 

. 5521 

.9184 

.0320 

. 3106 

. 6894 

41 

44 

20 

.24756 

.75244 

4.0394 

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3.9136 

1.0321 

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40 

40 

21 

. 4784 

. 5215 

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. 6880 

39 

36 

22 

. 4813 

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38 

32 

23 

. 4841 

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.0256 

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. 6865 

37 

28 

24 

. 4869 

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.0211 

. 5676 

.8947 

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. 3142 

. 6858 

36 

24 

25 

.24897 

.75103 

4.0105 

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3.8900 

1.0325 

.03149 

.96851 

35 

20 

26 

. 4925 

. 5075 

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. 6844 

34 

16 

27 

. 4953 

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33 

12 

28 

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32 

8 

29 

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3.9984 

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31 

4 

30 

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3.9939 

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3.8667 

1.0329 

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30 

3 

31 

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29 

56 

32 

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28 

52 

33 

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27 

48 

31 

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. 3214 

. 6785 

26 

44 

35 

.25179 

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3.9716 

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3.8436 

1.0333 

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25 

40 

36 

. 5207 

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24 

36 

37 

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23 

32 

38 

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22 

28 

39 

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21 

24 

40 

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3.9495 

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3.8208 

1.0337 

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20 

20 

41 

. 5348 

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19 

16 

42 

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18 

12 

43 

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17 

8 

44 

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16 

4 

45 

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3.9277 

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3.7983 

1.0341 

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15 

l 

46 

. 5488 

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14 

56 

47 

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13 

52 

48 

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. 4455 

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12 

48 

49 

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11 

44 

50 

25601 

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3.900 L 

.20483 

3.7759 

1.0345 

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10 

40 

51 

. 6629 

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. 3340 

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9 

36 

52 

. 6657 

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. 

. 6662 

8 

32 

63 

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7 

28 

54 

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6 

24 

55 

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3.8848 

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3.7539 

1.0349 

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.96630 

5 

20 

56 

. 5769 

. 4230 

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. 6670 

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. 3377 

. 6623 

4 

16 

57 

. 5798 

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.87 63 

. 6701 

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. 6610 

3 

12 

58 

. 5826 

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. 6608 

2 

8 

59 

. 5854 

. 4146 

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. 6764 

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. 3400 

. 6600 

1 

4 

60 

. 5882 

. 4118 

.8637 

. 6795 

.7320 

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. 3407 

. 6592 

0 

0 

M 

104° 

Gosiue. 

Vrs.Sin. 

Secaulc. 

Cotuug.iTangeut. 

Natural. 

Cosec’nt! 

Vrs. Cos 

Sine. 

M 

75° 

M.S. 

5 h 





























200 


Natural Lines. 


l h 

15° 

Natural Trig 

onometrical Functions. 

164° 

10 h 

M.S. 

M 

Sine. 

Vrs.Cos. 

Cosec'nte 

Tang. 

Cotang. 

Secatite. 

Vrs. Sin 

Cosine. 

M 

M.S. 

0 

0 

.25882 

.74118 

3.8637 

.26795 

3.7320 

1.0353 

.03407 

.96592 

60 

OO 

4 

1 

. 5910 

. 4090 

.8595 

. 6826 

.7277 

.0353 

. 3415 

. 6585 

59 

56 

8 

2 

. 5938 

. 4062 

.8553 

. 6857 

.7234 

.0354 

. 3422 

. 0577 

58 

52 

12 

3 

. 5966 

. 4034 

.8512 

. 6888 

.7191 

.0355 

. 3430 

. 6570 

57 

48 

16 

4 

. 59'. 14 

. 4006 

.8470 

. 6920 

.7147 

.0356 

. 3438 

. 0562 

56 

44 

20 

5 

.26022 

.73978 

3.8428 

.26951 

3.7104 

1.0357 

.0:5445 

.96555 

55 

40 

24 

6 

. 6050 

. 3949 

.8387 

. 6982 

.7062 

.0358 

. 3453 

. 6547 

54 

36 

28 

7 

. 6078 

. 3921 

.8346 

. 7013 

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. 3400 

. 6540 

53 

32 

32 

8 

. 6107 

. 3893 

.8304 

. 7044 

.6976 

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. 3468 

. 6532 

52 

28 

36 

9 

. 6135 

. 3865 

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. 7076 

.6933 

.0360 

. 3475 

. 6524 

51 

24 

40 

10 

.26163 

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3.8222 

.27107 

3.6891 

1.0361 

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.96517 

50 

20 

44 

11 

. 6191 

. 3809 

.8181 

. 7138 

.6848 

.0362 

. 3491 

. 6609 

49 

16 

48 

12 

. 6219 

. 3781 

.8140 

. 7169 

.6806 

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. 3498 

. 6502 

48 

12 

62 

13 

. 6247 

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. 7201 

.6764 

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47 

8 

66 

14 

. 6275 

. 3725 

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. 7232 

.6722 

.0364 

. 3514 

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46 

4 

1 

15 

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3.8018 

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3.6679 

1.0365 

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45 

59 

4 

16 

. 6331 

. 3669 

.7978 

. 7294 

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44 

56 

8 

17 

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43 

52 

12 

18 

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42 

48 

16 

19 

. 6415 

. 3585 

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. 3552 

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41 

44 

20 

20 

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3.7816 

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3.6470 

1.0889 

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40 

40 

24 

21 

.. 6471 

. 3528 

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. 7451 

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. 3567 

. 04:13 

39 

36 

28 

22 

. 6499 

. 3500 

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. 7482 

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. 3575 

. 6425 

38 

32 

32 

23 

. 6527 

. 3472 

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. 7513 

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. 3583 

. 6117 

37 

28 

36 

24 

. 6556 

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. 7544 

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. 3590 

. 6409 

36 

24 

40 

25 

.26584 

.73416 

3.7617 

.27676 

3.6263 

1.0373 

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.96402 

35 

20 

44 

26 

. 6612 

. 3388 

.7577 

. 7607 

.6222 

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. 36U0 

. 6394 

34 

16 

48 

27 

. 6640 

. 3360 

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. 7638 

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. 3614 

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33 

12 

62 

28 

. 6668 

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32 

8 

50 

29 

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. 7701 

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31 

4 

a 

30 

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3.7420 

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3.6059 

1.0377 

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30 

58 

4 

31 

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29 

56 

8 

32 

. 6780 

. 8220 

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. 7795 

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28 

52 

12 

33 

. 6808 

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. 3660 

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27 

48 

16 

34 

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26 

44 

20 

35 

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3.7224 

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3.5856 

1.0382 

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25 

40 

24 

36 

. 6892 

. 3108 

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. 7920 

.5816 

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. 3684 

. 6316 

24 

36 

28 

37 

. 6920 

. 3080 

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. 3691 

. 6308 

23 

32 

32 

38 

. 6948 

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. 3699 

. 6301 

22 

28 

36 

39 

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. 8014 

.5696 

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. 6293 

21 

24 

40 

40 

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3.7031 

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3.5656 

1.0386 

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20 

20 

44 

41 

. 7032 

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. 8077 

.5616 

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. 3723 

. 6277 

19 

16 

48 

42 

. 7060 

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. 8109 

.6576 

.0387 

. 3731 

. 6269 

18 

12 

52 

43 

. 7088 

. 2912 

.6917 

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. 3739 

. 6261 

17 

8 

56 

44 

. 7116 

. 2884 

.6878 

. 8171 

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16 

4 

3 

45 

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3.6840 

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3.5457 

1.0390 

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15 

57 

4 

46 

. 7172 

. 2828 

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. 8234 

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. 3762 

. 6238 

14 

56 

8 

47 

. 7200 

. 2800 

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. 3770 

. 6230 

13 

52 

12 

48 

. 7228 

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.6727 

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12 

48 

16 

49 

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. 6214 

11 

44 

20 

50 

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3.6651 

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3.5261 

1.0394 

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10 

40 

24 

51 

. 7312 

. 2688 

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9 

36 

28 

52 

. 7340 

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. 3810 

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8 

32 

32 

53 

. 7368 

. 2632 

.6539 

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. 6182 

7 

28 

36 

54 

. 7396 

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6 

24 

40 

65 

.27424 

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3.6464 

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3.6( 166 

1.0399 

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5 

20 

44 

56 

. 7452 

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. 8649 

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. 3842 

. 6158 

4 

16 

48 

57 

. 7480 

. 2520 

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. 8580 

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3 

12 

52 

58 

. 7508 

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. 8611 

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2 

8 

56 

59 

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1 

4 

4: 

60 

. 7564 

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. 8674 

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. 3874 

. 6126 

0 

50 

M.S. 

M 

Cosine. 

Yrs.Siu.I Secaute. 

Cotang. 

Tuugent, 

Conee'nt 

V rs.Coa 

Sine. 

M 

M.S. 

7 h 

i 

105° 



Natural. 




74° 

4 u 

■ „ 

























Natural Lines, 


2(U 


\ 


l h 

16 c 

Natural Trigonometrical 

Functions. 

163° 

10 h 

M.S. 

M 

Sine. 

Vrs.Cos. 

Cosec'nte 

Tang. 

Cotang. 

Seeaute. j Vrs.Siu 

Cosine. 

M 

M .S. 

4 

0 

.27564 

.72436 

3.6279 

.28674 

3.4874 

1.0403 

.03874 

.96126 

60 

5G 

4 

1 

. 7592 

. 2408 

.6243 

. 8706 

.4836 

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. 3882 

. 6118 

59 

56 

8 

2 

. 7620 

. 2380 

.6206 

. 8737 

.4798 

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. 3890 

. 6110 

58 

52 

12 

3 

. 7648 

. 2352 

.6169 

. 8769 

.4760 

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. 3898 

. 6102 

57 

48 

16 

4 

• i 07 o 

. 2324 

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. 8800 

.4722 

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. 3906 

. 6094 

56 

44 

20 

5 

.27703 

.72296 

3.6096 

.28832 

3.4684 

1.0407 

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55 

40 

24 

6 

. 7731 

. 2268 

.6060 

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54 

36 

28 

7 

. 7759 

. 2240 

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53 

32 

32 

8 

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. 2213 

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. 8926 

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. 3938 

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52 

28 

36 

9 

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51 

24 

40 

10 

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3.5915 

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3.4495 

1.(412 

.03954 

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50 

20 

44 

11 

. 7871 

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49 

16 

48 

12 

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48 

12 

52 

13 

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47 

8 

56 

14 

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46 

4 

5 

15 

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3.5736 

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3.4308 

1.0416 

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45 

55 

4 

16 

. 8011 

. 1989 

.5700 

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44 

56 

8 

17 

. 8039 

. 1961 

.5665 

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43 

52 

12 

18 

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42 

48 

16 

19 

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. 44)28 

. 5972 

41 

44 

20 

20 

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3.5559 

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3.4124 

1.0420 

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40 

40 

24 

21 

. 8150 

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. 5956 

39 

36 

28 

22 

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. 5948 

38 

32 

32 

23 

. 8206 

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. 4060 

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37 

28 

36 

24 

. 8234 

. 1766 

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. 9432 

.3977 

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36 

24 

40 

25 

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3.5383 

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3.3941 

1.(425 

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35 

20 

44 

26 

. 8290 

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34 

145 

48 

27 

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33 

12 

52 

28 

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32 

8 

56 

29 

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31 

4 

G 

30 

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3.5209 

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3.3759 

1.0429 

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30 

54 

4 

31 

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29 

56 

8 

32 

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28 

52 

12 

33 

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27 

48 

16 

34 

. 8513 

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. 5849 

26 

44 

20 

35 

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3.5037 

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3.3580 

1.0434 

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25 

40 

24 

36 

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. 4168 

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24 

36 

28 

37 

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. 9843 

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23 

32 

32 

38 

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22 

28 

36 

39 

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21 

24 

40 

40 

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3.4867 

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3.3402 

1.0438 

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20 

20 

44 

41 

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19 

16 

48 

42 

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18 

12 

52 

43 

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17 

8 

56 

44 

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16 

4 

7 

45 

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3.4698 

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3.3226 

1.4443 

.04243 

.95757 

15 

5.i 

4 

46 

. 8847 

. 1152 

.4665 

. 0128 


.4444 

. 4251 

. 5749 

14 

56 

8 

47 

. 8875 

. 1125 

.4632 

. 0160 

.3156 

.0445 

. 4260 

. 5740 

13 

52 

12 

48 

. 8903 

. 1097 

.4598 

. 0192 

.3121 

.0446 

. 4268 

. 5732 

12 

48 

v 16 

49 

. 8931 

. 1069 

.4565 

. 0223 

.3087 

.4447 

. 4276 

. 5723 

11 

44 

20 

50 

.28959 

.71041 

3.4532 

.30255 

3.3052 

1.(448 

.4)4285 

.95715 

10 

40 

24 

51 

. 8987 

. 1013 

.4498 

. 0287 

.3017 

.(448 

. 4293 

. 5707 

9 

36 

28 

52 

. 9014 

. 0985 

.4465 

. 0319 

.2983 

.4449 

. 4302 

. 5698 

8 

32 

32 

53 

. 9042 

. 0958 

.4432 

. 0350 

.2948 

.0450 

. 4310 

. 6690 

7 

28 

3(> 

54 

. 9(»70 

. 0930 

.4399 

. 0382 

.2914 

.0451 

. 4319 

. 5681 

6 

24 

40 

55 

.29098 

.70902 

3.4366 

.30414 

3.2879 

1.0452 

.04327 

.95673 

5 

20 

44 

56 

. 9126 

. 0874 

.4334 

. 0446 

.2845 

.0453 

. 4335 

. 5664 

4 

16 

48 

57 

. 9154 

. 0846 

.4301 

. 0478 

.2811 

.0454 

. 4344 

. 5656 

O 

12 

52 

58 

. 9181 

. 0818 

.4268 

. 0509 

.2777 

.(455 

. 4352 

. 6647 

2 

8 

56 

59 

. 9209 

. 0791 

.4236 

. 0541 

.2742 

.0456 

. 4361 

. 6639 

1 

4 

8 

60 

. 9237 

. 0763 

.4203 

. 0573 

.2708 

.0457 

. 4369 

. 5630 

0 

5« 

MS. 

7 h 

M 

10G C 

Gosiue. 

Vrs.Siu. 

Secante. 

Cotaug.jTangent. 

Natural. 

Cosec'nt 

Vrs.Cos 

Sine. 

M 

73° 

M.S. 

4 h 




























262 Natural Lines. 


l h 

17° 

Natural Trig 

onometrioal Functions 

162° 

M.S. 

M 

Sine. 

Vrs. Cos. 

Cosec’nte 

Tang. 

Cotang. 

Secat.ce. 

Vrs. Sin 

Cosine. 

M 

8 

0 

.29237 

.70763 

3.4203 

.30573 

3.2708 

1.0457 

.04369 

.95630 

60 

4 

1 

. 9265 

. 0735 

.4170 

. 0606 

.2074 

.0458 

. 4378 

. 5622 

59 

8 

2 

. 9293 

. 0707 

.4138 

. 0637 

.2640 

.0459 

. 4386 

. 5613 

58 

12 

3 

. 9321 

. 0679 

.4100 

. 0668 

.2607 

.0460 

. 4395 

. 5605 

57 

16 

4 

. 934S 

. 0651 

.4073 

. 0700 

.2573 

.0461 

. 4404 

. 5596 

66 

20 

5 

.29376 

.70624 

3.4041 

.30732 

3.2539 

1.0461 

.04412 

.95588 

55 

24 

6 

. 9404 

. 0596 

.4009 

. 0764 

.2505 

.0462 

. 4421 

. 5579 

54 

28 

7 

. 9432 

. 0568 

.3977 

. 0796 

.2472 

.0463 

. 4426 

. 5571 

53 

62 

8 

. 9460 

. 0540 

.3945 

. 0828 

.2438 

.0464 

. 4138 

. 5562 

52 

36 

9 

. 94^7 

. 0512 

.3913 

. 0859 

.2405 

.0465 

. 4446 

. 5554 

51 

' 40 

10 

.29515 

.70485 

3.3881 

.30891 

3.2371 

1.0466 

.04455 

.95545 

50 

44 

11 

. 9543 

. 0457 

.3849 

. 0923 

.2338 

.0467 

. 4463 

. 5536 

49 

48 

12 

. 9571 

. 0429 

.3817 

. 0955 

.2305 

.0468 

. 4472 

. 5528 

48 

52 

13 

. 9598 

. 0401 

.3785 

. 0987 

.2271 

.0469 

. 4481 

. 5519 

47 

56 

14 

. 9626 

. 0374 

.3751 

. 1019 

.2238 

.0470 

. 4489 

. 5511 

4G 

‘J 

15 

.29654 

.70346 

3.3722 

.31051 

3.2205 

1.0471 

.04498 

.95502 

45 

4 

16 

. 9682 

. 0318 

.3090 

. 1083 

.2172 

.0472 

. 4507 

. 5493 

44 

8 

17 

. 9710 

. 0290 

.3659 

. 1115 

.2139 

.0473 

. 4515 

. 5485 

43 

12 

IS 

. 9737 

. 0262 

.3627 

. 1146 

.2106 

.0474 

. 4524 

. 5476 

42 

16 

19 

. 9765 

. Q235 

.3596 

. 1178 

.2073 

.0475 

. 4532 

. 5467 

41 

20 

20 

.29793 

.70207 

3.3565 

.31210 

3.2041 

1.0476 

.04541 

.95459 

40 

24 

21 

. 9S21 

. 0179 

.3634 

. 1242 

.2008 

.0177 

. 4550 

. 5450 

39 

28 

22 

. 9848 

. 0151 

.3502 

. 1274 

.1975 

.0478 

. 4558 

. 5441 

38 

32 

23 

. 9876 

. 0124 

.3471 

. 1306 

.1942 

.0478 

. 4567 

. 5433 

37 

^() 

24 

. 9904 

. 0096 

.3440 

. 1338 

.1910 

.0479 

. 4576 

. 5424 

36 

40 

25 

.29932 

.70068 

3.34<i9 

.31370 

3.1877 

1.0480 

.04585 

.95415 

35 

44 

26 

. 9959 

. 0040 

.3378 

. 1402 

.1845 

.0481 

. 4593 

. 5407 

34 

48 

27 

. 9987 

. 0013 

.3347 

. 1434 

.1813 

.0482 

. 4602 

. 5398 

33 

52 

28 

.30015 

.69982 

.3316 

. 1466 

.1780 

.0483 

. 4611 

. 5389 

32 

56 

29 

.30043 

. 9951 

.3286 

. 1498 

.1748 

.0484 

. 4619 

. 5380 

31 

10 

30 

.30070 

.69929 

3.3255 

.31530 

3.1716 

1.0485 

.04628 

.95372 

30 

4 

31 

. 0098 

. 9902 

.3224 

. 1562 

.1684 

.0486 

. 4637 

. 5363 

29 

8 

32 

. 0126 

. 9874 

.3194 

. 1594 

.1652 

.0487 

. 4646 

. 5354 

28 

12 

33 

. 0154 

. 9846 

.3103 

. 1626 

.1620 

.0488 

. 4654 

. 5345 

27 

16 

34 

. 0181 

. 9818 

.3133 

. 1658 

.1588 

.0489 

. 4063 

. 5337 

26 

20 

35 

.30209 

.69791 

3.3102 

.31690 

3.1556 

1.0490 

.04672 

.95328 

25 

24 

36 

. 0237 

. 9763 

.3( 72 

. 1722 

.1524 

.0491 

. 4681 

. 5319 

24 

28 

37 

. 02*45 

. 9735 

.3042 

. 1754 

.1492 

.0492 

. 4690 

. 5310 

23 

32 

38 

. 0292 

. 9707 

.3011 

. 1786 

.1460 

.041*3 

. 4098 

. 5301 

22 

36 

39 

. 0320 

. 9680 

.2981 

. 1818 

.1429 

.0494 

. 4707 

. 5293 

21 

40 

40 

.30348 

.69652 

3.2951 

.31850 

3.1397 

1.0495 

.04716 

.95284 

20 

44 

41 

. 0375 

. 9624 

.2921 

. 1882 

.1366 

.0496 

. 4725 

. 5275 

19 

48 

42 

. 04 03 

. 9597 

.2891 

. 1914 

.1334 

.0497 

. 4734 

. 5266 

18 

52 

43 

. 0431 

. 9569 

.2861 

. 1946 

.1303 

.0498 

. 4743 

. 5257 

17 

56 

44 

. 0459 

. 9541 

.2831 

. 1978 

.1271 

.0499 

. 4751 

. 5248 

16 

31 

45 

.30486 

.69513 

3.2801 

.32010 

3.1240 

1.0500 

.04760 

.95239 

15 

4 

46 

. 0514 

. ‘i486 

.2772 

. 2042 

.1209 

.0501 

. 4769 

/ 5231 

14 

8 

47 

. 0542 

. 9458 

.2742 

. 2074 

.1177 

.0502 

. 4778 

. 5222 

13 

12 

48 

. 0569 

. 9430 

.2712 

. 2106 

.1146 

.0503 

. 4787 

. 5213 

12 

16 

49 

. 0597 

. 9403 

.2683 

2138 

.1115 

.0504 

. 4796 

. 5204 

11 

20 

50 

.30625 

.69375 

3.2653 

.32171 

3.1084 

1.0505 

.04805 

.95195 

10 

24 

51 

. 0653 

. 9347 

.2624 

. 2203 

.1053 

.0506 

. 4814 

. 5186 

9 

28 

52 

. 0680 

. 9320 

.2594 

. 2235 

.1022 

.0507 

. 4823 

. 5177 

8 

32 

53 

. 07 08 

. 9292 

.2565 

. 2267 

.0991 

.0508 

. 4832 

. 5108 

t 

36 

54 

. 0736 

. 9264 

.2535 

. 2299 

.0960 

.0509 

. 4840 

. 5159 

6 

40 

55 

.30763 

.69237 

3.2506 

.32331 

3.0930 

1.0510 

.04849 

.95150 

5 

44 

56 

. 0791 

. 9209 

.2477 

. 2363 

.0899 

.0511 

. 4858 

. 5141 

4 

48 

57 

. 0819 

. 9181 

.2448 

. 2395 

.0808 

.0512 

. 4867 

. 5132 

3 

52 

58 

. 0846 

. 9154 

.2419 

. 2428 

.0838 

.0513 

. 4876 

. 5124 

2 

56 

59 

. 0874 

. 9126 

.2390 

. 2460 

.0807 

.0514 

. 4885 

. 5115 

1 

15i 

60 

. 0902 

. 9098 

.2361 

. 2492 

.0777 

.0515 

. 4894 

. 51uG 

0 

M.S. 

M 

Cosine. 

Vrs. Sin. 

Secante. 

Cotang. 

Tangent. 

Cosec’nt 

Vrs.Cos 

Sine. 

M 

7“ 

107‘ 

D 



Natural. 




72° 






















Natural Lines. 


263 


l h 

18 

0 

Natural Trigonometrical Functions 

161° 

10 h 

M.S. 

M 

Sine. 

Vrs.Oos 

Cosec'nte 

Tang. 

Co tang. 

Secante. 

Vrs. Sin 

Cosine. 

M 

M.S. 

12 

0 

.30902 

.69098 

3.23G1 

.32492 

3.0777 

1.0515 

.04894 

.95106 

60 

4S 

4 

1 

. 0929 

. 9071 

.2332 

. 2524 

.0746 

.0516 

. 4908 

. 5097 

59 

56 

8 

2 

. 0957 

. 9043 

.2303 

. 2550 

.0716 

.0517 

. 4912 

. 5088 

58 

52 

12 

3 

. 0985 

. 9015 

.2274 

. 2588 

.0686 

.0518 

. 4921 

. 5079 

57 

48 

16 

4 

. 1012 

. 8988 

.2245 

. 2621 

.0655 

.0519 

. 4930 

. 5070 

56 

44 

20 

5 

.31040 

.68060 

3.2216 

.I >2653 

3.0625 

10520 

.04939 

195061 

55 

40 

24 

6 

. 1068 

. 8932 

.2L88 

. 2685 

.0595 

.0521 

. 4948 

. 5051 

54 

36 

28 

7 

. 1095 

. 8905 

.2159 

. 2717 

.0565 

.0522 

. 4957 

. 5042 

53 

32 

32 

8 

. 1123 

. 8877 

.2131 

. 2749 

.0535: 

.0523 

. 4966 

. 5033 

52 

28 

36 

9 

. 1150 

. 8849 

.2102 

. 2782 

.0505 

.0524 

. 4975 

. 5024 

51 

24 

40 

10 

.31178 

.68822 

3.2074 

.32814 

3.0475 

1.0525 

.04985 

.95015 

50 

20 

44 

11 

. 1206 

. 8794 

.2045 

. 2846 

.0445 

.0526 

. 4994 

. 5006 

49 

16 

48 

12 

. 1233 

. 87 00 

.2017 

. 2878 

.0415 

.0527 

. 5003 

. 4997 

48 

12 

52 

13 

. 1261 

. 8739 

.1989 

. 2910 

.0385 

.0528 

. 5012 

. 4988 

47 

8 

56 

14 

. 1289 

. 8711 

.1960 

. 2943 

.0356 

.0829 

. 5021 

. 4979 

46 

4 

13 

15 

.31316 

.68684 

3.1932 

.32975 

3.0326 

1.0530 

.05030 

.94970 

45 

47 

4 

16 

. 1344 

. 8050 

.1901 

. 3007 

.0296 

.0531 

. 5039 

. 4961 

44 

56 

8 

17 

. 1372 

. 8628 

.18/6 

. 3039 

.0267 

.0532 

. 5048 

. 4952 

43 

52 

12 

18 

. 1399 

. 860 L 

.1848 

. 3072 

.0237 

.0533 

. 5057 

. 4942 

42 

48 

16 

19 

. 1427 

. 8573 

.1820 

. 3104 

.0208 

.0534 

. 5066 

. 4933 

41 

44 

20 

20 

.31454 

.68545 

3.1792 

.33136 

3.0178 

1.0535 

.05076 

.94924 

40 

40 

24 

21 

. 1482 

. 8518 

.1764 

. 3169 

.0149 

.0536 

. 5085 

. 4915 

39 

36 

28 

22 

. 1510 

. 8490 

.1736 

. 3201 

.0120 

.0537 

. 5094 

. 4906 

38 

32 

32 

23 

. 1537 

. 8463 

.1708 

. 3233 

.0090 

.0538 

. 5103 

. 4897 

37 

28 

36 

24 

. 1565 

. 8435 

.1681 

. 3265 

.006 L 

.0539 

. 5 L12 

. 4888 

36 

24 

40 

25 

.31592 

.68407 

3.1653 

.33298 

3.0032 

1.0540 

.05121 

.94878 

35 

20 

‘14 

26 

. 1620 

. 8380 

.1025 

. 3330 

3.0003 

.0541 

. 5131 

. 4809 

34 

16 

48 

27 

. 1648 

. 8352 

.1598 

. 33)52 

2.9974 

.0542 

. 5140 

. 4860 

33 

12 

52 

28 

• 16 / 5 

. 8325 

.1570 

. 3395 

.9945 

.0543 

. 5149 

. 4851 

32 

8 

56 

29 

. 1703 

. 8207 

.1543 

. 3427 

.9916 

.0544 

. 5158 

. 4841 

31 

4 

14 

30 

.31730 

.68269 

3.1515 

.33459 

2.9887 

1.0545 

.05168 

.94832 

30 

4G 

4 

31 

. 1758 

. 8242 

.1488 

. 3492 

.9858 

.0546 

. 5177 

. 4823 

29 

56 

8 

32 

. 1786 

. 8214 

.1461 

. 3524 

.9829 

.0547 

. 5186 

. 4814 

28 

52 

12 

33 

. 1813 

. 8187 

.1433 

• 3ql) i 

.9800 

.0548 

. 5195 

. 4805 

27 

48 

16 

34 

. 1841 

. 8159 

.1406 

. 3589 

.9772 

.0549 

. 5205 

. 4795 

26 

44 

20 

35 

.31868 

.68132 

3.1379 

.33621 

2.9743 

1.0550 

.05214 

.94786 

25 

40 

24 

36 

. 1896 

. 8104 

.1352 

. 3654 

.9714 

.0551 

. 5223 

. 4777 

24 

36 

28 

37 

. 1923 

. 8o76 

.1325 

. 3686 

.9686 

.0552 

. 5232 

. 4767 

23 

32 

32 

38 

. 1951 

. 8049 

.1298 

. 3718 

.9657 

.0553 

. 5242 

. 4758 

22 

28 

36 

39 

. 1978 

. 8021 

.1271 

. 3751 

.9629 

.0554 

. 5251 

. 4749 

21 

24 

40 

40 

.32006 

.67904 

3.1244 

.33783 

2.9600 

1,0555 

.05260 

.94740 

20 

20 

44 

41 

. 2034 

. 7966 

.1217 

. 3816 

.9572 

.0556 

. 5270 

. 4730 

19 

16 

48 

42 

. 2061 

. 7939 

.1190 

. 3848 

.9544 

.0557 

. 5279 

. 4721 

18 

12 

52 

43 

. 2089 

. 7911 

.1163 

. 3880 

.9515 

.0558 

. 5288 

. 4712 

17 

8 

56 

44 

. 2116 

. 7884 

.1137 

. 3913 

.9487 

.0559 

. 5297 

. 47 u2 

16 

4 

15 

45 

.32141 

.67856 

3.1110 

.33945 

2.9459 

1.0560 

.05307 

..94693 

15 

45 

4 

46 

. 2171 

. 7828 

.1083 

. 3978 

.9481 

.0561 

. 5316 

. 4684 

14 

56 

8 

47 

. 2199 

. 78ol 

.1057 

. 4010 

.9403 

.0562 

. 5326 

. 46/4 

13 

52 

12 

48 

. 2226 

. 7773 

.1030 

. 4043 

.9375 

.0563 

. 5335 

. 4605 

12 

48 

16 

49 

. 2254 

. 7746 

.10U4 

4075 

.9347 

.0565 

. 5344 

. 4655 

11 

44 

20 

50 

* .32282 

.67718 

3.0977 

.34108 

2.9319 

1.05G6 

.0635 4 

.94646 

10 

40 

24 

51 

. 2309 

. 7691 

.0951 

. 4140 

.9291 

.0567 

. 5363 

. 4637 

9 

36 

28 

52 

. 2337 

. 7063 

.0925 

. 4173 

.9263 

.0568 

. 5373 

. 4627 

8 

32 

32 

53 

. 2364 

. 7636 

.0898 

. 4205 

.9235 

.0569 

. 5382 

. 4618 

7 

28 

36 

54 

. 23.*2 

. 7008 

.0872 

. 4238 

.9208 

..0570 

. 5391 

. 4608 

6 

24 

40 

55 

.32419 

.67581 

3.0846 

.34270 

2.9180 

1.0571 

.05401 

.94599 

5 

20 

'14 

56 

. 2447 

. 7653 

.0820 

. 4303 

.9152 

.0572 

. 5410 

. 4590 

4 

16 

48 

57 

. 2474 

. 7526 

.0793 

. 4335 

.9125 

.0573 

. 5420 

. 4580 

3 

12 

52 

58 

. 2502 

. 7498 

.0707 

. 4308 

.9097 

.0574 

. 5429 

. 4571 

2 

8 

56 

59 

. 2529 

. 7471 

.0741 

. 4400 

.9069 

.0575 

. 5439 

. 4561 

1 

4 

10 

60 

. 2557 

. 7443 

.0715 

. 4433 

.9042 

.0576 

. 5448 

. 4662 

0 

44 

M.S. 

M 

Cosine. 

Vrs.Siu.l 

Secaute. 

Cotaug.iTangeut. 

Cosec’ut IVrs.Cos 

Sine. 

M 

M.S. 

?h 

108 c 




Natural. 




71° 

4 U 











































264 


Natural Lines. 


l h 

19° 

Natural Trig 

onometrical Functions. 

160° 

10 h 

M.S. 

M 

Sine. 

Vrs.Cos. 

Cosec'nte 

1 Tang. 

Cotang. 

Secante.'Vrs.Sin 

Cosine. 

M 

M.S. 

16 

0 

.32557 

.67443 

3.0715 

.344433 

2.9042 

1.0576 

.05448 

.94552 

60 

44 

4 

1 

. 2584 

. 7416 

.0690 

. 4465 

.9015 

.0577 

. 5458 

. 4542 

59 

56 

8 

2 

. 2612 

. 7388 

.0664 

. 4498 

.8987 

.0578 

. 5467 

. 45433 

58 

52 

12 

Q 

O 

. 2639 

. 7361 

.0638 

. 4530 

.8960 

.0579 

. 5476 

. 4523 

57 

48 

16 

4 

. 2667 

. 7333 

.0612 

. 45643 

.89343 

.0580 

. 5486 

. 4514 

56 

44 

20 

5 

.32694 

.674506 

3.0586 

.34595 

2.8905 

1.0581 

.05495 

.94504 

55 

40 

24 

6 

. 2722 

. 7278 

.0561 

. 4628 

.8878 

.0582 

. 5505 

. 4495 

54 

36 

28 

7 

. 2749 

. 7251 

.0535 

. 4661 

.8851 

.0584 

. 5515 

. 4485 

53 

32 

32 

8 

. 2777 

. 7223 

.0509 

. 4693 

.8824 

,i i585 

. 5524 

. 4476 

52 

28 

36 

9 

. 2804 

. 7196 

.0484 

. 4726 

.8797 

.0586 

. 5534 

. 4466 

51 

24 

40 

10 

.32832 

.67168 

3.0458 

.34758 

2.8770 

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Cosine. 

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265 


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Natural Trigonometrical Functions 

159° 

10 h 

M. S 

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Sine. 

Vrs.Cos. 

Cosec'nte 

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Cotang. 

Secante. 

Vrs. Sin 

Cosine. 

M 

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Vrs.Sin.l 

Socaute. 

Cotang. 

Tangent. 

Cosec'nt 

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M 

M.S. 

7 h 

110° 



Natural. 




69° 

4 b 























Natural Lines. 


266 


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Natural Trigonometrical Functions. 

158° 

10 h 

M.S 

M 

Sine. 

Vrs.Cos. 

(Coeec'nte 

Tang. 

Cotnug. 

Secante. 

(Vrs.Sin 

Cosine. 

M 

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52 

12 

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2.5278 

1.0754 

.07012 

32988 

25 

40 

24 

36 

. 6812 

. 3187 

.7165 

. 9693 

.5257 

.0755 

. 7022 

. 2978 

24 

36 

28 

37 

. 6839 

. 3160 

.7145 

. 9626 

.5236 

.0756 

. 7033 

. 2967 

23 

32 

32 

38 

. 6666 

. 3133 

.7125 

. 9660 

.5214 

.0758 

. 7044 

. 2956 

22 

28 

36 

39 

. 6893 

. 3106 

.7105 

. 9694 

.5193 

.0759 

. 7051 

. 2945 

21 

24 

40 

40 

.36921 

.63079 

2.7085 

.39727 

2.5171 

1.0760 

.07065 

.92935 

20 

20 

44 

41 

. 6948 

. 3052 

:7066 

. 9761 

.5150 

.0761 

. 5<r76 

. 2924 

19 

16 

48 

42 

. 6975 

. 3025 

.7045 

. 9795 

.5129 

.U763 

. 7087 

. 2913 

18 

12 

52 

43 

. 7002 

. 2998 

.7026 

. 9828 

.5108 

.0764 

. 7097 

. 2902 

17 

8 

56 

44 

. 7029 

. 2971 

.7006 

. 9862 

.5086 

.0765 

. 7108 

. 2892 

10 

4 

27 

45 

.37066 

.62944 

2.6986 

.39896 

2.5005 

1.6766 

.07119 

.92881 

15 

33 

4 

46 

. 7083 

. 2917 

.6967 

. 9930 

.5044 

.0768 

. 7130 

. 2870 

14 

56 

8 

47 

. 7110 

. 2800 

.6947 

. 9963 

.5023 

.0769 

. 7141 

. 2859 

13 

52 

12 

48 

. 7137 

. 2863 

.6927 

. 9997 

.5002 

.0770 

. 7151 

. 2848 

12 

48 

16 

49 

. 7164 

. 2836 

•G908 

.40031 

.4981 

.6771 

. 7162 

. 2838 

11 

44 

2C 

50 

.37191 

.62809 

2.6888 

.40065 

2.4960 

1.0773 

.07173 

.92827 

10 

40 

24 

51 

. 7218 

. 2782 

.6869 

. 0098 

.4939 

.0774 

. 7184 

. 2S16 

9 

36 

28 

52 

. 7245 

. 2755 

.6849 

. 0132 

.4918 

.0775 

. 7195 

. 2805 

8 

32 

32 

53 

. 7272 

. 2728 

.6830 

. 0116 

.4897 

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. 7205 

. 2794 

7 

28 

36 

54 

. 7299 

. 2701 

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. 0200 

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.6778 

. 721G 

. 2784 

G 

24 

4o 

55 

.37326 

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2.6791 

.40233 

2.4855 

1.0779 

.07227 

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5 

20 

44 

56 

. 7353 

. 2647 

.6772 

. 0267 

.4834 

.0780 

. 7238 

. ‘2762 

4 

16 

48 

57 

. 7380 

. 2620 

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. 0301 

.4813 

.0781 

. 7249 

. 2751 

3 

12 

52 

5S 

. 7407 

. 2593 

.6733 

. 0335 

.4792 

.0783 

. 7260 

. 2740 

2 

8 

56 

59 

. 7434 

. 2566 

.6714 

. 0369 

.4772 

.0784 

. 7271 

. 2729 

1 

4 

38 

60 

. 740 L 

. 2539 

.6695 

. 0403 

.4761 

.0785 

. 7282 

. 2718 

0 

32 

M. S. 

7 h 

M 

lir 

Cosine. 

Vis. Sin. 

Secant e. 

Cotang. Tangeui. 

Natural. 

Cosec'utj Vrs.Cos 

Sine. 

M 

68° 

M.S. 

4 h 





























Natural Lines, 


267 


l h 

OQO 

Natural Trigonometrical Functions 

157° 

10 h 

M.S 

M 

Sine. 

Vrs. Cos 

Cosec'nte 

Tang. 

Cotang. 

Secante.jVrs. Sin 

Cosine. 

M 

M.S. 

38 

0 

.37461 

.62539 

2.6695 

.40403 

2.4751 

1.0785 

.07282 

.92718 

60 

33 

4 

1 

. 7488 

. 2512 

.6675 

. 0436 

.1730 

.0787 

. 7292 

. 2707 

59 

56 

8 

2 

. 7514 

. 2485 

.6656 

. 0470 

.4709 

.0788 

. 7303 

. 2696 

58 

52 

12 

3 

. 7541 

. 2458 

.6637 

. 0501 

.4689 

.0789 

. 7314 

. 2686 

57 

48 

16 

4 

. 7568 

. 2431 

.6018 

. 0538 

.4668 

.0790 

. 7325 

. 2675 

56 

44 

20 

5 

.37595 

.62404 

2.6599 

.4057 2 

2.4647 

1.0792 

.07336 

.92664 

55 

40 

24 

6 

. 7622 

. 2377 

.6580 

. 0606 

.4627 

.0793 

. 7347 

. 2653 

54 

36 

28 

7 

. 7619 

. 2351 

.6561 

. 0610 

.4606 

.0794 

. 7358 

. 2642 

53 

32 

32 

8 

. 7676 

. 2324 

.6542 

. 0673 

.4586 

.0795 

. 7369 

. 2631 

52 

28 

36 

9 

. 7703 

. 2297 

.6523 

. 0707 

.4565 

.0797 

. 7380 

. 2620 

51 

24 

40 

10 

.37730 

.62270 

2.6504 

.40741 

2.4545 

1.0798 

.07391 

.92609 

50 

20 

44 

11 

. 7757 

. 2213 

.6485 

. 0775 

.4525 

.0799 

. 7402 

. 2598 

49 

16 

48 

12 

. 7784 

. 2216 

.6466 

. 0809 

.4504 

.0801 

. 7413 

. 2587 

48 

12 

52 

13 

. 7811 

. 2189 

.6447 

. 0843 

.4484 

.0802 

. 7424 

. 2576 

47 

8 

56 

14 

. 7838 

. 2162 

.6428 

. 0877 

.4463 

.0803 

. 7435 

. 2565 

46 

4 

30 

15 

.37865 

.62135 

2.6410 

.409 LI 

2.4443 

1:0804 

.07446 

.9255 1 

45 

31 

4 

16 

. 7892 

. 2108 

.6391 

. 0945 

.4423 

.0S06 

. 7457 

. 2543 

44 

56 

8 

17 

. 7919 

. 20S1 

.6372 

. 0979 

.4403 

.0807 

. 7468 

. 2532 

43 

52 

12 

18 

. 7946 

. 2054 

.6353 

. 1013 

.4382 

.0808 

. 7479 

. 2521 

42 

48 

16 

19 

. 7972 

. 2027 

.6335 

. 1047 

.4362 

.0810 

. 7490 

. 2510 

41 

44 

20 

20 

.37999 

.62000 

2.6316 

.41081 

2.4342 

1.0811 

.07501 

.92499 

40 

40 

24 

21 

. 8026 

. 1974 

.6297 

. 1115 

.4322 

.0812 

. 7512 

. 2488 

39 

36 

28 

22 

. 8053 

. 1947 

.6279 

. 1149 

.4302 

.0813 

. 7523 

. 2477 

38 

32 

32 

23 

. 8080 

. 1920 

.6260 

. 1183 

.4282 

.0815 

. 7534 

. 2466 

37 

28 

36 

24 

. 8107 

. 1893 

.6242 

. 1217 

.4262 

.0816 

. 7545 

. 2455 

36 

24 

40 

25 

.38134 

.61866 

2.G223 

.41251 

2.4242 

1.0817 

.07556 

.92443 

35 

20 

44 

26 

. 8161 

. 1839 

.6205 

. 1285 

.4222 

.0819 

. 7567 

. 2432 

34 

16 

48 

27 

. 8188 

. 1812 

.6186 

. 1319 

.4202 

.0820 

. 7579 

. 2421 

33 

12 

52 

28 

. 8214 

. 1785 

.6168 

. 1353 

.4182 

.0821 

. 7590 

. 2410 

32 

8 

56 

29 

. 8211 

. 1758 

.6150 

. 1387 

.4162 

.0823 

. 7601 

.2399 

31 

4 

30 

30 

.38268 

.61732 

2.6131 

.41421 

2.4142 

1.0824 

.07612 

.9238,S 

30 

30 

4 

31 

. 8295 

. 1705 

.6113 

. 1455 

.4122 

.0825 

. 7623 

. 2377 

29 

56 

8 

32 

. 8322 

. 1678 

.6095 

. 1489 

.4102 

.0826 

. 7634 

. 2366 

28 

52 

12 

33 

. 8349 

. 1651 

.6076 

. 1524 

.4083 

.0828 

. 7645 

. 2354 

27 

48 

16 

34 

. 8376 

. 1624 

.6058 

. 1558 

.4063 

.0829 

. 7657 

. 2343 

2G 

44 

20 

35 

.38403 

.61597 

2.6040 

.41592 

2.4043 

1.0830 

.07668 

.92332 

25 

40 

24 

36 

. 8429 

. 1570 

.6022 

. 1626 

.4023 

.0832 

. 7679 

. 2321 

24 

36 

28 

37 

. 8456 

. 1544 

.6003 

. 1660 

.4004 

.0833 

. 7690 

. 2310 

23 

32 

32 

38 

. 8483 

. 1517 

.5985 

. 1694 

.3984 

.0834 

. 7701 

. 2299 

22 

28 

36 

39 

. 8510 

. 1490 

.5967 

. 1728 

.3964 

.0836 

. 7712 

. 2287 

21 

24 

40 

40 

.38537 

.61463 

2.5949 

.41762 

2.3945 

1.0837 

.07724 

.92276 

20 

20 

44 

41 

. 8564 

. 1436 

.5931 

. 1797 

.3925 

.0838 

. 7735 

. 2265 

19 

16 

48 

42 

. 8591 

. 1409 

.5913 

. 1831 

.3906 

.0840 

. 7746 

. 2254 

18 

12 

52 

43 

. 8617 

. 1382 

.5895 

. 1805 

.3886 

.0841 

. 7757 

.• 2242 

17 

8 

56 

44 

. 8614 

. 1356 

.5877 

. 1899 

.3867 

.0842 

. 7769 

. 2231 

16 

4 

31 

45 

.38671 

.61329 

2.5859 

.41933 

2.3847 

1.0844 

.07780 

.92220 

15 

30 

4 

46 

. 8698 

. 1302 

.5841 

. 1968 

.3*28 

.0845 

. 7791 

. 2209 

14 

56 

8 

47 

. 8725 

. 1275 

.5823 

. 2002 

.3808 

.0846 

. 7802 

. 2197 

13 

52 

12 

48 

. 8751 

. 1248 

.5805 

. 2036 

.3789 

.0847 

. 7814 

. 2186 

12 

48 

16 

49 

. 8778 

. 1222 

.5787 

2070 

.3770 

.0849 

. 7825 

. 2175 

11 

44 

20 

50 

.38805 

.61195 

2.5770 

.42105 

2.3750 

1.0850 

.07836 

.92164 

10 

40 

24 

51 

. 8832 

. 1168 

.5752 

. 2139 

.3731 

.0851 

. 7847 

. 2152 

9 

36 

28 

52 

. 8859 

. 1141 

.5734 

. 2173 

.3712 

.0853 

. 7859 

. 2141 

8 

32 

32 

53 

. S886 

. 1114 

.5716 

. 2207 

.3692 

.0854 

. 7870 

. 21% 

7 

28 

36 

54 

. 8912 

. 1088 

.5699 

. 2242 

.3673 

.0855 

. 7881 

. 2118 

6 

24 

40 

55 

.38939 

.61061 

2.5681 

.42276 

2.3654 

1.0857 

.07893 

.92107 

5 

20 

4-1 

56 

. 8961) 

. 1034 

.5663 

. 2310 

.3635 

.0858 

. 7904 

. 2096 

4 

16 

48 

57 

. 8993 

. 1007 

.5646 

. 2344 

.3616 

.0859 

. 7915 

. 2084 

3 

12 

52 

58 

. 9019 

. 0980 

.5628 

. 2379 

.3597 

.0861 

. 7927 

. 2073 

2 

8 

66 

59 

. 9046 

. 0954 

.5610 

. 2413 

.3577 

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. 7938 

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1 

4 

32 

60 

. 9073 

. 0927 

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. 2447 

.3558 

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. 2050 

0 

38 

M.S. 

M 

Cosine. 

Vrs. Sin. 

Secante. 

Cotang. 

rangent. | 

Cosec’utiVrs.Cos 

Sine. 

M 

M.S. 

7 h 

112° 



Natural. 




67° 

4 b 




































268 


Natural Lines, 


l h 

23° Natural Trig 

onometrical 

Functions. 

156° 

10 h 

M.S. 

M 

Sine. 

Vrs.Cos. 

Cosec'nte 

Tang. 

Cotang. 

Secante. 

Vrs.Sin 

Cosine. 

M 

M.S. 

33 

0 

.39073 

.60927 

2.5593 

.42-147 

2.3558 

1.0864 

.07949 

.92050 

60 

38 

4 

1 

. 9100 

. 0900 

.5575 

. 2482 

.3539 

.0865 

. 7961 

. 2039 

59 

56 

8 

2 

. 9126 

. 0873 

.6558 

. 2516 

.3520 

.0866 

. 7972 

. 2028 

58 

52 

12 

3 

. 9153 

. 0846 

.6540 

. 2550 

.3501 

.0868 

. 7984 

. 2016 

57 

48 

16 

4 

. 9180 

. 0820 

.5523 

. 2585 

.3482 

.0869 

. 7995 

. 2005 

56 

44 

20 

5 

.39207 

.60793 

2.5506 

.42619 

2.3463 

1.0870 

.08006 

.91993 

55 

40 

24 

6 

. 9234 

. 0766 

.5488 

..2654 

.3445 

.0872 

. 8018 

. 1982 

54 

36 

28 

7 

. 9260 

. 0739 

.5471 

. 2688 

.3426 

.0873 

. 8029 

. 1971 

53 

32 

32 

8 

. 9287 

. 0713 

.5453 

. 2722 

.3407 

.0874 

. 8041 

. 1959 

52 

28 

36 

9 

. 9314 

. 0686 

.5436 

. 5757 

.3388 

.0876 

. 8052 

. 194S 

51 

24 

40 

10 

.39341 

60659 

2.6419 

,42791 

2.3369 

1.0877 

.08063 

.91936 

50 

20 

44 

11 

. 9367 

. 0632 

.5402 

. 2826 

.3350 

.0878 

. 8075 

. 1925 

49 

16 

48 

12 

. 9394 

. 0606 

.5384 

. 2860 

.3332 

.0880 

. S( 186 

. 1913 

48 

12 

62 

13 

. 9421 

. 0579 

.5367 

. 2894 

.3313 

.0881 

. 8098 

. 1902 

47 

8 

56 

14 

. 9448 

. 0552 

.5350 

. 2929 

.3214 

.0882 

. 8109 

. 1891 

46 

4 

33 

15 

.39474 

.60526 

2.5333 

.42963 

2.3276 

1.0884 

.08121 

.91879 

45 

37 

4 

16 

. 9501 

. 0499 

.5316 

. 2998 

.3257 

.0885 

. 8132 

. 1868 

44 

56 

8 

17 

. 9528 

. 0452 

.5299 

. 3032 

.3238 

.0886 

. S144 

. 1856 

43 

52 

12 

18 

. 9554 

. 0445 

.5281 

. 3067 

.3220 

.0888 

. 8155 

. 1845 

42 

48 

16 

19 

. 9581 

. 0419 

.5264 

. 3101 

.3201 

.0889 

. 8167 

. 1833 

41 

44 

20 

20 

.39608 

.60392 

2.5247 

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2.3183 

1.0S91 

.08178 

.91822 

40 

40 

24 

21 

. 9635 

. 0365 

.5230 

. 3170 

.3161 

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. 8190 

. 1810 

39 

36 

28 

22 

. 9661 

. (1339 

.5213 

. 3205 

.3145 

.0S93 

. 8201 

. 1798 

38 

32 

32 

23 

. 9688 

. 0312 

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. 3239 

.3127 

.0895 

. 8213 

. 1787 

37 

28 

36 

24 

. 9715 

. 0285 

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. 3274 

.3109 

.0896 

. 8224 

. 1775 

36 

24 

40 

25 

.39741 

.60258 

2.5163 

.43308 

2.3090 

1.0897 

.08236 

.91764 

35 

20 

44 

26 

. 9768 

. 0232 

.6146 

. 3343 

.3072 

.0899 

. 8248 

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34 

16 

48 

27 

. 9795 

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. 3377 

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1741 

33 

12 

52 

28 

. 9821 

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32 

8 

56 

29 

. 9848 

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. 8282 

. 1718 

31 

4 

3 4 

30 

.39875 

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2.5078 

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2.2998 

1.0904 

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30 

3(i 

4 

31 

. 9901 

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29 

56 

8 

32 

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23045 

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28 

52 

12 

33 

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.2941 

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27 

48 

16 

34 

. 9981 

. 0018 

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. 3620 

.2925 

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. 8340 

. 1659 

26 

44 

20 

35 

.40008 

.59992 

2.4995 

.43654 

2.2907 

1.0911 

.08352 

.91648 

25 

40 

24 

36 

. 0035 

. 9965 

.4978 

. 3689 

.28 S9 

.0913 

. 8364 

. ?636 

21 

30 

28 

37 

. 0061 

. 9938 

.4961 

. 3723 

.2871 

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. 8375 

. 1625 

23 

32 

32 

38 

. 0088 

. 9912 

.4945 

. 3758 

.2853 

.0915 . 

< 8387 

. 1613 

22 

28 

36 

39 

. 0115 

. 9385 

.4928 

. 3793 

.2835 

.0917 

. 8399 

. 1601 

21 

21 

40 

40 

.40141 

.59858 

2.4912 

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2.2817 

1.0918 

.08410 

.91590 

20 

20 

44 

41 

. 0168 

. 9832 

.4895 

. 3862 

.2794 

.0920 

. 8422 

. 1578 

19 

16 

48 

42 

. 0195 

. 9805 

.4879 

. 3897 

.2781 

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. 8434 

. 1566 

18 

12 

52 

43 

. 0221 

. 9778 

.4862 

. 3932 

.2763 

.0922 

. 8445 

. 1554 

17 

S 

56 

44 

. 0248 

. 9752 

.4846 

. 3966 

.2745 

.0924 

. 8157 

. 1543 

16 

4 

35 

45 

.40275 

.59725 

2.4829 

.44001 

2.27 27 

1.0925 

.08469 

.91531 

15 

35 

4 

46 

. 0801 

. 9699 

.4813 

. 4036 

.2709 

.0927 

. 8480 

. 1519 

14 

56 

8 

47 

. 3328 

. 9672 

.4797 

. 4070 

.2691 

.0928 

. 8492 

. 1508 

13 

52 

12 

48 

. 0354 

. 9645 

.4780 

. 4105 

.2673 

.0929 

. 85(4 

. 1480 

12 

48 

16 

49 

. 0381 

. 9619 

.47 64 

. 4140 

.2655 

.0931 

. 8516 

. 1484 

11 

44 

20 

50 

.40408 

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2.4748 

.44175 

2.2637 

1.0932 

.08527 

.91472 

10 

40 

24 

61 

. 0434 

. 9560 

.4731 

. 4209 

.2619 

.0934 

. 8539 

. 1461 

9 

36 

28 

52 

. 0461 

. 9539 

.4715 

. 4244 

.2602 

.0935 

. 8551 

. 1449 

8 

32 

32 

53 

. 0487 

. 9512 

.4699 

. 4279 

.2584 

.0936 

. 8563 

. 14:57 

7 

28 

36 

54 

. 0514 

. 9486 

.4083 

. 4314 

.2566 

.0938 

. 8575 

. 1425 

6 

24 

40 

55 

.40541 

.59469 

2.4666 

.44349 

2.2548 

1.0939 

.08586 

.91414 

5 

20 

44 

56 

. 0567 

. 9433 

.4650 

. 4383 

.2531 

.0941 

. 8598 

. 1402 

4 

10 

48 

5 7 

. 0594 

. 9406 

.46:54 

. 4418 

.2513 

.0942 

. 8610 

. 1390 

3 

12 

52 

58 

. 0620 

. 9379 

.4618 

. 4453 

.2495 

.0943 

. 8622 

. 137S 

2 

8 

56 

59 

. 0647 

. 9353 

.4602 

. 4488 

.2478 

.0945 

. 8634 

. 1366 

1 

4 

30 

60 

. 0674 

. 9326 

.4586 

. 4523 

.2460 

.0946 

. 8646 

. 1354 

0 

34 

M.S. 

M 

Cosine. 

Yrs.Sin. 

Seoaute. 

Cotaug.lTangeut. 

Coaec'ut 

Vrs.Cos 

Sine. 

M 

M.S. 

7 h 

113 

D 



Natural. 




06° 

4 h 



























Natural Lines. 


269 


l h 

24 

O 

Natural Trigonometrical Functions 

155 c 

10“ 

M.S 

M 

Sine. 

Vrs.Cos 

. Cosec’nt 

e Tang. 

Cotang 

. Sccante 

• jVrs.Sir 

Cosine 

M 

M.S. 

3fi 

0 

.40674 

.69326 

2.4586 

.44523 

2.2460 

1.0946 

.08645 

.91354 

60 

21 

4 

1 

. 0700 

. 9300 

.4570 

. 4558 

.2443 

.0948 

. 8657 

. 1343 

59 

56 

8 

12 

2 

. 0727 

. 9273 

.4554 

. 4593 

.2425 

.0949 

. 8669 

. 1331 

58 

52 

3 

. 0753 

. 9247 

.4538 

. 4627 

.24 OS 

.0951 

. 8681 

. 1319 

57 

48 

10 

4 

. 0780 

. 9220 

.4522 

. 4662 

.2390 

.0952 

. 8693 

. 1307 

56 

44 

20 

5 

.40800 

.59193 

2.4506 

.44697 

2.2373 

1.0953 

.08705 

.91295 

55 

40 

24 

6 

. 0833 

. 9167 

.4490 

. 4732 

.2355 

.0955 

. 8716 

. 1283 

54 

30 

32 

2S 

7 

. 0860 

. 9140 

.4474 

. 4767 

.2338 

.0956 

. 8728 

. 1271 

53 

32 

36 

8 

. 0886 

. 9114 

.4458 

. 4802 

.2320 

.0958 

. 8740 

. 1260 

52 

28 

9 

10 

. 0913 

. 9087 

.4442 

. 4837 

.2303 

.0959 

. 8752 

. 1248 

51 

24 

40 

.40939 

.59061 

2.4426 

.44872 

2.2286 

1.0901 

.08764 

.91236 

50 

20 

41 

ii 

. 0966 

. 9034 

.4411 

. 4907 

.2268 

.0962 

. 8776 

. 1224 

49 

16 

48 

12 

. 0992 

. 9008 

.4395 

. 4942 

.2251 

.0963 

. 8788 

. 1212 

48 

12 

52 

13 

. 1019 

. 8981 

.4379 

. 4977 

.2234 

.0965 

. 8800 

. 1200 

47 

8 

56 

14 

. 1045 

. 8955 

.4363 

. 5012 

.2216 

.0966 

. 8812 

. 1188 

46 

4 

37 

15 

.41072 

.58928 

2.4347 

.45047 

2.2199 

1.0968 

.08824 

.91176 

45 

23 

4 

16 

. 1098 

. 8901 

.4332 

. 5082 

.2182 

.0969 

. 8836 

. 1164 

44 

50 

8 

1 / 

. 1125 

. 8875 

.4316 

. 5117 

.2165 

.0971 

. 884 8 

. 1152 

43 

52 

12 

IS 

. 1151 

. 8848 

.4300 

. 5152 

.2147 

.0972 

. 8860 

. 1140 

42 

48 

16 

19 

. 1178 

. 8822 

.4285 

. 5187 

.2130 

.0973 

. 8872 

. 1128 

41 

44 

20 

20 

.41204 

.68795 

2.4269 

.45222 

2.2113 

1.0975 

.08884 

.91116 

40 

40 

24 

21 

. 1201 

. 8769 

.4254 

. 5257 

.2096 

.0976 

. 8896 

. 1104 

39 

36 

28 

22 

. 1257 

. 8742 

.4238 

. 5292 

.2079 

.0978 

. 8908 

. 1092 

38 

32 

32 

23 

. 1284 

. 8716 

.4222 

. 5327 

.2062 

.0979 

. 8920 

. 1080 

37 

28 

36 

24 

. 1310 

. 8689 

.4207 

. 6362 

.2045 

.0981 

. 8932 

. 1068 

36 

24 

40 

25 

.41337 

.68663 

2.4191 

.45397 

2.2028 

1.0982 

.08944 

.91,056 

35 

20 

44 

26 

. 1363 

. 8636 

.4176 

. 6432 

.2011 

.0984 

. 8956 

. 1044 

34 

16 

48 

27 

. 1390 

. 8610 

.4160 

. 6467 

.1994 

.0985 

. 8968 

1032 

33 

12 

52 

28 

. 1416 

.*8584 

.4145 

. 5502 

.1977 

.0986 • 

. 8980 

. 1020 

32 

8 

56 

29 

. 1413 

. 8557 

.4130 

. 5537 

.1960 

.0988 

. 8992 

. 1008 

31 

4 

3S 

30 

.41469 

.68531 

2.4114 

.45573 

2.1943 

1.0989 

.09(104 

.90996 

30 

22 

4 

31 

. 1496 

. 8501 

.4099 

. 6608 

.1926 

.0991 

. 9016 

. 0984 

29 

56 

8 

32 

. 1522 

. 8478 

.4083 

. 5643 

.1909 

.0992 

. 9028 

. 0972 

28 

52 

12 

33 

. 1549 

. 8451 

.4068 

. 6678 

.1892 

.0994 

. 9040 

. 0960 

27 

48 

16 

31 

. 1575 

. 8425 

.4053 

. 5713 

.1875 

.0995 

. 9052 

. 0948 

26 

44 

20 

35 

.41602 

.58398 

2.4037 

.45748 

2.1859 

1.0997 

.09064 

.90936 

25 

40 

24 

36 

. 1628 

. 8372 

.4022 

. 5783 

.1842 

.0998 

. 9076 

. 0924 

24 

36 

28 

37 

. 1654 

. 8345 

.4007 

. 5819 

.1825 

.1000 

. 908S 

. 0911 

23 

32 

32 

38 

. 1681 

. 8319 

.3992 

. 5854 

.1808 

.1001 

. 9101 

. 0899 

22 

28 

36 

39 

. 1707 

. 8292 

.3976 

. 5889 

.1792 

.1003 

. 9113 

. 08S7 

21 

24 

40 

40 

.41734 

.58266 

2.39G1 

.45924 

2.1775 

1.1004 

.09125 

.90875 

20 

20 

44 

41 

. 1760 

. 8240 

.3946 

. 5960 

.1758 

.1005 

. 9137 

. 0863 

19 

16 

48 

42 

. 1787 

. 8213 

.3931 

. 5995 

.1741 

.1007 

. 9.149 

. 0851 

18 

12 

52 

43 

. 1813 

. 8187 

.3916 

. 6030 

.1725 

.1008 

. 9161 

. 0839 

17 

8 

56 

44 

. 1839 

. 8160 

.3901 

. 6065 

.1708 

.1010 

. 9173 

. 0826 

10 

4 

39 

45 

.41866 

.58134 

2.3886 

.46101 

2.1692 

1.1011 

.09186 

.90814 

15 

21 

4 

46 

. 1892 

. 8108 

.3871 

. 6136 

.1675 

.1013 

. 9198 

. 0802 

14 

56 

8 

47 

. 1919 

. 8081 

.3856 

. 6171 

.1658 

.1014 

. 9210 

. 0790 

13 

52 

12 

48 

. 1945 

. 8055 

.3841 

. 6206 

.1642 

.1016 

. 9222 

. 0778 

12 

48 

16 

49 

. 1972 

. 8028 

.3826 

. 6242 

.1625 

.1017 

. 9234 

. 0765 

11 

44 

20 

50 

.41998 

.68002 

2.3811 

.46277 

2.1609 

1.1019 

.09247 

.90753 

10 

40 

24 

51 

. 2024 

. 7975 

.3796 

. 6312 

.1592 

.1020 

. 9259 

. 0741 

9 

36 

28 

52 

. 2051 

. 7949 

.3781 

. 6348 

.1576 

.1022 

. 9271 

. 0729 

8 

32 

32 

53 

. 2077 

. 7923 

.3766 

. 6383 

.1559 

.1023 

. 9283 

. 0717 

7 

28 

36 

54 

. 2103 

. 7896 

.3751 

. 6418 

.1543 

.1025 

9296 

. 0704 

6 

24 

40 

55 

.42130 

.57870 

2.3736 

.46454 

2.1527 

1.1026 

.09308 

.90692 

5 

20 

41 

56 

. 2156 

. 7844 

.3721 

. 6489 

.J510 

.1028 

9320 

. 0680 

4 

16 

48 

57 

. 2183 

. 7817 

.3706 

. 6524 

.1494 

.1029 

9332 

. 0668 

3 

12 

52 

58 

. 2209 

. 7791 

.3691 

. 6560 

.1478 

.1031 

9345 

0655 

2 

8 

56 

59 

. 2235 

. 7764 

.3677 

. 6595 

.1461 

.1032 

9357 

0643 

1 

4 

40 

60 

. 2262 

. 7738 

.3662 

. 6631 

.1445 

.1034 

9369 

0631 

0 

20 

M. S. 

7 h 

M 

14° 

Cosine. 

frs.Sin. 

Became. 

Cotang. 1 

Natu 

’angem. 

ral. 

Bosee'nti Vrs.Cos | 

Sine. 

1 

M 

35° 

l.S. 

4 h 
















































270 


Natural Lines. 


l h 

25° 

Natural Trig 

onometrical Functions. 

154° 

10 h 

M.S. 

M 

Sine. 

Vrs.Cos. 

Cosec’nte 

Tang. 

Cotang. 

Secante. 

Vrs. Sin 

Cosine. 

M 

M.S. 

40 

0 

.42262 

.57738 

2.3662 

.46631 

2.1445 

1.10:34 

.09369 

.90631 

60 

40 

4 

1 

. 2288 

. 7712 

.3647 

. 6666 

.1429 

.1035 

. 9381 

. 0618 

59 

56 

8 

2 

. 2314 

. 7685 

.3632 

. 6702 

.1412 

.1037 

. 9394 

. 0606 

58 

52 

12 

3 

. 2:141 

. 7659 

.3618 

. 6737 

.1396 

.1038 

. 9406 

. 0594 

57 

48 

18 

4 

. 2367 

. 7633 

.3603 

. 6772 

.1380 

.1040 

. 9418 

. 0581 

56 

44 

20 

5 

.42394 

.57606 

2.3588 

.46808 

2.1364 

1.1041 

.09431 

.90569 

55 

40 

24 

6 

. 2420 

. 7580 

.3574 

. 6843 

.1348 

.1043 

. 9443 

. 0557 

54 

36 

28 

7 

. 2446 

. 7554 

.3559 

. 6879 

.1331 

.1044 

. 9455 

. 0544 

53 

32 

32 

8 

. 2473 

. 7527 

.3544 

. 6914 

.1315 

.1046 

. 9468 

. 0532 

52 

28 

36 

9 

. 2499 

. 7501 

.3530 

. 6950 

.1299 

.1047 

. 9480 

. 0520 

51 

24 

40 

10 

.42825 

.57475 

2.3515 

.46985 

2.1283 

1.1649 

.09492 

.90507 

50 

20 

44 

11 

. 2552 

. 7448 

.3501 

. 7021 

.1267 

.1050 

. 9505 

. 0495 

49 

16 

48 

12 

. 2578 

. 7422 

.3486 

. 7056 

.1251 

.1052 

. 9517 

. 0483 

48 

12 

52 

13 

. 2604 

. 7396 

.3472 

. 7092 

.1235 

.1053 

. 9530 

. 0470 

47 

8 

56 

14 

. 2630 

. 7369 

.3457 

. 7127 

.1219 

.1055 

. 9542 

. 0458 

46 

4 

41 

15 

.42657 

.57343 

2.3443 

.47163 

2.1203 

1.1056 

.09554 

.90445 

45 

19 

4 

16 

. 2683 

. 7317 

.3428 

. 7199 

.1187 

.1058 

. 9567 

. 0433 

44 

56 

8 

17 

. 2709 

. 7290 

.3414 

. 7234 

.1171 

.1059 

. 9579 

. 0421 

43 

52 

12 

18 

. 2736 

. 7264 

.3399 

. 7270 

.1155 

.1061 

. 9592 

. 0408 

42 

4.3 

16 

19 

. 2762 

7238 

.3385 

. 7305 

.1139 

.1062 

. 96(4 

. 0396 

41 

44 

20 

20 

.42788 

237212 

2.3371 

.47341 

2.1123 

1.1064 

.09617 

.90383 

40 

40 

24 

21 

. 2815 

. 7185 

.3356 

. 7376 

.1107 

.1065 

. 9629 

. 0371 

39 

36 

28 

22 

. 2841 

. 7159 

.3342 

. 7412 

.1092 

.1067 

. 9641 

. 0358 

38 

32 

32 

23 

. 2867 

. 7133 

.3328 

. 7448 

.1076 

.1068 

. 9654 

. 0346 

37 

28 

36 

24 

. 2893 

. 7106 

.3313 

. 74«3 

.1060 

.1070 

. 9666 

. 0333 

36 

24 

40 

25 

.42920 

.57080 

2.3299 

.47519 

2.1044 

1.1072 

.09679 

.90321 

35 

20 

44 

26 

. 2946 

. 7054 

.3285 

. 7555 

.1028 

.1073 

. 9691 

. 0308 

34 

16 

48 

27 

. 2972 

. 7028 

.3271 

. 7690 

.1013 

.1075 

. 9704 

. 0296 

33 

12 

52 

28 

. 2998 

. 7001 

.3256 

. 7626 

.0997 

.1076 

. 9716 

. 0283 

32 

8 

56 

29 

. 302-5 

. 6975 

.3242 

. 7662 

.0981 

.1078 

. 9729 

. 0271 

31 

4 

4 a 

30 

.43051 

.56949 

2.3228 

.47697 

2.6965 

1.1079 

.09741 

.90258 

30 

18 

4 

31 

. 3077 

. 6923 

.3214 

. 7733 

.0950 

.1081 

. 9754 

. 0246 

29 

56 

8 

32 

. 3L04 

. 6896 

.3200 

. 7769 

.0934 

.1082 

. 9766 

. 0233 

28 

52 

12 

33 

. 3130 

. 6870 

.31S6 

. 7805 

.09 L8 

.1084 

. 9779 

. 0221 

27 

48 

16 

34 

. 3156 

. 6844 

.3172 

. 7840 

.0903 

.1085 

. 9792 

. 0008 

26 

44 

20 

35 

.43182 

.56818 

2.3158 

.47876 

2.0887 

1.108? 

.09814 

.90196 

25 

40 

24 

36 

. 3208 

. 6791 

.3143 

: 7912 

.0872 

.1088 

. 9817 

. 0183 

24 

36 

28 

37 

. 3235 

. 6765 

.3129 

. 7948 

.0856 

.1090 

. 9829 

. 0171 

23 

32 

32 

38 

. 3261 

. 6739 

.3115 

. 7983 

.0840 

.1092 

. 9842 

. 0158 

‘22 

28 

36 

39 

. 3287 

. 6713 

.3101 

. 8019 

.0825 

.1093 

. 9854 

. 0145 

21 

24 

40 

40 

.43313 

.56686 

2.3087 

.48055 

2.0809 

1.1095 

.09867 

.90133 

20 

20 

44 

41 

. 3340 

. 6660 

.3073 

. 8091 

.0794 

.1096 

. 9880 

. 0120 

19 

16 

48 

42 

. 3366 

. 6634 

.3059 

. 8127 

.0778 

.1098 

. 9892 

. 0108 

18 

12 

52 

43 

. 3392 

. 6608 

.3046 

. 8162 

.0763 

.1099 

. 9905 

. 0095 

17 

8 

56 

44 

. 3418 

. 6582 

.3032 

. 8198 

.0747 

.1101 

. 0917 

. 0082 

16 

4 

4.4 

45 

.48444 

.66555 

2.3018 

.48234 

2.0732 

1.1102 

.09930 

.90070 

15 

17 

4 

46 

. 3471 

. 6529 

.3004 

. 8270 

.0717 

.1104 

. 9943 

. 0057 

14 

56 

8 

47 

. 3497 

. 6503 

.2990 

. 8300 

.070 L 

.1106 

. 9955 

. 0044 

13 

52 

12 

48 

. 3523 

. (1177 

.2976 

. 8342 

.0686 

.1107 

. 9968 

. 0032 

12 

48 

16 

49 

. 3.549 

. 6451 

.2962 

8378 

.0671 

.1109 

. 9981 

. 0019 

11 

44 

20 

60 

.43575 

.56424 

2.2949 

.48414 

2.0655 

1.1110 

.09993 

.90006 

10 

40 

24 

61 

. 3602 

. 6398 

.2935 

. 8440 

.0640 

.1112 

.10006 

.89994 

9 

36 

28 

52 

. 3628 

. 6372 

.2921 

. 8485 

.0625 

.1113 

. 0010 

. 99S1 

8 

32 

32 

53 

. 3651 

. 6346 

.2907 

. 8521 

.0609 

.1115 

. 0031 

. 9968 

7 

28 

36 

.54 

. 3680 

. 6320 

.2894 

. 8557 

.0594 

.1116 

. 0044 

. 9956 

6 

24 

40 

56 

.43706 

.56294 

2.2880 

.48593 

2.0579 

1.1118 

.10057 

.89943 

5 

20 

44 

56 

, 3732 

. 6207 

.2866 

. 8629 

.0564 

.1120 

. 0070 

. 9930 

4 

16 

48 

57 

• 8759 

. 6241 

.2853 

. 8665 

.0518 

.1121 

. 0082 

. 9918 

3 

12 

52 

58 

. 3785 

. 6215 

.2839 

. 8701 

.0533 

.1123 

. 0095 

. 9905 

2 

8 

66 

59 

. 3811 

. 6189 

.2825 

. 8737 

.0518 

.1124 

. 0108 

. 9892 

1 

4 

44 

60 

. 3837 

. 6163 

.2812 

. 8773 

.0503 

.1126 

. 0121 

. 9879 

0 

10 

M.S. 

M 

Cosine. 

Vrs.Sin.i Secante. 

Cotang. 

Tangent. 

Coseo’nt 

Vrs.Cos 

Sine. 

M 

M.S. 

7 u 

115 




Natural. 




G4° 

4 U 

















Natural Lines. 


271 


l h 

26 

D 

Natural Trigonometrical Functions 

153 c 

10 h 

M.S. 

M 

Sine. 

Yrs.Cos. 

Cosec'nte 

Tang. 

Cotang. 

Secante. 

Yrs. Sin 

Cosine. 

M 

M.S. 

44 

0 

.43837 

.56163 

2.2812 

.48773 

2.0503 

1.1126 

.10121 

•89879 

60 

1<» 

4 

1 

. 3863 

. 6137 

.2798 

. 8809 

.0488 

.1127 

. 0133 

. 98t»7 

59 

56 

8 

2 

. 3889 

. 6111 

.27 S4 

. 8815 

.0473 

.1129 

. 0146 

. 9854 

58 

52 

12 

3 

. 3915 

. 6084 

.2771 

. 8881 

.0458 

.1131 

. 0159 

. 9841 

57 

48 

16 

4 

. 3942 

. 6058 

.2757 

. 8917 

.0443 

.1132 

. 0172 

. 9828 

56 

44 

20 

5 

.43968 

.561 >02 

2.2744 

.48953 

2.0427 

1.1184 

.10184 

.89815 

55 

40 

24 

6 

. 3994 

. 6006 

.2730 

. 8989 

.0412 

.1135 

. 0197 

. 9803 

54 

30 

28 

7 

. 4020 

. 5980 

.2717 

. 9025 

.0397 

.1137 

. 0210 

. 9790 

53 

32 

32 

8 

. 4046 

. 5954 

.2703 

. 9062 

.0382 

.1139 

. 0223 

. 9777 

52 

28 

36 

9 

. 4072 

. 5928 

.2690 

. 9098 

.0367 

.1140 

. 0236 

. 9764 

51 

24 

40 

10 

.44098 

.55902 

2.2676 

.49131 

2.0352 

1.1142 

.10248 

.89751 

50 

20 

44 

11 

. 4124 

. 5875 

.2663 

. 9170 

.0338 

.1143 

. 0261 

. 9739 

49 

16 

48 

12 

. 4150 

. 5849 

.2650 

. 9206 

.0323 

.1145 

. 0274 

. 9726 

48 

12 

62 

13 

. 4177 

. 5823 

.2636 

. 9242 

.0308 

.1147 

. 0287 

. 9713 

47 

8 

56 

14 

. 4203 

. 5797 

.2623 

. 9278 

.0293 

.1148 

. 0300 

. 9700 

46 

4 

45 

15 

.44229 

.55771 

2.2610 

.49314 

2.0278 

1.1150 

10313 

.89687 

45 

15 

4 

1G 

. 4255 

. 5745 

.2596 

. 9351 

.0263 

.115L 

. 0316 

. 9674 

44 

56 

8 

17 

. 4281 

. 5719 

.2583 

. 9387 

.0248 

.1153 

. 0338 

. 9661 

43 

52 

12 

18 

. 4307 

. 5693 

.2570 

. 9423 

.0233 

.1155 

. 0351 

. 9649 

42 

48 

16 

19 

. 4333 

. 5667 

.2656 

. 9159 

.0219 

.1156 

. 0364 

. 9636 

41 

44 

20 

20 

.44359 

.55641 

2.2543 

.49495 

2.0204 

1.1158 

.10377 

.89623 

40 

40 

24 

21 

. 4385 

. 5615 

.2530 

. 9532 

.0189 

.1159 

. 03J0 

. 9610 

39 

36 

28 

22 

. 4411 

. 5589 

.2517 

. 9568 

.0174 

.1161 

. 0403 

. 9597 

38 

32 

32 

23 

. 4137 

. 5562 

.2503 

. 9604 

.0159 

.1163 

. 0416 

. 9584 

37 

28 

36 

24 

. 4463 

. 5536 

.2490 

. 9640 

.0145 

.1164 

. 0429 

. 9571 

36 

24 

40 

25 

.44489 

.55510 

2.2477 

.49677 

2.0130 

1.1166 

10442 

.89558 

35 

20 

44 

26 

. 4516 

. 5484 

.2461 

. 9713 

.0115 

.1167 

. 0455 

. 9545 

34 

16 

48 

27 

. 4542 

. 5458 

.2451 

. 9749 

.0101 

.1169 

. 0468 

. 9532 

33 

12 

52 

28 

. 4568 

. 5432 

.2438 

. 9785 

.0086 

.1171 

. 0481 

. 9519 

32 

8 

56 

29 

. 4594 

. 5406 

.2425 

. 9822 

.0071 

.1172 

. 0493 

. 9506 

31 

4 

4»> 

30 

.44620 

.55380 

2.2411 

.49858 

2.0057 

1.1174 

.10506 

.89493 

30 

11 

4 

31 

. 4646 

. 5354 

.2398 

. 9894 

.0042 

.1176 

. 0519 

. 9480 

29 

56 

8 

32 

. 4672 

. 5328 

.2385 

. 9931 

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. 0532 

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28 

52 

12 

33 

. 4698 

. 5302 

.2372 

. 9967 

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. 0545 

. 9454 

27 

48 

16 

34 

. 4724 

. 5276 

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1.9998 

.1180 

. 0568 

. 9441 

20 

44 

20 

35 

.44750 

.55250 

2.2346 

.50010 

1.9984 

1.1182 

.10571 

.89428 

25 

40 

24 

36 

. 4776 

. 5224 

.2333 

. 0076 

.9969 

.1184 

. 0584 

. 9415 

24 

36 

28 

37 

. 4802 

. 5198 

.2320 

. 0113 

.9955 

.1185 

. 0598 

. 9402 

23 

32 

32 

38 

. 4828 

. 5172 

.2307 

. 0149 

.9940 

.1187 

. 0611 

. 9389 

22 

28 

36 

39 

. 4854 

. 5116 

.2294 

. 0185 

.9926 

.1189 

. 0624 

. 9376 

21 

24 

4o 

40 

.44880 

.55120 

2.2282 

.50222 

1.9912 

1.1190 

.10637 

.89363 

20 

20 

44 

41 

. 4906 

. 5094 

.2269 

. 0258 

.9897 

.1192 

. 0650 

. 9350 

19 

16 

48 

42 

. 4932 

. 5068 

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. 0295 

.9883 

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. 0683 

. 9337 

18 

12 

52 

43 

. 4958 

. 5042 

.2243 

. 0331 

.9868 

.1195 

. 0676 

. 9324 

17 

8 

56 

44 

. 4984 

. 5016 

.2230 

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.9854 

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. (J689 

. 9311 

10 

4 

47 

45 

.45010 

54990 

2.2217 

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1.9840 

1.1198 

.10702 

.89298 

15 

13 

4 

46 

. 5036 

. 4964 

.2204 

. 0141 

.9825 

.1200 

. 0715 

.. 9285 

14 

56 

8 

47 

. 5062 

. 4938 

.2192 

. 0477 

.9811 

4202 

. 0728 

. 9272 

13 

52 

12 

48 

. 5088 

. 4912 

.2179 

. 0514 

.9797 

.1203 

. 0741 

. 9258 

12 

48 

16 

49 

. 5114 

. 4886 

.2166 

. 0550 

.9782 

.1205 

. 0754 

. 9245 

11 

44 

20 

50 

.45140 

.54860 

2.2153 

.50587 

1.9768 

1.1207 ' 

.10768 

.89232 

10 

40 

24 

51 

. 5166 

. 4834 

.2141 

. 0623 

.9754 

.1208 

. 0781 

. 9219 

9 

36 

28 

62 

. 5191 

. 4808 

.2128 

. 0660 

.9739 

.1210 

. 0794 

. 9206 

8 

32 

32 

53 

. 5217 

. 4782 

.2115 

. 0696 

.9725 

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. 0807 

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7 

2X 

36 

54 

. 5243 

. 4756 

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. 0733 

.9711 

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. 0820 

. 9180 

6 

24 

40 

55 

.45269 

.54730 

2.2090 

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1.9697 

1.1215 

.10833 

.89166 

5 

20 

44 

56 

. 5295 

. 4705 

.2077 

. 0806 

.9683 

.1217 

. 0841i 

. 9153 

4 

16 

48 

57 

. 5321 

. 4679 

.2065 

. 0843 

.9668 

.1218 

. 0860 

. 9140 

3 

12 

62 

58 

. 5347 

. 4653 

.2052 

. 0879 

.9654 

.1220 

. 0873 

. 9127 

2 

8 

56 

59 

. 5373 

. 4627 

.2039 

. 0916 

.9640 

.1222 

. 08 SO 

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1 

4 

48 

60 

. 5399 

. 4601 

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. 0899 

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0 

12 

M. 8 . 
7 h 

M 

116" 

Cosine. IVrs.Sin. 

:> 

Seuanto. | 

Co tang. 

Nall 

Taugent. 

iral. 

Cosec'ut.Vrs. Cos I 

Sine. 

M 

63° 

M.S. 

4 U 






























272 


Natural Lines. 


l h 

27° 

Natural Trig 

onometrical 

Functions. 

152° 

10 h 


M.S. 

M 

Sine. 

Vrs.Cos, 

Cosec’nte 

Tang. 

Cotang, 

Secante. 

Vrs. Sin 

Cosine. 

M 

M.S. 


4 8 

0 

.45399 

.54601 

2.2027 

.50952 

1.9626 

1.1223 

.10899 

.89101 

60 

14 


4 

1 

. 5425 

. 4575 

.2014 

. 0989 

.9612 

.1225 

. 0912 

. 9087 

59 

56 


8 

2 

. 6451 

. 4549 

.2002 

. 1026 

.9598 

.1226 

. 0926 

. 9074 

68 

52 


12 

3 

. 6477 

. 4523 

.1989 

. 1062 

.9584 

.1228 

. 0939 

. 9061 

57 

48 


ie 

4 

. 5563 

. 4497 

.1977 

. 1099 

.9.70 

.1230 

. 0952 

. 9048 

60 

44 


2C 

5 

.45528 

.54171 

2.1964 

.51136 

1.9556 

1.1231 

.10965 

.89034 

65 

40 


24 

6 

. 5554 

. 4445 

.1952 

. 1172 

.9542 

.1233 

. 0979 

. 9021 

54 

36 


28 

7 

. 5580 

. 4420 

.1939 

. 1209 

.9528 

.1235 

. 0992 

. 9008 

53 

32 


32 

8 

. 5C.OG 

. 4394 

.1927 

. 1246 

.9514 

.1237 

. 1005 

. 8995 

52 

28 


3<j 

9 

. 5632 

. 43G8 

.1914 

. 1283 

.9500 

.1238 

. 1018 

. 8981 

51 

24 


40 

10 

.45658 

.54342 

2.1902 

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1.9486 

1.1240 

.11032 

.88968 

50 

20 


44 

11 

. 5684 

. 4316 

.1889 

. 1350 

.9472 

.1242 

. 1045 

. 8955 

49 

16 


48 

12 

. 5710 

. 4290 

.1877 

. 1393 

.9458 

.1243 

. 1058 

. 8942 

48 

12 


52 

13 

. 5736 

. 4264 

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. 1430 

.9444 

.1245 

. 1072 

. 8928 

47 

8 


56 

14 

. 5761 

. 4238 

.1852 

. 1466 

.9430 

.1247 

. 1085 

. 8915 

46 

4 


49 

15 

.45787 

.54213 

2.1840 

.51503 

1.9416 

1.1248 

.11098 

.88902 

45 

11 


4 

16 

. 5813 

. 4187 

.1828 

. 1540 

.9402 

.1250 

. 1112 

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44 

56 


8 

17 

. 5839 

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. 8875 

43 

52 


12 

18 

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1138 

. 8862 

42 

48 


16 

19 

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. 4109 

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. 1651 

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41 

44 


20 

20 

.45917 

.54083 

2.1778 

.51687 

1.9347 

1.1257 

.11165 

.88835 

40 

40 


24 

21 

. 5942 

. 4057 

.1766 

. 1724 

.9333 

.1258 

. 1178 

. 8822 

39 

36 


28 

22 

. 5968 

. 4032 

.1764 

. 1761 

.9319 

.1260 

. 1192 

. 8808 

38 

32 


32 

23 

. 5994 

. 4006. 

.1742 

. 1798 

.9306 

.1262 

. 1205 

. 8795 

37 

28 


36 

24 

. 6620 

. 3980 

.1730 

. 1835 

.9292 

.1264 

. 1218 

. 8781 

36 

24 


40 

25 

.46046 

.53954 

2.1717 

.51872 

1.9278 

1.1265 

.11232 

.S8768 

35 

20 


41 

26 

. 6072 

. 3928 

.1705 

. 1909 

.9264 

.1267 

. 1245 

. 8755 

34 

16 


48 

27 

. 6097 

. 3902 

.1693 

. 1946 

.9251 

.1269 

. 1259 

. 8741 

33 

12 


52 

28 

. 6123 

. 3877 

.1681 

. 1983 

.9237 

.1270 

. 1272 

. 8728 

32 

8 


56 

29 

. 6149 

. 3851 

.1669 

. 2020 

.9223 

.1272 

. 1285 

. 8714 

31 

4 


50 

30 

.46175 

.53825 

2.1657 

.52057 

1.9210 

Vf274 

.11299 

.88701 

30 

10 


4 

31 

. 6201 

. 3799 

.1645 

. 2094 

.9196 

.1275 

. 1312 

. S6SS 

29 

56 


8 

32 

• 6226 

. 3773 

.1633 

. 2131 

.9182 

.1277 

. 1326 

. 8674 

28 

52 


12 

33 

. 6262 

• 3* 48 

•1620 

. 2168 

.9169 

.1279 

. 1339 

. 8661 

27 

48 


16 

34 

. 6278 

. 3722 

.1608 

. 2205 

.9155 

.1281 

. 1353 

. 8647 

26 

44 


20 

35 

.46304 

.53696 

2.1596 

.52242 

1.9142 

1.1282 

.11366 

.88634 

25 

40 


24 

36 

. 6330 

. 3670 

.1584 

. 2279 

.9128 

.1284 

. 1380 

. 8620 

24 

36 


28 

37 

. 6355 

. 3645 

.1572 

. 2316 

.9115 

.1286 

1393 

. 8607 

23 

32 


32 

38 

• 6381 

. 3G19 

.1560 

. 2353 

.9101 

.1287 

. 1407 

. 8593 

22 

28 


36 

39 

. 6407 

. 3593 

.1548 

. 2390 

.9088 

.1289 

. 1420 

. 8580 

21 

24 


40 

40 

.46133 

.53567 

2.1536 

.52427 

1.9074 

1.1291 

.11434 

.SSoGO 

20 

20 


44 

41 

. 645S 

. 3541 

.1525 

. 2464 

.9061 

.1293 

. 1447 

. 8553 

19 

16 


48 

42 

. 6484 

. 3516 

.1513 

. 2501 

.9047 

.1294 

. 1461 

. 8539 

18 

12 


52 

43 

. 6510 

. 3490 

.1501 

. 2538 

.9034 

.1296 

. 1474 

. 8526 

17 

S 


56 

44 

. 6536 

. 3464 

.1489 

. 2575 

.9020 

.1298 

. 1488 

. 8512 

16 

4 


51 

45 

.46561 

.53438 

2.1477 

.52612 

1.9007 

1.1299 

.11501 

.88499 

15 

9 


4 

46 

. 6587 

. 3413 

.1465 

. 2650 

.8993 

.1301 

. 1515 

. 8485 

14 

56 


8 

47 

• 6613 

. 3387 

.1453 

. 2687 

.8980 

.1303 

. 1528 

. 8472 

13 

52 


12 

48 

. 6639 

. 3361 

.1441 

. 2724 

.8967 

.1305 

. 1542 

. 8458 

12 

4S 


16 

49 

• 6664 

. 3336 

.1430 

. 2761 

.8953 

.1306 

. 1555 

. 8444 

11 

44 


20 

50 

.46690 

.53310 

2.1418 

.52798 

1.8940 

1.1308 

.11569 

.88431 

10 

40 


24 

51 

. 6716 

. 3284 

.1406 

. 2836 

.8927 

.1310 

. 1583 

. 8417 

9 

36 


28 

52 

. 6741 

. 3258 

.1394 

. 2873 

.8913 

.1312 

. 1596 

. 8404 

8 

32 


32 

53 

. 6767 

. 3233 

•13S2 

. 2910 

.8900 

.1313 

. 1610 

. 8390 

7 

28 


36 

54 

. 6793 

. 3207 

.1371 

. 2947 

.8887 

.1315 

. 1623 

. 8376 

6 

24 


40 

55 

.46819 

.53181 

2.1359 

.52984 

1.8873 

1.1317 

.11637 

.88363 

5 

20 


44 

56 

. G844 

. 3156 

.1347 

. 3022 

.8860 

.1319 

. 1651 

. 8349 

4 

16 


48 

57 

. 6870 

. 3130 

.1335 

. 3059 

.8847 

.1320 

. 1664 

. 8336 

3 

12 


52 

58 

. 6896 

. 3104 

.1324 

. 3096 

.8834 

.1322 

. 1678 

. 8322 

2 

8 


56 

59 

. 6921 

. 3078 

.1312 

. 3134 

.8820 

.1324 

. 1691 

. 8308 

i 

4 


54 

GO 

. 6947 

. 3053 

.1300 

. 3171 

.8807 

.1326 

. 1705 

. 8295 

0 

8 


M.S. 

. 

Cosine. 

Vrs. Sin. 

Secautc. 

Cotang. 

Tangent. 

Cosec’nt-Vrs. Cos 

Sine. 

M 

MS. 


7“ 

117° 



Natural. 




62° 

4 h 



























Natural Lines. 


273 


l h 

28 

3 

Natural Trigonometrical Functions 

151° 

10 h 

M.S. 

M 

Sine. 

Vrs.Cos. 

Cosec'nte 

Tang. 

Cotaug. 

Secante. 

Vrs. Sin 

Cosine. 

M 

M.S. 

5 Si 

0 

.46947 

.53053 

2.1300 

.53171 

1.8807 

1.1326 

.11705 

.88295 

60 

8 

4 

1 

. 0973 

. 3027 

.1289 

. 3208 

.8794 

.1327 

. 1719 

. 8281 

59 

66 

8 

2 

. 0998 

. 3001 

.1277 

. 3245 

.8781 

.1329 

. 1732 

. 8267 

58 

52 

12 

3 

. 7024 

. 2976 

.1266 

. 3283 

.8708 

.1331 

. 1746 

. 8254 

57 

48 

16 

4 

. 7050 

. 2950 

.1254 

. 3320 

.8754 

.1333 

. 1760 

. 8240 

56 

44 

20 

5 

.47075 

.52924 

2.1242 

.53358 

1.8741 

1.1334 

.11774 

.88226 

55 

40 

24 

6 

. 7101 

. 2899 

.1231 

. 3395 

.8728 

.1336 

. 1787 

. 8213 

54 

36 

28 

7 

. 7127 

. 2873 

.1219 

. 3432 

.8715 

.1338 

. 1801 

. 8199 

53 

32 

32 

8 

. 7152 

. 2847 

.1208 

. 3470 

.8702 

.1340 

. 1815 

. 8185 

52 

28 

3G 

9 

. 7178 

. 2822 

.1196 

. 3507 

.8689 

.1341 

. 1828 

. 8171 

51 

24 

40 

10 

.47204 

.52796 

2.1185 

.53545 

1.8676 

1.1343 

.11842 

.SS158 

50 

20 

44 

11 

. 7229 

. 2770 

.1173 

• 3o82 

.8663 

.1345 

. 1856 

. 8144 

49 

16 

48 

12 

« 7—«jo 

. 2745 

.1162 

. 3619 

.8650 

.1347 

. 1870 

. 8130 

48 

12 

02 

13 

. 7281 

. 2719 

.1150 

. 3657 

.8637 

.1349 

. 1883 

. 8117 

47 

8 

50 

14 

. 7300 

. 2694 

.1139 

. 3694 

.8624 

.1350 

. 1897 

. 8103 

46 

4 

53 

15 

.47332 

.52668 

2.1127 

.53732 

1.8611 

1.1352 

.11911 

.88089 

45 

7 

4 

16 

. 7357 

. 2042 

.1110 

. 3769 

.8598 

.1354 

. 1925 

. 8075 

44 

56 

8 

17 

. 73S3 

. 2617 

.1104 

. 3807 

.8585 

.1356 

. 1938 

. 8061 

43 

52 

12 

18 

. 7409 

. 2591 

.1093 

. 3844 

.8572 

.1357 

1952 

. 8048 

42 

48 

10 

19 

. 7434 

. 2565 

.1082 

. 3882 

.8559 

.1359 

. 1966 

. 8034 

41 

44 

20 

20 

.47400 

.52540 

2.1070 

.53919 

1.8546 

1.1361 

.11980 

.88020 

40 

40 

24 

21 

. 7486 

. 2514 

.1059 

. 3957 

.8533 

.1363 

. 1994 

. 8006 

39 

36 

28 

22 

. 7511 

-. 2489 

.1018 

. 3995 

.8520 

.1365 

. 2007 

. 7992 

38 

32 

32 

23 

• 7 oo / 

. 2463 

.1036 

. 4032 

.8507 

.1366 

. 2021 

. 7979 

37 

28 

36 

24 

. 7562 

. 2437 

.1025 

. 4070 

.8495 

.1368 

. 2035 

. 7965 

36 

24 

40 

25 

.47588 

.52412 

2.1014 

.54107 

1.8482 

1.1370 

.12049 

.87951 

35 

20 

44 

20 

. 7013 

. 2386 

.1002 

. 4145 

.8469 

.1372 

. 2063 

. 7937 

34 

1G 

48 

27 

. 7639 

. 236 L 

.0991 

. 4183 

.8456 

.1373 

. 2077 

. 7923 

33 

12 

52 

28 

. 7005 

. 2335 

.0980 

. 4220 

.8443 

.1375 

. 2090 

. 7909 

32 

8 

56 

29 

. 7690 

. 2310 

.0969 

. 4258 

.8430 

.1377 

. 2104 

. 7895 

31 

4 

54: 

30 

.47716 

.52284 

2.0957 

.54295 

1.8418 

1.1379 

.12118 

.87882 

30 

6 

4 

31 

. 7741 

. 2258 

.0946 

. 4333 

.8405 

.1381 

. 2132 

. 7868 

29 

56 

8 

32 

. 7767 

. 2233 

.0935 

. 4371 

.8392 

.1382 

.. 2146 

. 7854 

28 

52 

12 

33 

. 7792 

. 2207 

.0924 

. 4409 

.8379 

.1384 

. 2160 

. 7840 

27 

48 

16 

34 

. 7818 

. 2182 

.0912 

. 4446 

.8307 

.1386 

. 2174 

. 7826 

26 

44 

20 

35 

.47844 

.52156 

2.0901 

.54484 

1.8354 

1.1388 

.12188 

.87812 

25 

40 

24 

36 

. 7869 

. 2131 

.0890 

. 4522 

.8341 

.1390 

. 2202 

. 7798 

24 

36 

28 

37 

. 7895 

. 2105 

.0879 

. 4559 

.8329 

.1391 

2216 

. 7784 

23 

32 

32 

38 

. 7920 

. 2080 

.0808 

. 4597 

.8316 

.1393 

. 2229 

. 7770 

22 

28 

36 

39 

. 7946 

. 2054 

.0857 

. 4635 

.8303 

.1395 

. 2243 

. 7756 

21 

24 

40 

40 

.47971 

.52029 

2.0846 

.5467 3 

1.S291 

1.1397 

.12257 

.87742 

20 

20 

44 

41 

. 7997 

. 2003 

.0835 

. 4711 

.8278 

.1399 

. 2271 

. 7728 

19 

16 

48 

42 

. 8022 

. 1978 

.0824 

. 4748 

.8265 

.14 0L 

. 2285 

. 7715 

18 

12 

52 

43 

. 8048 

. 1952 

.0812 

. 4786 

.8253 

.1402 

. 2299 

. 7701 

17 

8 

56 

44 

. 8073 

. 1927 

.0801 

. 4824 

.8240 

.1404 

. 2313 

. 7687 

16 

4 

55 

45 

.48099 

.51901 

2.0790 

.54862 

1.8227 

1.1406 

.12327 

.87673 

15 

5 

4 

46 

. 8124 

. 1876 

.0779 

. 4900 

.8215 

.1408 

. 2341 

. 7659 

14 

56 

8 

47 

. 8150 

. 1850 

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M.S. 

M 

Cosine. 

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Secaute. 

Cotaug. 

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Cosec’ut.VTs. Cos 

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M 

M.S. 

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118 ( 

3 



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274 


Natural Lines, 


l h 

29° 

Natural Trigonometrical Functions 

150° 

10 h 

M. S. 

M 

Sine. 

Vra.Cos. 

Cosec’nte 

Tang. 

Cotang. 

Secante. 

Vrs. Sin 

Cosine. 

M 

M.S. 

50 

0 

.48181 

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2.0627 

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1.8040 

1.1433 

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4 

4 

1 

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. 5469 

.8028 

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56 

8 

2 

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. 1468 

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. 5507 

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.1437 

. 2566 

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58 

52 

42 

3 

. 8557 

. 1443 

.0594 

. 5545 

.8003 

.1439 

. 2580 

. 7420 

57 

43 

16 

4 

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. 1417 

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. 5583 

.7991 

.1441 

. 2594 

. 7405 

56 

44 

20 

5 

.48608 

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2.0573 

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1.7979 

1.1443 

.12609 

.87391 

55 

40 

24 

6 

. 8633 

. 1366 

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.7966 

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54 

36 

28 

7 

. 8659 

. 1341 

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53 

32 

32 

8 

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28 

36 

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1.7917 

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56 

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4 

57 

15 

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2.0466 

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1.7856 

1.1461 

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3 

4 

16 

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56 

8 

17 

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; 6079 

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52 

12 

18 

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48 

16 

19 

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1.7795 

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36 

28 

22 

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. 6270 

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38 

32 

32 

23 

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37 

28 

36 

24 

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30 

24 

40 

25 

.49116 

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2.0360 

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1.1480- 

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35 

20 

44 

26 

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. 6424 

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. 2907 

. 7093 

34 

16 

48 

27 

. 9166 

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33 

12 

52 

28 

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32 

8 

56 

29 

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31 

4 

58 

30 

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2.0308 

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1.7675 

1.1489 

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40 

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O 

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Secantc. 

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Sine. 

M 

MS. 

j 7“ 

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Natural. 




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4 h 





























Natural Lines. 


275 


2 h 

30° 

Natural Trigonometrical Functions. 

149° 

9 h 

M.S 

M 

Sine. 

Vrs.Cos 

Cosec'nte 

Tang. 

Cotang 

Secante 

Vrs.Siu 

Cosine. 

M 

M.S 

0 

0 

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2.0000 

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1.7320 

1.1547 

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00 

4 

1 

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1.9990 

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56 

8 

2 

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58 

52 

12 

3 

. 0075 

. 9924 

.9970 

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.7286 

.1553 

. 3441 

. 6559 

57 

48 

1G 

4 

. 0101 

. 9899 

.9960 

. 7890 

.7274 

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. 3456 

. 6544 

56 

44 

20 

5 

.50126 

.49874 

1.9950 

.57929 

1.7262 

1.1557 

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.86530 

55 

40 

24 

6 

. 0151 

. 9849 

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54 

36 

28 

7 

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32 

82 

8 

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28 

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10 

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1.9900 

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1.7205 

1.1566 

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20 

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11 

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16 

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12 

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13 

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8 

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4 

1 

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1.9850 

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59 

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56 

M.S. 

M 

Cosine. 

Vrs.Siu. 

Secante. 1 

CotaugjTaugeut. 1 

Coscc'nt 

Vrs. Cos 

Sine. 

M 

M.S. 

8“ 

120 c 




Natural. 



59 u 

3 h 




































276 


Natural Lines. 


2 h 

31 c 

Natural Trigonometrical Functions. 

148° 

9 h 

Af.S. 

M 

Sine. 

Vrs.Cos. 

Cosec'nte 

Tang. 

Cotang. 

Secaute. 

Vrs. Sin 

Cosine. 

M 

M.S. 

4 

0 

.51504 

.48496 

1.9416 

.60080 

1.0643 

1.1666 

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00 

5G 

4 

l 

. 1529 

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59 

50 

8 

2 

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58 

52 

12 

3 

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.6610 

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57 

48 

16 

4 

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5G 

44 

20 

5 

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1.9369 

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1.6588 

1.1070 

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55 

40 

24 

6 

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54 

30 

28 

7 

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32 

32 

8 

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. 1927 

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M. S. 

M 

Cosine. 

Vrs. Sin. 

Secaute. 

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Tangent. 

Coseo’ut 

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M 

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s u 

121 

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277 


1 2 h 

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Natural Trigonometrical Functions. 

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44 

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1.8827 

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1.1802 

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55 

40 

24 

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4S 

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M 

Cosine. 

Vrs.Siu- 

Secuute. 

Cotaug. 

Tangent. 

Sosec'ut! 

Vrs.Cos 

Sine. 

M 

M.S. 

8 h 

122° 



Natural. 



57 

3 h 






































278 


Natural Lines. 


2 h 

33 c 

Natural Trigonometrical 

Functions. 

146° 

9 h 

M.S. 

M 

Sine. 

Vrs.Cos. 

Cosec’nte 

Tang. 

Cotang. 

Secante. 

Vrs.Sin 

Cosine. 

M 

M.S. 

13 

0 

.54404 

.45536 

1.8361 

.64911 

1.5399 

1.1924 

.16133 

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60 

■48 

4 

1 

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59 

66 

8 

2 

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58 

52 

12 

3 

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. 6065 

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. 6180 

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57 

48 

16 

4 

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. 5438 

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. 6196 

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56 

44 

20 

5 

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1.8320 

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1.5350 

1.1935 

.16212 

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55 

40 

24 

6 

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. 6189 

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54 

36 

28 

7 

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.5330 

.1939 

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63 

32 

32 

8 

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52 

28 

36 

0 

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51 

24 

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1.8279 

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1.5301 

1.1946 

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50 

20 

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16 

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48 

12 

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47 

8 

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4 

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1.8238 

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1.5252 

1.1958 

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47 

4 

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44 

56 

8 

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43 

52 

12 

18 

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42 

48 

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41 

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20 

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1.1969 

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24 

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. 6467 

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39 

36 

28 

22 

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.8182 

. 5854 

.5185 

.1974 

. 6483 

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38 

32 

32 

23 

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. 4976 

.8174 

. -9896 

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.1976 

. 6499 

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37 

28 

36 

24 

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. 5938 

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. 6515 

. 3485 

36 

24 

40 

25 

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1.8158 

.65930 

1.5156 

1.1980 

.16531 

.83469 

35 

20 

44 

26 

. 5097 

. 4903 

.8150 

. 6021 

.5147 

.1983 

. 6547 

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34 

16 

48 

27 

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. 6063 

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. 6563 

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33 

12 

52 

28 

. 5145 

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. 6105 

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.1987 

. 6579 

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32 

8 

56 

29 

. 5169 

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.5118 

.1990 

. 6595 

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31 

4 

14 

30 

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1.8118 

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1.5108 

1.1992 

.16611 

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30 

40 

4 

31 

. 6218 

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. 6230 

.5099 

.1994 

. 6627 

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29 

56 

8 

32 

. 5242 

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.5089 

.1997 

. 6613 

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28 

52 

12 

33 

. 6266 

. 4733 

.8094 

. 6314 

.5080 

.1999 

. 6660 

. 3340 

27 

48 

16 

34 

. 6291 

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.5070 

.2001 

. 6676 

. 3324 

20 

44 

20 

35 

.65315 

.44685 

1.8078 

.66398 

1.5061 

1.2004 

.16692 

.83308 

25 

40 

24 

36 

. 5339 

. 4661 

.8070 

. 6440 

.5051 

.2006 

. 6708 

. 3202 

24 

36 

28 

37 

. 6363 

. 4687 

.8062 

. 6482 

.5042 

.2008 

. 6724 

. 3276 

23 

32 

32 

38 

. 5388 

. 4612 

.8054 

. 6524 

.5032 

.2010 

. 6740 

. 3260 

22 

28 

36 

39 

. 6412 

. 4588 

.8047 

. 6560 

.5023 

.2013 

. 6756 

. 3244 

21 

24 

40 

40 

.55436 

.44564 

1.8039 

.0(1008 

1.5013 

1.2015 

.16772 

.83228 

20 

20 

44 

41 

. 5460 

. 4540 

.8031 

. 6050 

.5004 

.2017 

. 6788 

. 3211 

19 

16 

48 

42 

. 5484 

. 4515 

.8(123 

. 6692 

.4934 

.2020 

. 6804 

. 3195 

18 

12 

62 

43 

. 5509 

. 4191 

.8015 

. 6734 

.4985 

.2022 

. 6821 

. 3179 

17 

8 

56 

44 

. 6533 

. 4467 

.8007 

. 6776 

.4975 

.2024 

. 6837 

. 3163 

16 

4 

15 

45 

.55557 

.41443 

1.7999 

.00818 

1.4966 

1.2027 

.16853 

.83147 

15 

45 

4 

46 

. 5581 

. 4119 

.7992 

. 6860 

.4957 

.2029 

. 6869 

. 3131 

14 

56 

8 

47 

. 5605 

. 4395 

.7984 

. 6902 

.4947 

.2031 

. 6885 

. 3115 

13 

62 

12 

48 

. 5629 

. 4370 

.7976 

. 6944 

.4938 

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. 6901 

. 3098 

12 

48 

16 

49 

. 5054 

. 4346 

.7*368 

6986 

.4928 

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. 30*2 

11 

44 

20 

50 

.55678 

.44322 

1.7960 

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1.4919 

1.2039 

.16934 

.83066 

10 

40 

24 

51 

. 5702 

. 4298 

.7953 

. 7071 

.4910 

.2041 

. 6 *50 

. 3050 

9 

36 

28 

52 

. 5726 

. 4274 

.7945 

. 7113 

.4980 

.2043 

. 6966 

. 3034 

8 

32 

32 

53 

. 6750 

. 4250 

.7937 

. 7155 

.489 L 

.2046 

. 6982 

. 3017 

7 

28 

36 

54 

. 5774 

. 4225 

.7929 

. 7197 

.4881 

.2048 

. 6999 

. 3001 

6 

24 

40 

55 

.55799 

44201 

1.7921 

.67239 

1.1872 

1.2050 

.17015 

.82985 

5 

20 

44 

56 

. 5823 

. 4177 

.7914 

. 7282 

.4863 

.2053 

. 7031 

. 200*0 

4 

16 

48 

57 

. 6847 

. 4153 

.7906 

. 7321 

.4863 

.2055 

. 7047 

. 2952 

3 

12 

52 

58 

. 5871 

. 4129 

.7898 

. 7366 

.4844 

.2057 

. 7064 

. 2936 

2 

8 

56 

59 

. 5895 

. 4.105 

.7891 

. 7408 

.48:15 

.2060 

. 7080 

. 2920 

1 

4 

10 

60 

. 5919 

. 4081 

.7883 

. 7451 

.4826 

.2062 

. 7006 

. 2904 

0 

44 

M.S. 

M 

Cosine. 

Vrs.Siu. 

Secanlc. 

Cotaug. 

Tangent. 

Cosec’nt 

Vrs.Cos 

Sine. 

M 

M.S. 

8 h 

123° 



Natural. 




56° 

3 h 
























Natural Lines, 


279 


2 h 

34° 

Natural Trigonometrical Functions. 

145° 

9 h 

M.S 

M 

Sine. 

Vrs.Cos 

Cosee’nte 

Tang. 

Cotang 

Sccante.lVrs.Sin 

Cosine. 

M. 

M.S. 

10 

0 

.55919 

.44081 

1.7883 

.67451 

1.4826 

1.2062 

.17096 

.82904 

60 

44 

4 

1 

. 6943 

. 4057 

.7875 

. 7493 

.4816 

.2064 

. 7112 

. 2887 

59 

56 

8 

2 

. 6967 

. 4032 

.7867 

. 7535 

.4807 

.2667 

. 7129 

. 2871 

58 

52 

12 

3 

. 5902 

. 4008 

.7860 

. 7578 

.4798 

.2069 

. 7145 

. 2855 

57 

48 

10 

4 

. 6016 

. 3984 

.7852 

. 7620 

.4788 

.2072 

. 7161 

. 2839 

56 

44 

20 

5 

.56040 

.43960 

1.7844 

.67663 

1.4779 

1.2074 

.17178 

.82822 

65 

40 

24 

6 

. 6064 

. 3936 

.7837 

. 7705 

.4770 

.2076 

. 7194 

. 2806 

54 

36 

28 

7 

. 6088 

. 3912 

.7829 

. 7747 

.4761 

.2079 

. 7210 

. 2790 

53 

32 

32 

8 

. 6112 

. 3888 

.7821 

. 7790 

.4751 

.2081 

. 7227 

. 2773 

52 

28 

36 

9 

. 6136 

. 3864 

.7814 

. 7832 

.4742 

.2083 

. 7243 

. 2757 

51 

24 

40 

10 

.56160 

.43840 

1.7806 

.67875 

1.4733 

1.2086 

.17259 

,82741 

50 

20 

44 

11 

. 6184 

. 3816 

.7798 

. 7917 

.4724 

.2088 

. 7276 

. 2724 

49 

16 

48 

12 

. 6208 

. 3792 

.7791 

. 7960 

.4714 

.2001 

. 7292 

. 2708 

48 

12 

52 

13 

. 6232 

. 3768 

.7783 

. 8002 

.4705 

.2093 

. 7308 

. 2692 

47 

8 

56 

14 

. 6256 

. 3743 

.7776 

. 8045 

.4696 

.2095 

. 7325 

. 2675 

46 

4 

IT 

15 

.56280 

.43719 

1.7768 

.68087 

1.4687 

1.2098 

.17341 

.82669 

45 

43 

4 

16 

. 6304 

. 3695 

.7760 

. 8130 

.4678 

.2100 

. 7357 

. 2643 

44 

56 

8 

17 

. 6328 

. 3671 

.7753 

. 8173 

.4669 

.2103 

. 7374 

. 2626 

43 

52 

12 

18 

. 6353 

. 3647 

.7745 

. 8215 

.4659 

.2105 

. 7390 

. 2610 

42 

48 

1G 

19 

. 6377 

. 3623 

.7738 

. 8258 

.4650 

.2107 

. 7406 

. 2593 

41 

44 

20 

20 

.56401 

.43599 

1.7730 

.68301 

1.4G41 

1.2110 

.17423 

.82577 

40 

40 

24 

21 

. 6425 

. 3575 

.7723 

. 8343 

.4632 

.2112 

. 7439 

. 2561 

39 

36 

28 

22 

. 6449 

. 3551 

.7715 

. 8386 

.4623 

.2115 

. 7456 

. 2544 

38 

32 

32 

23 

. 6473 

. 3527 

.7708 

. 8429 

.4614 

.2117 

. 7472 

. 2528 

37 

28 

36 

24 

. 6497 

. 3503 

.7700 

. 8471 

.4605 

.2119 

. 7489 

. 2511 

36 

24 

40 

25 

.56521 

.43479 

1.7693 

.68514 

1.4595 

1.2122 

.17505 

.82495 

35 

20 

44 

26 

. 6545 

. 3455 

.7685 

. 8557 

.4586 

.2124 

. 7521 

. 2478 

34 

16 

48 

27 

. 6569 

. 3131 

.7678 

. 8600 

.4577 

.2127 

. 7538 

. 2462 

33 

12 

52 

28 

. 6593 

. 3407 

.7670 

. 8642 

.4568 

.2129 

. 7554 

. 2445 

32 

8 

56 

29 

. 6617 

. 8383. 

.7663 

. 8685 

.4559 

.2132 

. 7571 

. 2429 

31 

4 

18 

30 

.56641 

.43359 

1.7655 

.68728 

1.4550 

1.2134 

.17587 

.82413 

30 

4 2 

4 

31 

. 666-4 


.7648 

. 8771 

.4541 

.2136 

. 7604 

. 2396 

29 

66 

8 

32 

. 6688 

. 3311 

.7640 

. 8814 

.4632 

.2139 

. 7620 

. 2380 

28 

52 

12 

33 

. 6712 

. 3287 

.7633 

• 8857 

.4523 

.2141 

. 7637 

. 2363 

27 

48 

16 

34 

. 6736 

. 3263 

.7(125 

. 8899 

.4514 

.2144 

. 7653 

. 2347 

26 

44 

20 

35 

.56760 

.43239 

1.7GIS 

.68942 

1.4505 

1.2146 

.17670 

.82330 

25 

40 

24 

36 

. 6784 

. 3216 

.7610 

. 8985 

.441-6 

.2149 

. 7C8G 

. 2314 

24 

36 

28 

37 

. 6808 

. 3192 

.7603 

. 9028 

.4487 

.2151 

. 7703 

. 2297 

23 

32 

32 

38 

. 6832 

. 3168 

.7593 

. 9071 

.4-478 

.2153 

. 7719 

. 2280 

22 

28 

36 

39 

. 6856 

. 3144 

.7588 

. 9114 

.44G3 

.2156 

. 7736 

. 2264 

21 

24 

40 

40 

.56880 

.43120 

1.7581 

.69157 

1.4460 

1.2158 

.17752 

.82247 

20 

20 

44 

41 

. 6904 

. 3096 

.7573 

. 9200 

.4451 

.2161 

. 7769 

. 2231 

19 

16 

48 

42 

. 6y28 

. 3072 

.7566 

. 9243 

.4442 

.2163 

. 7786 

. 2214 

18 

12 

52 

43 

. 6952 

. 3048 

.7559 

. 9286 

.4433 

.2166 

. 7802 

. 2198 

17 

8 

56 

44 

. G976 

. 3024 

.7551 

. 9329 

.4424 

.2168 

. 7819 

. 2181 

16 

4 

19 

45 

.57000 

.43000 

1.7544 

.69372 

1.4415 

1.2171 

.17835 

.82165 

15 

41 

4 

46 

. 7023 

. 2976 

.7537 

. 9415 

.4406 

.2173 

. 7852 

. 214X 

14 

56 

8 

47 

. 7047 

. 2952 

.7529 

. 9459 

.4397 

.2175 

. 7868 

. 2131 

13 

52 

12 

48 

. 7071 

. 2929 

.7522 

. 9502 

.43S8 

.2178 

. 7885 

. 2115 

12 

48 

16 

49 

. 7065 

. 2905 

.7514 

. 9545 

.4379 

.2180 

. 7902 

. 2098 

11 

44 

20 

50 

.57119 

.42881 

1.7507 

.69588 

1.4370 

1.2183 

.17918 

.82082 

10 

40 

24 

51 

. 7143 

. 2857 

.7500 

. 9G31 

.4361 

.2185 

. 7935 

. 2065 

9 

36 

28 

52 

. 7167 

. 2833 

.7493 

. 9G74 

.4352 

.2188 

. 7951 

. 2048 

8 

32 

32 

53 

. 7191 

. 2S09 

.7485 

. 9718 

.4243 

.2190 

. 7968 

. 2032 

7 

28 

36 

54 

. 7214 

. 27S5 

.7478 

. 9761 

.4335 

.2193 

. 7985 

. 2015 

6 

24 

40 

55 

.57238 

.42761 

1.7471 

.69804 

1.4326 

1.2195 

.18001 

.81998 

5 

20 

44 

56 

. 7262 

. 2738 

.7463 

. 9847 

.4317 

.2198 

. 8018 

. 1982 

4 

16 

48 

57 

. 72S6 

. 2714 

.7456 

. 9891 

.4308 

.2260 

. 8035 

. 1965 

3 

12 

52 

58 

. 7310 

. 2690 

.7449 

. 9934 

.4299 

.2203 

. 8051 

. 1948 

2 

8 

56 

59 

. 7334 

. 2666 

.7442 

. 9377 

.4230 

.2205 

. 8(J6S 

. 1932 

1 

4 

20 

60 

. 7358 

. 2642 

.7434 

.70021 

.4281 

.2208 

. 8085 

. 1915 

0 

4 0 

M. S. 

8 h 

M 

L24° 

Cosine. 

i'rs.Siu. 

Secaute. 

CotaugJTangeut. 

Natural. 

Coseout 1 

Vrs.Cos 

Sine. 

M 

55° 

M.S. 

3 h 






























280 


Natural Lines. 


2" 

35° Natural Tri" 

onometrleal 

Functions. 

144° 

9 h 

M.S. 

M 

Sine. 

Yrs.Oos. 

Cosec'nte 

Tang. 

Cotang. 

Secaute. 

Yrs. Sin 

Cosine. 

M 

M.S. 

20 

0 

.57:358 

.42642 

1.7434 

.70021 

1.4281 

1.2208 

.18085 

.81915 

60 

40 

4 

1 

. 738 L 

. 2618 

.7427 

. 0064 

.4273 

.2210 

. 8101 

. 1898 

59 

56 

8 

2 

. 7405 

. 2595 

.7420 

. 0107 

.4264 

.2213 

. 8118 

. 1882 

58 

52 

12 

3 

. 7429 

. 2571 

.7413 

. 0151 

.4255 

.2215 

. 8135 

. 1865 

57 

48 

16 

4 

. 7453 

. 2547 

.7405 

. 0194 

.4246 

.2218 

. 8151 

. 1848 

5G 

44 

20 

5 

.57477 

.42523 

1.7398 

.70238 

1.4237 

1.2220 

.18168 

.81832 

55 

4o 

24 

6 

. 7500 

. 2499 

.7391 

. 6281 

.4228 

.2223 

. 8185 

. 1815 

54 

36 

28 

7 

. 7524 

. 2476 

.7384 

. 0325 

.4220 

.2225 

. 8202 

. 1798 

53 

32 

32 

8 

. 7548 

. 2452 

.7377 

. 0368 

.4211 

.2228 

. 8218 

. 178L 

52 

28 

36 

9 

. 7572 

. 2428 

.7369 

. 0412 

.4202 

.2230 

. 8235 

. 1765 

51 

24 

40 

10 

.57596 

.42404 

1.7362 

.70455 

1.4193 

1.2233 

.18252 

.81748 

50 

20 

14 

11 

. 7619 

. 2380 

.7355 

. 0499 

.4185 

.2235 

. 8269 

. 1731 

49 

16 

48 

12 

. 7643 

. 2357 

.7348 

. 0542 

.4176 

.2238 

. 8285 

. 1714 

48 

12 

02 

13 

. 7667 

. 2333 

.7341 

. 0586 

.4167 

.2240 

. 8302 

. 1698 

47 

8 

56 

14 

. 7691 

. 2309 

.7334 

. 0629 

. .4158 

.2243 

. 8319 

. 1681 

46 

4 

21 

15 

.57714 

.42285 

1.7327 

.70673 

1.4150 

1.2245 

.18336 

.81664 

45 

39 

4 

16 

. 7738 

. 2262 

.7319 

. 0717 

.4141 

.2248 

. 8353 

. 1647 

44 

56 

8 

17 

. 7762 

. 2238 

.7312 

. 0760 

.4132 

.2250 

. 8369 

. 1630 

43 

52 

12 

18 

. 7786 

. 2214 

.7305 

. 0804 

.4123 

.2253 

. 8386 

. 1614 

42 

48 

16 

19 

. 7809 

2190 

.7298 

. 0848 

.4115 

.2255 

. 8403 

. 1597 

41 

44 

20 

20 

.57833 

.42167 

1.7291 

.7089 L 

1.4106 

1.2258 

.18420 

.81580 

40 

40 

24 

21 

. 7857 

.2143 

.7284 

. 0935 

.4097 

.2260 

. 8437 

. 15(i3 

39 

36 

28 

22 

. 7881 

. 2119 

.7277 

.13979 

.4089 

.2263 

• b45'} 

. 1546 

38 

32 

32 

23 

. 7904 

. 2096 

.7270 

. 1022 

.4080 

.2265 

. 8470 

. 1530 

37 

28 

3(5 

24 

. 7928 

. 2072 

.7263 

. 1066 

.407 L 

.2268 

. 8487 

. 1513 

36 

24 

40 

25 

.57952 

.42048' 

1.725G 

.71110 

1.4063 

1.2270 

.18504 

.81496 

35 

20 

44 

26 

. 7975 

. 2024 

.7249 

. 1154 

.4054 

.2273 

. 8521 

. 1479 

34 

16 

48 

27 

. 7999 

. 2001- 

.7242 

. 119S 

.4045 

.2276 

. 8538 

. 1462 

33 

12 

52 

28 

. 8023 

. 3977 

.7234 

. 1241 

.4037 

.2278 

. 8555 

. 1445 

32 

8 

56 

29 

. 8047 

. 1953 

.7227 

. 128.5 

.4028 

.2281 

. 8571 

. 1428 

31 

4 

22 

30 

.58070 

.41930 

1.7220 

.71329 

1.4019 

122283 

.18588 

.81411 

30 

38 

4 

31 

. 8094 

. 1906 

.7213 

• lo73 

.4011 

.2286 

. 8605 

. 1395 

29 

56 

8 

32 

. 8118 

. 1882 

.7206 

. 1417 

.4002 

222S8 

. 8622 

. 1378 

28 

52 

12 

33 

. 8141 

. 1859 

.7199 

. 1461 

.3994 

.2291 

. 8639 

. 1361 

27 

48 

16 

34 

. 8165 

. 1835 

.7192 

. 1505 

.3985 

.‘>293 

. 8656 

. 1344 

26 

44 

20 

35 

.58189 

.41811 

1.7185 

.71549 

1.3976 

1.2296 

.18673 

.81327 

25 

40 

24 

36 

. 8212 

. 1788 

.7178 

. 1593 

.3968 

.2298 

. 8690 

. mo 

24 

36 

28 

37 

. 8236 

. 1764 

.7171 

. 1637 

.3959 

.2301 

. ,S707 

. 1293 

23 

32 

32 

38 

. 8259 

. 1740 

.7161 

. 1681 

.3951 

.2304 

. 8724 

. 1276 

22 

28 

36 

39 

. 8283 

. 1717 

.7157 

. 1725 

.3942 

.2300 

. 8741 

. 1259 

21 

24 

40 

40 

.58307 

.41693 

1.7151 

.71769 

1.3933 

1.2309 

.18758 

.81242 

20 

20 

44 

41 

. 8330 

. 1669 

.7114 

. 1SI3 

.3925 

.2311 

. 8775 

. 1225 

19 

16 

48 

42 

. 8351 

. 1646 

.7137 

. 1857 

.3916 

.2314 

. 8792 

. 1208 

18 

12 

52 

43 

. 8378 

. 1622. 

.7130 

. 1901 

.3908 

.2316 

. 8809 

. 1191 

17 

8 

56 

44 

. 8401 

. 1599 

.7123 

. 1945 

.3899 

.2319 

. 8826 

. 1174 

16 

4 

23 

45 

.58425 

.41576 

1.7LIG 

.71990 

1.3891 

1.2322 

.18843 

.81157 

15 

37 

4 

46 

. 8448 

. 1551 

.7109 

. 2034 

.38S2 

.2324 

. 8860 

. 1140 

14 

56 

8 

47 

. 8472 

. 1528 

.7102 

. 2078 

.3874 

22327 

. 8877 

. 1123 

13 

52 

12 

48 

. 8496 

. 1504 

.7095 

. 2122 

.3865 

.23.9 

. 8894 

. 1106 

12 

48 

16 

49 

. 8519 

. 1481 

.7088 

21G6 

.3857 

.2332 

. 8911 

. 1089 

11 

44 

20 

50 

.58543 

.41457 

1.7081 

.72211 

1.3848 

1.2335 

.18928 

.81072 

10 

40 

24 

51 

. 8566 

. 1433 

.7075 

. 2255 

.3840 

.2337 

. 8945 

. 1055 

9 

36 

28 

52 

. 8590 

. 1410 

.7068 

. 2299 

.3831 

.2340 

. 8962 

. 1038 

8 

32 

32 

53 

. 8614 

. 1886 

.7061 

. 2314 

.3823 

.2342 

. 8979 

. 1021 

7 

2S 

36 

54 

. 8637 

. 1363 

.7051 

. 2388 

.3814 

.2345 

. 8996 

. 1004 

6 

24 

40 

55 

.58661 

.41339 

1.7047 

.72432 

1.3806 

1.2348 

.19013 

.80987 

5 

20 

44 

56 

. 8684 

. 1316 

.7010 

. 2477 

.3797 

.2350 

. 9030 

. 0970 

4 

16 

48 

57 

. 8708 

. 1292 

.7033 

. 2521 

.3789 

.2353 

. 9047 

. 0953 

3 

12 

52 

58 

. 8731 

. 1268 

.7027 

. 2565 

.3781 

.2355 

. 9064 

. 0936 

2 

8 

56 

59 

. 8755 

. 1245 

.7020 

. 2610 

.3772 

.2358 

. 9081 

. 0919 

1 

4 

24: 

60 

. 8778 

. 1221 

.7013 

. 2654 

.3764 

.2361 

. 9398 

. 0902 

0 

30 

M.S. 

M 

Cosine. 

Yis. Sin. 

Secaute. 

Cotuug. 

I Tangent. 

Coseo’m • Yrs.Cos 

Sine. 

1 M 

M.S. 

8 h 

125° 



Natural. 




54° 

3“ 
























Natural Lines. 281 


2 h 

36° 

Natural Trigonometrical Functions. 143° 

9" 

M.S 

M 

Sine. 

Vrs.Oos 

Cosee'nte 

Tang. 

Cotang. 

Secante. 

Vrs. Sin 

Cosine. 

M 

M.S. 

31 

0 

.58778 

.41221 

1.7013 

.72654 

1.3704 

1,2361 

.19098 

.80902 

60 

36 

4 

1 

• 8802 

. 1198 

.7006 

. 2699 

.3755 

.2363 

. 9115 

. 0885 

59 

56 

8 

2 

. 8825 

. 1174 

.6999 

. 2743 

.3747 

.2366 

. 9132 

. 0867 

58 

52 

12 

3 

. 8849 

. 1151 

.6903 

. 2788 

.3738 

.2368 

. 9150 

. 0850 

57 

48 

16 

4 

. 8873 

. 1127 

.6986 

. 2832 

.3730 

.2371 

. 9167 

. 0833 

56 

44 

20 

5 

.58896 

.41104 

1 6979 

.72877 

1.3722 

1.2374 

.19184 

.80816 

55 

40 

24 

C 

. 8920 

. 1080 

.6972 

. 2921 

.3713 

.2376 

. 9201 

. 9799 

54 

36 

28 

7 

. 8943 

. 1057 

.6965 

. 2966 

.3705 

.2379 

. 9218 

. 0782 

53 

32 

32 

8 

• 8967 

. 1033 

.6959 

. 3010 

.••'697 

.2382 

. 9235 

. 07('5 

52 

28 

36 

9 

. 8990 

. 1010 

.6952 

. 3055 

.3688 

.2384 

. 9252 

. 0747 

51 

24 

40 

10 

.59014 

.40986 

1.6945 

.73100 

1.3 .80 

1.2387 

.19270 

.80730 

50 

20 

44 

11 

. 9037 

. 0903 

.6938 

. 3144 

..”672 

.2389 

. 9287 

. 0713 

49 

16 

48 

12 

. 9060 

. 0939 

.6932 

. 3189 

.2603 

.2392 

. 9301 

. 0696 

48 

12 

52 

13 

. 9084 

. 0916 

.6925 

. 3234 

.3655 

.2305 

. 9321 

. 0679 

47 

8 

56 

14 

. 9107 

. 0892 

.6918 

• 3278 

.3617 

.2397 

. 9338 

. 0662 

40 

4 

45 

15 

.59131 

.40869 

1.6912 

.73323 

1.3638 

1.2400 

.19355 

.80644 

45 

35 

4 

16 

• 9154 

. 0845. 

.6905 

. 3368 

.363! I 

.2403 

. 9373 

. 0627 

44 

56 

8 

17 

. 9178 

. 0822 

.6898 

. 3412 

.3622 

.2405 

. 9390 

. 0610 

43 

52 

12 

IS 

. 92ol 

. 0799 

.6891 

. 3457 

21613 

.2408 

. 9407 

. 0593 

42 

48 

16 

19 

. 9225 

. 0775 

•68c 5 

. 3502 

.3605 

.2411 

. 9424 

. 0576 

41 

44 

20 

20 

.59248 

.40752 

1.6878 

.73547 

1.3597 

1.2413 

.19412 

.80558 

40 

40 

24 

21 

. 9272 

. 0728 

.6871 

. 3592 

.3588 

.2416 

. 9459 

. 0541 

39 

36 

28 

22 

. 9295 

. 0705 

.6365 

. 3637 

.3580 

.2419 

. 9476 

. 0524 

28 

32 

32 

23 

. 9318 

. 0681 

.6858 

. 3681 

.3572 

.2421 

. 9493 

. 0507 

37 

28 

36 

24 

. 9342 

. 0658 

.6851 

. 3726 

.3564 

.2424 

. 9511 

. 0489 

36 

24 

40 

25 

.59365 

.40635 

1.6845 

.73771 

1.3555 

1.2427 

.19528 

.80472 

35 

20 

. 44 

26 

. 9389 

. 0611 

.6838 

. 3816 

.3517 

.2429 

. 9545 

. 0465 

34 

10 

48 

27 

. 9412 

. 0588 

•6S31 

. 3861 

.3539 

.2482 

. 9562 

. 0437 

33 

12 

52. 

28 

. 9435 

. 0564 

.6825 

. 3906 

.8531 

.2435 

. 9580 

. 0420 

32 

8 

56 

29 

. 9459 

. 0541 

.GS18 

. 3951 

.3522 

.2437 

. 9597 

. 0403 

31 

4 

£0 

30 

.59182 

.40518 

1.6812 

.73996 

1.3514 

1.2440 

.19614 

.80386 

30 

34 

4 

31 

. 9506 

. 0494 

.6805 

. 4041 

.3506 

.2443 

. 9632 

. 0368 

29 

56 

8 

32 

. 9529 

. 0471 

.6798 

. 4086 

.3498 

.2445 

. 9649 

. 0351 

28 

52 

12 

33 

. 9552 

. 0447 

•6792 

. 4131 

.3489 

•244S 

. 9666 

. 0334 

27 

48 

16 

34 

. 9576 

. 0424 

.6785 

. 4176 

.34S1 

.2451 

. 9683 

. 0316 

26 

41 

20 

35 

.59599 

.40401 

1.6779 

.74221 

1.3473 

1.2153 

.19701 

.80299 

25 

40 

24 

36 

. 9622 

. 0377 

■ 0772 

. 4266 

.3465 

.2456 

. 9718 

. 0282 

24 

36 

28 

37 

. 9646 

. 0354 

.6766 

. 4312 

.3457 

.2459 

9736 

. 0264 

23 

32 

32 

38 

. 9669 

. 0331 

.6759 

. 4357 

.3449 

.2461 

. 9753 

. 0247 

22 

28 

36 

39 

- 9692 

. 0307 

.0752 

. 4402 

.3440 

.2464 

. 9770 

. 0230 

•A 

24 

40 

40 

.59716 

.40284 

1.0746 

.74447 

1.3432 

1.2467 

.19788 

.80212 

20 

20 

44 

41 

. 9739 

. 0261 

.6739 

. 44:2 

.3424 

.2470 

. 9805 

. 0195 

19 

16 

48 

42 

. 9762 

. 0237 

.6733 

. 4538 

.3416 

.2472 

. 9822 

. 0177 

18 

12 

52 

43 

. 9786 

. 0214 

.6726 

. 4583 

.3408 

.2475 

. 9,-40 

. 0160 

17 

8 

56 

41 

. 9809 

. 0191 

.6720 

. 4628 

.3400 

.2178 

. 9857 

. 0143 

10 

4 

27 

45 

.59832 

.40167 

1.6713 

.74673 

1.3392 

1.2480 

.19875 

.80125 

15 

33 

4 

46 

. 9856 

. 0144 

.6707 

. 4719 

.3383 

.2183 

. 9892 

. 0108 

14 

56 

8 

47 

. 9879 

. 0121 

.6700 

. 4764 

.3375 

.24 v 6 

. 9909 

. 0090 

13 

52 

12 

48 

. 9902 

. 0098 

.6694 

. 4809 

.3367 

.2488 

. 9927 

. 0073 

12 

48 

16 

49 

. 9926 

. 0074 

.6687 

. 4855 

.3359 

.2495 

. 9944 

. 0056 

11 

44 

20 

50 

.59949 

.40051 

1.6681 

.74900 

1.3351 

1.2494 

.19962 

.80038 

10 

40 

24 

51 

. 9972 

. 0028 

.6674 

. 4946 

.3343 

.2497 

. 9979 

. 0021 

9 

36 

28 

52 

. 9995 

. 0004 

.66GS 

. 4991 

.3335 

.2499 

. 9997 

. 0003 

8 

32 

32 

53 

.60019 

.39981 

.6661 

. 5037 

.3327 

.2502 

.20014 

.79986 

7 

28 

36 

54 

. 0042 

. 9958 

.6655 

. 5082 

.3319 

.2505 

. 0031 

. 9968 

6 

24 

40 

55 

.60065 

.39935 

1.6048 

.75128 

1.3311 

1.2508 

.20049 

.79951 

5 

20 

41 

56 

. 0088 

. 9911 

.6642 

. 5173 

.3303 

.2510 

. 0066 

. 9933 

4 

16 

48 

•57 

. 0112 

. 9888 

.6636 

. 5219 

.3294 

.2513 

. 0084 

. 9916 

3 

12 

62 

58 

. 0135 

. 9865 

.6629 

. 5264 

.3286 

.2516 

. 0101 

. 9898 

2 

8 

56 

59 

. 0158 

. 9842 

.6623 

. 5310 

.3278 

.2519 

. 0119 

. 9881 

1 

4 

28 

60 

. 0181 

. 9818 

.6616 

. 5355 

.3270 

.2521 

. 0136 

. 9863 

0 

34 

M.S. 

M 

Cosine. 

Vrs.Sin 

Secante. 

lotJlIlg. 

1’au gent. 

Cosec’nt. 

yrs. Cos 

Sine. 

M 

MS. 

8 h 

126° 




Natural. 



53° 

o h 





























282 


Natural Links. 


2 h 

87° 

Natural Trigonometrical Functions 

142° 

9 h 

M.S. 

M 

Sine. 

Vl’.s. Cos. 

Cosec’ute 

Tang. 

Cotang. 

Secaute. 

Vrs. Sin 

1 Cosine. 

M 

M.S. 

38 

0 

.60181 

.39818 

1.6016 

•75355 

1.3270 

1.2521 

.20136 

.79868 

GO 

33 

4 

l 

. 0205 

. 9795 

.6010 

. 5401 

.3262 

.2524 

. 0154 

. 9846 

59 

56 

8 

2 

. 0228 

. 9772 

.6603 

. 5447 

.3254 

.2527 

. 0171 

. 9828 

58 

52 

12 

3 

. 0251 

. 9719 

.6597 

. 5492 

.3246 

.2530 

. 0189 

. 9811 

57 

48 

Ki 

4 

. 0274 

. 9726 

.6591 

. 5538 

.3238 

.2532 

. 0206 

. 9793 

56 

44 

20 

5 

.63298 

.39702 

1.6584 

.75584 

1.3230 

1.2535 

.20224 

.79776 

55 

40 

24 

6 

. 0320 

. 9679 

.6578 

. 5629 

.3222 

.2538 

. 0212 

. 9758 

54 

3 6 

28 

7 

. 0344 

. 9656 

.6572 

. 5675 

.3214 

.2541 

. 0259 

. 9741 

53 

32 

32 

8 

. 0367 

. 9633 

.6565 

. 5721 

.3206 

.2543 

. 0277 

. 9723 

52 

28 

36 

9 

. 0390 

. 9610 

.6659 

. 5767 

.3198 

.2546 

. 0294 

. 9706 

51 

24 

■JO 

10 

.60413 

.39586 

1.6552 

.75812 

1.3190 

1.25,9 

.20312 

.79688 

50 

20 

44 

11 

. 0437 

. 9563 

.6546 

. 5858 

.3182 

.2552 

. 0329 

. 9670 

49 

16 

48 

12 

. 0460 

. 9540 

.6540 

. 5904 

.3174 

.2554 

. 0547 

. 9653 

48 

12 

52 

13 

. 0483 

. 9517 

.6533 

. 6950 

.3166 

.2557 

. 0865 

. 9635 

47 

8 

56 

14 

. 0506 

. 9494 

.6527 

. 5996 

.3159 

.2560 

. 0382 

. 9618 

4G 

4 

39 

15 

.60529 

.39471 

1.6521 

•76042 

1.3151 

1.2563 

.20400 

.79600 

45 

31 

4 

16 

. 0552 

. 9447 

.6514 

. 0088 

.3143 

.2565 

. 0417 

. 9582 

44 

56 

8 

17 

. 0576 

. 9424 

.6508 

. 6134 

.3135 

.2568 

. 0435 

. 9565 

43 

52 

12 

18 

. 0599 

. 9401 

.6502 

. 6179 

.3127 

.2571 

. ( 453 

. 9547 

42 

48 

16 

19 

. 0622 

. 9378 

.6496 

. 6225 

.3119 

.2574 

. 0470 

. 9580 

41 

44 

20 

20 

.6' 1646 

.39355 

1.6189 

.7627 

1.3111 

1.2577 

.204SS 

.79512 

40 

40 

24 

21 

. 0668 

. 9332 

.6183 

. 6317 

.3103 

.2579 

. 0505 

. 9494 

o9 

36 

28 

22 

. 0691 

. 9309 

.6477 

. 0364 

.3095 

.2582 

. 0523 

. 9477 

38 

32 

32 

23 

. 0714 

. 9285 

.6470 

. 0410 

.3087 

.2585 

. 0541 

. 9459 

37 

28 

30 

24 

• 0737 

. 9202 

.6464 

. 6450 

. .3079 

.2588 

. 0558 

. 9441 

36 

24 

40 

26 

.60761 

.39239 

1.6458 

•76502 

1.3071 

1.2591 

.20576 

.79424 

35 

20 

41 

26 

• 0784 

. 9216 

.6452 

. 6548 

.3064 

.2593 

. 0594 

. 9406 

34 

16 

48 

27 

• 0807 

. 9193 

.6445 

. 0594 

.8056 

.2596 

. 0611 

. 9388 

33 

12 

52 

28 

. 08:0 

. 9170 

.6439 

. 6(440 

.3048 

.2599 

. 0629 

. 9371 

32 

8 

56 

29 

. 0853 

. 9147 

.6433 

• 66S6 

.3040 

.2602 

. 0647 

. 9353 

31 

* 4 

30 

30 

•60876 

.39121 

1.6427 

.76733 

1.8032 

1.2605 

.20665 

.79335 

30 

30 

4 

31 

. 0899 

. 9101 

.6420 

. 6779 

.3024 

.2607 

. 0682 

. 9318 

29 

56 

8 

32 

. 0922 

. 9078 

.6414 

. 6825 

.3016 

.2610 

. 0700 

. 9300 

28 

52 

12 

33 

. 0945 

. 8065 

.6408 

. 6871 

.3009 

.2613 

. 0713 

. 9282 

27 

48 

1G 

34 

. 0963 

. 9031 

.6402 

. 0918 

.3001 

.2616 

. 0735 

. 9264 

2G 

41 

20 

35 

•60991 

.39008 

1.6396 

.76964 

1.2993 

1.2619 

.20753 

.79247 

25 

40 

24 

36 

• 1014 

. 8985 

.6389 

. 7010 

.2985 

.2622 

. 0771 

. 9.29 

24 

36 

28 

37 

• 1 (i3 1 

. 8962 

.6383 

. 7057 

.2977 

.2624 

0789 

. 9211 

23 

32 

32 

38 

. 1061 

. 8939 

.6377 

. 7103 

.2970 

.2627 

. 08UG 

. 9193 

22 

28 

36 

:m 

• 1081 

. 8916 

.637 L 

. 7149 

.2962 

.2630 

. 0824 

. 9176 

21 

24 

40 

40 

.61107 

.3" 893 

1.0365 

.77190 

1.2954 

1.26:13 

.20842 

.79158 

20 

20 

44 

41 

. 1130 

. 8870 

.6359 

. 7242 

.2946 

.2636. 

. 0860 

. 9140 

19 

16 

48 

42 

. 1153 

. 8847 

.6352 

. 7289 

.2938 

.2639 

. 0878 

. 9122 

18 

12 

52 

43 

. 117G 

. 8824 

.6346 

. 7335 

22931 

.2641 

. 0895 

. 9104 

17 

8 

50 

41 

. 1199 

. 8801 

.G340 

. 7382 

.2923 

.2644 

. 0913 

. 9087 

16 

4 

31 

45 

.61222 

.38778 

1.6334 

.77428 

1.2915 

1.2647 

20931 

.79069 

15 

39 

4 

46 

. 1245 

. 8755 

.6328 

. 7475 

.2907 

.2650 

. 0949 

. 9051 

14 

50 

8 

47 

. 1268 

. 8732 

.6322 

. 7521 

.2900 

.2653 

. 0967 

. 9033 

13 

52 

12 

48 

. 1290 

. 8709 

.6316 

. 7568 

.2892 

.2656 

. 0984 

. 9015 

12 

48 

10 

49 

. 1314 

. 8686 

.6309 

. 7614 

.2884 

.2659 

. 1002 

. 899S 

11 

44 

20 

50 

■61337 

.38663 

1.6303 

.77661 

1.2876 

1.2661 

.21020 

.78980 

10 

40 

24 

51 

. 1369 

. 8040 

.6297 

. 7708 

.2869 

.2664 

. 1038 

. 8962 

9 

36 

28 

52 

. 1383 

. 8617 

.6291 

. 7764 

.2861 

.2667 

. 1056 

. 8914 

8 

32 

32 

53 

• 1405 

. 8594 

.6285 

. 7801 

.2853 

.2670 

. 1074 

. 8926 

7 

28 

36 

54 

. 1428 

. 8571 

.6279 

. 7848 

.2345 

.2673 

. 1091 

. 8908 

6 

24 

40 

55 

.61451 

.38548 

1 6273 

.77895 

1.2838 

1.2676 

.21109 

.78880 

5 

20 

41 

50 

. 1474 

. 8525 

.6267 

. 7941 

.2830 

.2679 

. 1127 

. 8873 

4 

16 

48 

57 

. 1497 

. 8503 

.6261 

. 7988 

.2822 

.2681 

. 1145 

. 8855 

3 

12 

52 

58 

. 1520 

. 8480 

.6255 

. 8035 

.2815 

.2684 

. 1163 

. 8837 

2 

8 

56 

59 

. 1543 

. 8467 

.6249 

. S0S2 

.2807 

.2687 

. 1181 

. 8819 

1 

4 

32 

60 

. 1560 

. 8434 

.6243 

. 8128 

.2799 

.2690 

. 1199 

. 8801 

0 

38 

M.S. 

M 

Cosine. 

Vrs.Sin. 

Secaute. 

Cotang. 

Tii agent. 

Cosec ut. 

Vrs. Cos 

Sine. 

M 

M . S. 

86 

127° 



Natural. 




52° 

3 h 
























Natural Lines 


283 


2 h 

38° 

Natural Trigonometrical Functions 

141° 

9 h 

M.S 

; m 

Sine. 

Vrs.Cos 

Coscc'nte 

Tang. 

Cotang 

Sccante.jVrs.Sii 

Cosine. 

M 

M.S. 

33 

0 

.61566 

.38434 

1.6243 

.78128 

1.2799 

1.2690 

.21199 

.78801 

GO 

a 8 

4 

1 

. .1589 

. 8411 

.6,*:;? 

. 8175 

.2792 

.2693 

. 1217 

. 8783 

59 

56 

8 

2 

. 1612 

. 83 v 8 

.6231 

. 8222 

.2784 

.2696 

. 1235 

. 8765 

58 

52 

12 

3 

. 1635 

. 8365 

.6224 

. 8269 

.2776 

.2699 

. 1253 

. 8747 

57 

48 

16 

4 

. 1658 

. 8342 

.6218 

. 8316 

.2769 

.2702 

. 1271 

. 8729 

56 

44 

20 

5 

.61681 

.38319 

1.6212 

.78363 

1.2761 

1.2705 

.21288 

.78711 

55 

40 

24 

6 

. 1703 

. 8296 

.6206 

. 8410 

.2753 

.2707 

. 1306 

. 8693 

54 

36 

28 

7 

. 1726 

. 8273 

.6200 

. 8457 

.2746 

.2710 

. 1324 

. 8675 

53 

32 

32 

8 

. 1749 

. 8251 

.6194 

. 8504 

.2738 

.2713 

. 1342 

. 8657 

52 

28 

36 

9 

. 1772 

. 8228 

.6188 

. 8561 

.2730 

.2716 

. 1300 

. 8640 

51 

24 

40 

10 

.61795 

.382(35 

1.6182 

.78598 

1.2723 

1.2719 

.21378 

.78622 

•50 

20 

44 

11 

. 1818 

. 8182 

.6176 

. 8645 

.2715 

.2722 

. 1396 

. 8604 

49 

16 

48 

12 

. 1841 

. 8159 

.6170 

. 8692 

.2708 

.2725 

. 1414 

. 8586 

48 

32 

52 

13 

. 1864 

. 8136 

.6164 

• 8739 

.2700 

.2728 

. 1432 

. 8568 

47 

8 

56 

14 

. 1886 

. 8113 

.6159 

. 8786 

.2092 

.2731 

. 1450 

. 8550 

46 

4 

33 

15 

.61909 

.38091 

1.6153 

.78834 

1.2685 

1.2734 

.21468 

.78532 

45 

37 

4 

16 

. 1932 

. 8068 

.6147 

. 8881 

.2677 

.2737 

. 1486 

. 8514 

44 

56 

8 

17 

. 1955 

. 8045 

.6141 

. 8928 

.2670 

.2739 

. 1504 

. 8496 

43 

52 

12 

18 

. 1978 

. 8022 

.6135 

. 8975 

.2662 

.2742 

. 1522 

. 8478 

42 

48 

16 

19 

. 2001 

. 7990 

.6129 

. 9022 

.2655 

.2745 

. 1540 

. 8460 

41 

44 

20 

20 

.62023 

.37976 

1.6123 

.79070 

1-.2047 

1.2748 

.21558 

J8441 

40 

40 

24 

21 

. 2046 

. 7954 

.6117 

. 9117 

.2639 

.2751 

. 1576 

. 8423 

39 

36 

28 

22 

. 2069 

. 7931 

.6111 

. 9164 

.2632 

.2754 

. 1594 

. 8405 

38 

32 

32 

23 

. 2092 

. 7908 

.6105 

. 9212 

.2G24 

.2757 

. 1612 

. 8387 

37 

28 

36 

24 

. 2115 

. 78S5 

.6099 

. 9259 

.2617 

.2760 

. 1631 

. 8369 

36 

24 

40 

25 

.62137 

.37862 

1.6093 

.79306 

1.2609 

1.2763 

.21649 

.78351 

35 

20 

44 

26 

. 2160 

. 7840 

.6087 

. 9354 

.2602 

.27G6 

. 1667 

. 8333 

34 

16 

48 

27 

. 2183 

. 7817 

.eosi 

. 94ol 

.2594 

.2769 

. 1685 

. 8315 

33 

12 

52 

2S 

. 2206 

. 7794 

.6077 

. 9449 

.2587 

.2772 

. 1703 

. 8297 

32 

8 

56 

29 

. 2229 

. 7771 

.6070 

. 9496 

.2579 

.2775 

. 1721 

. 8279 

31 

4 

34 

30 

.62251 

.37748 

1.C0G4 

.79543 

1.2572 

1.2778 

.21739 

.78261 

30 

30 

4 

31 

. 2274 

. 7726 

.6058 

. 9591 

.2564 

.2781 

. 1757 

. 8243 

29 

56 

8 

32 

. 2297 

. 7703 

.6052 

. 9639 

.2557 

.2784 

. 1775 

. 8224 

28 

52 

12 

33 

. 2320 

. 7680 

.6046 

. 9686 

.2549 

.2787 

. 1793 

. 8206 

27 

48 

16 

34 

. 2342 

. 7657 

.6040 

. 9734 

.2542 

.2790 

. 1812 

. 8188 

26 

44 

20 

35 

.62365 

.37 635 

1.6034 

.79781 

1.2534 

1.2793 

.21830 

.78170 

25 

40 

24 

36 

. 2388 

. 7612 

.6029 

. 9829 

.2527 

.2795 

. 1848 

. 8152 

24 

36 

28 

37 

. 2411 

. 7589 

.6023 

. 9876 

.2519 

.2798 

. I860 

. 8134 

23 

32 

32 

38 

. 2433 

. 7566 

.6017 

. 9924 

.2512 

.2801 

. 1884 

. 8116 

22 

28 

36 

39 

. 2456 

. 7544 

.6011 

. 9972 

.2504 

.2804 

. 1902 

. 8097 

21 

24 

40 

40 

.62479 

.37521 

1.6005 

.80020 

1.2497 

1.2807 

.21921 

.78079 

20 

20 

44 

41 

. 2501 

. 7498 

.6000 

. 0067 

.2489 

.2810 

. 1939 

. 8061 

19 

16 

48 

42 

. 2524 

. 7476 

.6994 

. 0115 

.2482 

.2813 

. 1957 

. 8043 

18 

12 

52 

43 

. 2547 

. 7453 

.6988 

. 0163 

.2475 

.2816 

. 1975 

. 8025 

17 

8 

56 

44 

. 2570 

. 7430 

.5982 

. 0211 

.2467 

.2819 

. 1993 

. 8007 

16 

4 

35 

45 

.62592 

.37408 

1.5976 

.80258 

1.2460 

1.2822 

.22011 

.7 7 OSS 

15 

35 

4 

46 

. 2615 

• 4 

.5971 

. 0306 

.2452 

.2825 

. 2030 

. 7970 

14 

56 

8 

47 

. 2638 

. 7362 

.5965 

. 0354 

.2445 

.2828 

. 2048 

. 7952 

13 

52 

12 

48 

. 2660 

. 7340 

.5959 

. 0402 

.2437 

.2831 

. 2066 

. 7934 

12 

48 

16 

49 

. 2683 

. 7317 

.5963 

. 0450 

.2430 

.2834 

. 2084 

. 7915 

11 

44 

20 

50 

.62706 

.37294 

1.5947 

.80498 

1.2423 

1.2837 

.22103 

.77897 

10 

40 

24 

51 

. 2728 

. 7*72 

.5942 

. 0546 

.2415 

.2840 

. 2121 

. 7879 

9 

36 

28 

52 

. 2751 

. 7249 

.5936 

. 0594 

.2408 

.2843 

. 2139 

. 7861 

8 

32 

32 

53 

. 2774 

. 7226 

.5930 

. 0642 

.2400 

.2846 

. 2157 

. 7842 

7 

28 

36 

54 

. 2796 

. 7204 

.6924 

. 0690 

.2393 

.2849 

. 2176 

. 7824 

6 

24 

40 

55 

.62819 

.37181 

1.5919 

.80738 

1.2386 

1.2852 

.22194 

.77806 

5 

20 

44 

56 

. 2841 

. 7158 

.5913 

. 0786 

.2378 

.2855 

. 2212 

. 7788 

4 

16 

48 

57 

. 2861 

. 7136 

.5907 

. 0834 

.2371 

.2858 

. 2230 

. 7709 

3 

12 

52 

58 

. 2887 

. 7113 

.5901 

. 0882 

.2364 

.2861 

. 2249 

. 7751 

2 

8 

56 

59 

. 2909 

. 7090 

.6896 

. 0930 

.2356 

.2864 

. 2267 

. 7733 

1 

4 

JO 

60 

. 2932 

. 7068 

.5890 

. 0978 

.2349 

.2867 

. 2285 

. 7715 

0 

34 

. s. 

M 

Cosine. 

Vis.Sin. 

Secantc. 

Co tang. (Tangent. 

Coseo’ntl Vrs.Cos 

Sine. 

M 

M.S. 





































284 


Natural Lines. 


Oh 

39° Natural Trig 

ouometrleal Functions. 140° 

9 h 

M.S. 

M 

Sine. 

Vrs.Oos. 

Cosec’nte 

Tang. 

Cotang. 

Secaute. 

Vrs. Sin 

Cosiue. 

M 

M.S. 

30 

0 

.02932 

.37068 

1.5890 

.80978 

1.2349 

1.2867 

.22285 

.77715 

60 

34 

4 

1 

.-2955 

. 7045 

.6884 

. 1026 

.2342 

.2871 

. 2304 

. 7696 

59 

56 

8 

2 

. 2977 

. 7023 

.587y 

. 1075 

.2334 

.2874 

. 2322 

. 7678 

58 

52 

12 

3 

. 3U0i» 

. 7000 

.6873 

. 1123 

.2327 

.2877 

. 2340 

. 7660 

67 

48 

16 

4 

. 9022 

. 6977 

.6S67 

. 1171 

.2320 

.2880 

. 2359 

. 7641 

66 

44 

20 

5 

.G3045 

.36955 

1.5862 

.81219 

1.2312 

1.2'88 

.22377 

.770-23 

55 

40 

24 

6 

. 3007 

. 6932 

.6850 

. 1208 

.2305 

.2886 

. 2395 

. 7005 

54 

36 

28 

7 

. 31)90 

. 6910 

.5850 

. 1316 

.2297 

.2889 

. 2414 

. 7586 

53 

32 

32 

8 

. 3113 

. 6887 

.6845 

. 1364 

.2290 

.2892 

. 2432 

. 7508 

52 

28 

36 

9 

. 3135 

. 6865 

.5839 

. 1413 

.2283 

.2895 

. 2450 

. 7549 

51 

24 

40 

10 

.63158 

.30S42 

1.5833 

.81461 

1.2276 

1.2898 

.2 .'469 

.77531 

50 

20 

44 

11 

. 3180 

. 6820 

.6828 

. 1509 

.2208 

.2901 

. 2487 

. 7513 

49 

16 

48 

12 

. 3203 

. 6797 

.5822 

. 1558 

.2201 

.2904 

. 2505 

. 7494 

48 

12 

52 

13 

. 3225 

. 6774 

.5810 

. 1606 

.2254 

.2907 

. 2524 

. 7476 

47 

8 

56 

14 

. 3218 

. 6752 

•5sll 

. 1655 

.2247 

.2910 

. 2542 

. 7458 

46 

4 

37 

15 

.03270 

.36729 

1.5805 

.81703 

1.2239 

1.2913 

.22561 

.7743*9 

45 

33 

4 

16 

. 3293 

. 6707 

.5799 

. 1752 

.2232 

.2916 

. 2579 

. 7421 

44 

56 

8 

.17 

. 3315 

. 66S4 

.5794 

. 180i» 

.2225 

.2919 

. 2597 

. 7402 

43 

52 

12 

18 

. 3338 

. 6662 

•57 v 8 

. 1849 

.2218 

.2922 

. 2616 

. 7:384 

42 

48 

16 

19 

. 3360 

6639 

.5783 

. 1898 

.2210 

.2926 

. 26:14 

. 7365 

41 

44 

20 

20 

.63383' 

.30617 

1.5777 

.81946 

1.2203 

1.2929 

.22653 

.77347 

40 

40 

24 

21 

. 3405 

. 6594 

.5771 

. 1995 

.2190 

.2932 

. 2671 

. 7329 

39 

36 

28 

22 

. 3428 

. 6572 

.5706 

. 2043 

.2189 

.2936 

. 2690 

. 7310 

38 

32 

32 

23 

. 3450 

. 0549 

.5760 

. 2092 

.2181 

.2938 

. 2708 

. 7292 

37 

28 

36 

24 

. 3473 

. G527 

•5755 

. 2141 

.2174 

.2941 

. 2727 

. 7273 

36 

24 

40 

25 

.63495 

.36504 

1.5749 

.82190 

1.2167 

1.2944 

.22745 

.77255 

35 

20 

44 

26 

. 3518 

. 6482 

.6743 

. 2238 

.2160 

.2947 

. 2763 

. 7236 

34 

16 

48 

27 

. 3510 

. 6459 

•5738 

. 2287 

.2152 

.2950 

. 2782 

. 7218 

33 

12 

52 

28 

. 3503 

. 6437 

.5732 

. 2336 

.2145 

.2953 

. 2800 

. 7199 

32 

8 

56 

29 

• 3ob5 

. 6415 

.5727 

. 2385 

.213S 

.2956 

. 2819 

. 7181 

31 

4 

38 

30 

.63608 

.36392 

1.5721 

.82434 

1.2131 

1.2960 

.22837 

.77L62 

30 

33 

4 

31 

. 3630 

. 6370 

.5716 

. 2482 

.2124 

.2983 

. 2856 

. 7144 

29 

56 

8 

32 

. 3053 

. 6347 

.5710 

. 2531 

.2117 

.2906 

. 2874 

. 7125 

28 

52 

12 

33 

. 3675 

. 6325 

.5705 

. 2580 

.2109 

.2909 

. 2893 

. 7107 

27 

48 

16 

34 

. 3697 

. 6302 

.5699 

. 2629 

.2102 

.2972 

. 2912 

. 7088 

26 

44 

20 

35 

.63720 

.30280 

1.5694 

.82678 

1.2095 

1.2975 

.22930 

.77070 

25 

40 

24 

36 

. 3742 

. 6258 

.5088 

. 2727 

.2088 

.2978 

. 2949 

. 7061 

24 

36 

28 

37 

. 3765 

. 6235 

.6683 

. 2776 

.2081 

.2981 

. 2967 

. 7033 

23 

32 

32 

38 

. 3787 

. 6213 

.6677 

. 2825 

.2074 

.2985 

. 2986 

. 7014 

22 

28 

36 

39 

. 3810 

. 6190 

.5672 

. 2874 

.2066 

.2988 

. 3004 

. 6996 

21 

24 

40 

40 

.63832 

.36168 

1.5GCG 

.82923 

1.2059 

1.2991 

.23023 

.76977 

20 

20 

44 

41 

. 3854 

. 6146 

.5601 

. 2972 

.2052 

.2994 

. 3041 

. 6958 

19 

16 

48 

42 

. 3877 

. 0123 

.6665 

. 3022 

.2045 

.2997 

. 3060 

. 6940 

18 

1-2 

52 

43 

. 3899 

. 6101 

.5650 

. 3071 

.2038 

.3000 

. 3079 

. 6921 

17 

8 

56 

44 

. 3921 

. 6078 

.5644 

. 312U 

.2031 

.3003 

. 3097 

. 0903 

16 

4 

39 

45 

.63944 

.36056 

1.5639 

.83169 

1.2024 

1.3006 

.23110 

.76884 

15 

31 

4 

46 

. 3966 

. 0034 

.5633 

. 3218 

.2010 

.3010 

. 3134 

. 6865 

14 

56 

8 

47 

. 3989 

. 6011 

.6628 

. 32H7 

.2009 

.3013 

. 3153 

. 6847 

13 

52 

12 

48 

. 4011 

. 5989 

.5622 

. 3317 

.2002 

.3016 

. 3172 

. 6828 

12 

48 

16 

49 

. 4033 

. 5967 

.5617 

. 3366 

.1995 

.3019 

. 3190 

6810 

11 

44 

20 

50 

.64056 

.35944 

1.56 LI 

.83415 

1.1988 

1.3022 

.23209 

.76791 

10 

40 

24 

51 

. 4078 

. 6922 

.5606 

. 3465 

.1981 

.3025 

. 3227 

. 6772 

9 

36 

2S 

62 

. 4100 

. 5900 

.5600 

. 3514 

.1974 

.3029 

. 3246 

. 6754 

8 

32 

32 

53 

. 4123 

. 5877 

.5595 

. 3563 

.1967 

.3032 

. 3266 

. 6735 

7 

28 

36 

64 

. 4145 

. 5855 

.6590 

. 3613 

.1960 

.3035 

. 3283 

. 6716 

6 

24 

40 

55 

.64167 

.35833 

1.5584 

.83662 

1.1953 

1.3038 

.23302 

.76698 

5 

20 

44 

56 

. 4189 

. 5810 

.6579 

. 3712 

.1940 

.3041 

. 3321 

. 6679 

4 

16 

48 

57 

. 4212 

. 578S 

.5573 

. 3761 

.1939 

.3044 

. 3339 

. 6660 

3 

12 

52 

58 

. 4234 

. 5700 

.5568 

. 3811 

.1932 

.3048 

. 3358 

. 6642 

2 

8 

56 

59 

. 4256 

. 5743 

.6563 

. 3800 

.1924 

.3051 

. 3377 

. 6623 

1 

4 

10 

60 

. 4279 

. 5721 

.5557 

. 3910 

.1917 

.3054 

. 3395 

6604 

0 

30 

M. S. 

8 h 

M 

129‘ 

Cosiue. 

D 

Yrs.Sin. 

Secaute. 

Cotaug. 

Natn 

Tangent. 

iral. 

Coscc’ut 

Vrs.Uos 

Sine. 

M 

50° 

M.S. 

3 b 



















Natural Lines. 


285 


2 h 

o 

o 

Natural Trigonometrical Functions. 

139° 

9 h 

M.S 

M 

Sine. 

Vrs.Cos.'Cosce'nte 

Tang. 

Cotang. 

Secaute 

Vrs.Sin 

Cosine. 

M 

M.S. 

40 

0 

.64279 

.35721 

1.5557 

.83910 

1.1917 

1.3054 

.23:195 

.76604 

60 

20 

4 

1 

. 4301 

. 5699 

.5552 

. 3959 

.1910 

.3057 

. 3414 

. 6586 

59 

56 

8 

2 

. 4323 

. 5677 

.5546 

. 4009 

.1903 

.3060 

. 3433 

. 6567 

58 

52 

12 

3 

. 4345 

. 5654 

.5541 

. 4059 

.1896 

.3064 

. 3452 

. 6548 

57 

48 

10 

4 

. 4368 

. 5632 

.5536 

. 4108 

.1889 

,3( )67 

. 3470 

. 6530 

56 

44 

20 

5 

.64390 

.35610 

1.5530 

.84158 

1.1882 

1.3070 

.23489 

.76511 

55 

40 

24 

6 

. 4412 

. 5588 

.5525 

. 4208 

.1875 

.3073 

. 3508 

. 6492 

54 

36 

28 

7 

. 4435 

. 5565 

.5520 

. 4257 

.1868 

.3076 

. 3527 

. 6473 

53 

32 

82 

8 

. 4457 

. 5543 

.5514 

. 4307 

.1861 

.3080 

. 3545 

. 6455 

52 

28 

3G 

9 

. 4479 

. 5521 

.5509 

. 4357 

.1854 

.3083 

. 3564 

. 6436 

51 

24 

40 

10 

.64501 

.35499 

1.5503 

.84407 

1.1847 

1.3086 

.23583 

.76417 

50 

20 

44 

11 

. 4523 

. 5476 

.5498 

. 4457 

.1840 

.3089 

. 3602 

. 6398 

49 

16 

48 

12 

. 4546 

. 5454 

.5493 

. 4506 

.1833 

21092 

. 3620 

. 6380 

48 

12 

52 

13 

. 4568 

. 5432 

.5487 

. 4556 

.1826 

.3096 

. 3639 

. 6361 

47 

8 

5G 

14 

. 4590 

. 5410 

.5482 

. 4606 

.1819 

.3099 

. 3658 

. 6342 

46 

4 

41 

15 

.64612 

.35388 

1.5477 

.84656 

1.1812 

1.3102 

.23677 

.76323 

45 

19 

4 

16 

. 4635 

. 5365 

.5471 

. 4706 

.1805 

.3105 

. 3695 

. 6304 

44 

56 

8 

17 

. 4657 

. 5343 

.5466 

. 4756 

.1798 

.3109 

. 3714 

. 6286 

43 

52 

12 

18 

. 4679 

. 5321 

.5461 

. 4806 

.1791 

.3112 

. 3733 

. 6267 

42 

48 

1G 

19 

. 4701 

. 5299 

.5456 

. 4856 

.1785 

.3115 

. 3752 

. 6248 

41 

44 

20 

20 

.64723 

.35277 

1.5450 

.84906 

1.1778 

1.3118 

.23771 

.76229 

40 

40 

24 

21 

. 4745 

. 5254 

.5445 

. 4956 

.1771 

.3121 

. 3790 

. 6210 

39 

30 

28 

22 

. 4768 

. 5232 

.5440 

.. 5006 

.1764 

.3125 

. 3808 

. 6191 

38 

32 

32 

23 

. 4790 

. 5210 

.5434 

. 5056 

.1757 

.3128 

. 3827 

. 6173 

37 

28 

36 

24 

. 4812 

. 6188 

.5429 

. 5107 

.1750 

.3131 

. 3846 

. 6154 

36 

24 

40 

25 

.64834 

.35166 

1.5424 

.85157 

1.1743 

1.3134 

.23865 

.76135 

35 

20 

44 

26 

. 4856 

. 5144 

.5419 

. 5 207 

.1736 

.3138 

. 3884 

, 6116 

34 

16 

48 

27 

. 4878 

. 5121 

.5413 

. 5257 

.1729 

.3141 

. 3903 

. 6097 

33 

12 

52 

28 

. 4900 

. 5099 

.5408 

. 5307 

.1722 

.3144 

. 3922 

. 6078 

32 

8 

56 

29 

. 4923 

. 5077 

.5403 

. 5358 

.1715 

.3148 

. 3940 

. G059 

31 

4 

43 

30 

.64945 

.35055 

1.5398 

.85408 

1.1708 

1.3151 

.23959 

.76041 

30 

18 

4 

31 

. 4967 

. 5033 

.5392 

. 5458 

.1702 

.3154 

. 3978 

. 6022 

29 

56 

8 

32 

. 4989 

. 5011 

.5387 

. 5509 

.1695 

.3157 

. 3997 

. 6003 

28 

52 

12 

33 

. 5011 

. 4989 

.5382 

. 5059 

.1688 

.3161 

. 4016 

. 5984 

27 

48 

16 

31 

. 5033 

. 4967 

.5377 

. 5609 

.1681 

.3164 

. 4035 

. 5965 

26 

44 

20 

35 

.65055 

.34945 

1.5371 

.85660 

1.1674 

1.3167 

.24054 

.75946 

25 

40 

24 

36 

. 5i)77 

. 4922 

. .5366 

. 5710 

.1667 

.3170 

. 4073 

. 5927 

24 

36 

28 

37 

. 5099 

. 4900 

.5361 

. 5761 

.1660 

.3174 

. 4092 

. 5908 

23 

32 

32 

38 

. 5121 

. 4878 

.5356 

. 5811 

.1653 

.3177 

. 4111 

. 58.89 

22 

28 

36 

39 

. 5144 

. 4856 

.5351 

. 5S62 

.1647 

.3180 

. 4130 

. 5870 

21 

21 

40 

40 

.65166 

.34834 

1.5345 

.85912 

1.1640 

1.3184 

.24149 

.75851 

20 

20 

44 

41 

. 5188 

. 4812 

.0346 

. 5963 

.1633 

.3187 

. 4168 

. 5832 

19 

16 

48 

42 

. 5210 

. 4790 

.5335 

. 6013 

.1626 

.3190 

. 4186 

. 5813 

18 

12 

52 

43 

. 5232 

. 4768 

.5330 

. 6064 

.1619 

.3193 

. 4205 

. 5794 

17 

8 

56 

44 

. 5254 

. 4746 

.5325 

. 6115 

.1612 

.3197 

. 4224 

. 5775 

16 

4 

43 

45 

.65276 

.34724 

1.5319 

.86165 

1.1605 

1.3200 

.24243 

.75756 

15 

17 

4 

46 

. 529S 

. 4702 

.5314 

. 6216 

.1599 

.3203 

. 4262 

. 5737 

14 

56 

8 

47 

. 5320 

. 4680 

.53 U9 

. 6267 

.1592 

.3207 

. 4281 

. 571S 

13 

52 

12 

48 

. 5342 

. 4658 

.5304 

. 6318 

.1585 

.3210 

. 4300 

. 5699 

12 

48 

16 

49 

. 5364 

. 4636 

.5299 

. 6368 

.1578 

.3213 

. 4319 

. 5680 

11 

44 

20 

50 

.65386 

.34614 

1.5294 

.86419 

1.1571 

1.3217 

.24338 

.75661 

10 

40 

24 

51 

. 5108 

. 4592 

.5289 

. 6470 

.1565 

.3220 

. 4357 

. 5042 

9 

36 

28 

52 

. 5430 

. 4570 

.5283 

. 6521 

.1558 

.3223 

. 4376 

. 5623 

8 

32 

32 

53 

. 5452 

. 4548 

.5278 

. 6572 

.1551 

.3227 

. 4396 

. 5604 

7 

28 

36 

54 

. 5474 

. 4526 

.5273 

. 6623 

.1544 

.3230 

. 4415 

. 5585 

6 

24 

40 

55 

.65496 

.34504 

1.5268 

.86674 

1.1537 

1.3233 

.24434 

.75566 

5 

20 

44 

56 

. 5518 

. 4482 

.5263 

. 6725 

.1531 

.3237 

. 4453 

. 5547 

4 

16 

48 

57 

. 5540 

. 4460 

.5258 

. 6775 

.1524 

.3240 

. 4472 

. 5528 

3 

12 

52 

58 

. 5562 

. 4438 

.5253 

. 6826 

.1517 

.3243 

. 4491. 

. 5509 

2 

8 

56 

59 

. 5584 

. 4416 

.5248 

. 6878 

.1510 

.3247 

. 4510 

. 5490 

1 

4 

44 

60 

. 56o6 

. 4394 

.5242 

. 6929 

.1504 

.3250 

. 4529 

. 5471 

0 

10 

M. 8. 

8 b 

M 

130° 

Cosine. 

Vrs.Siu. 

Secuute. 

Cotaug.lTangeut. 

Natural. 

3oseo’nt 

Vrs.Cos 

Sine. 

M 

49° 

M.S. 

3 b 




























286 


Natural Lines. 


2 h 

41° 

Natural Trig 

ouometrica] 

Functions. 

138° 

9 h 

M.S. 

M 

Sine. 

Vrs.Cos. 

Cosec'nte 

Tang. 

Cotang. 

Seoante. 

Vrs. Sin 

Cosine. 

M 

M.S. 

44 

0 

.65606 

.34394 

1.5242 

.86929 

1.1504 

1.3250 

.24529 

.75471 

60 

10 

4 

1 

. 6628 

. 4372 

.5237 

. 6980 

.1497 

.3253 

. 4548 

. 5452 

59 

56 

8 

2 

. 5650 

. 4350 

.5232 

. 7031 

.1490 

.3257 

. 4567 

. 5433 

58 

52 

12 

3 

. 5672 

. 4328 

.5227 

. 7082 

.1483 

.3260 

. 4586 

. 5414 

57 

48 

16 

4 

. 5694 

. 4306 

.5222 

. 7133 

.1477 

.3263 

. 4605 

. 5394 

56 

44 

20 

5 

.65716 

.34284 

1.5217 

.87184 

1.1470 

1.3267 

.24624 

.75376 

55 

40 

24 

6 

• 5 1 ' > i 

. 4262 

.5212 

. 7235 

.1463 

.3270 

. 4044 

. 5356 

54 

36 

28 

7 

. 5759 

. 4240 

.5207 

. 7287 

.1456 

.3274 

. 4668 

. 6337 

63 

32 

32 

8 

. 5781 

. 4219 

.5202 

. 7338 

.1450 

.3277 

. 4682 

. 5318 

52 

28 

36 

9 

. 5803 

. 4197 

.5197 

. 7389 

.1443 

.3280 

. 4701 

. 5299 

61 

24 

40 

10 

.65825 

.34175 

1.5192 

.87441 

1.1436 

1.3284 

.24720 

.75280 

50 

20 

44 

11 

. 5S47 

. 4153 

.5187 

. 7492 

.1430 

.3287 

. 4739 

. 5261 

49 

16 

48 

12 

. 5869 

. 4131 

.5182 

. 7543 

.1423 

.3290 

. 4758 

. 5241 

48 

12 

52 

13 

. 5891 

. 4109 

.5177 

. 7595 

.1416 

.3294 

. 4778 

. 5222 

47 

8 

56 

14 

. 5913 

. 4087 

.5171 

. 7646 

.1409 

.3297 

. 4797 

. 5203 

46 

4 

43 

15 

.65934 

.34065 

1.5166 

.87698 

1.1403 

1.3301 

.24816 

.75184 

45 

15 

4 

16 

. 5956 

. 4043 

.5161 

. 7749 

.1396 

.3304 

. 4835 

. 5165 

44 

50 

8 

17 

. 5978 

. 4022 

.5156 

. 7801 

.1389 

.3307 

. 4854 

. 5146 

43 

52 

12 

18 

. 6000 

. 4000 

.5151 

. 7852 

.1383 

.3311 

. 4878 

. 5126 

42 

48 

16 

19 

. 6022 

. 3978 

.5146 

. 7904 

.1376 

.3314 

. 4893 

. 5107 

41 

44 

20 

20 

.66044 

.33956 

1.5141 

.87 955 

1.1369 

1.3318 

.24912 

.75088 

40 

40 

24 

21 

. 6066 

. 3934 

.5136 

. 8007 

.1363 

.3321 

. 4931 

. 5069 

39 

36 

28 

22 

. 6087 

. 3912 

.5131 

. 8058 

.1356 

.3324 

. 4950 

. 5049 

38 

32 

32 

23 

. 6109 

. 3891 

.5126 

. 8110 

.1349 

.3328 

. 4970 

. 5030 

37 

28 

36 

24 

. 6131 

. 3869 

.5121 

. 8162 

.1343 

.3331 

. 4*89 

. 5011 

36 

24 

40 

25 

.66153 

.33847 

1.5116 

.88213 

1.1336 

1.3335 

.25008 

.74992 

35 

20 

44 

26 

. 6175 

. 3825 

.5111 

. 8265 

.1329 

.3338 

. 5027 

. 4973 

34 

16 

48 

27 

. Cl 97 

. 3803 

.5106 

. 8317 

.1323 

.3342 

. 5047 

. 4958 

33 

12 

52 

28 

. 6218 

. 3781 

.5101 

. 8809 

.1316 

.3345 

. 5066 

. 4934 

32 

8 

56 

29 

. 6240 

. 3760 

.5096 

. 8421 

.1309 

.3348 

. 5085 

. 4915 

31 

4 

40 

30 

.66262 

.33738 

1.5092 

.88472 

1.1303 

1.3352 

.25104 

.74895 

30 

14 

4 

31 

. 6284 

. 3716 

.5087 

. 8524 

.1296 

.3355 

. 5124 

. 4876 

29 

66 

8 

32 

. 6305 

. 3691 

.5082 

. 8576 

.1290 

.3359 

. 5143 

. 4857 

28 

52 

12 

33 

. 6327 

. 3673 

.5077 

. 8628 

.1283 

.3362 

. 5162 

. 48ii8 

27 

48 

16 

34 

. 6349 

. 3651 

.5072 

. 8680 

.1276 

.3366 

. 5181 

. 4818 

26 

44 

20 

35 

.66371 

.33629 

1.5067 

.88732 

1.1270 

1.3309 

.25201 

.74799 

25 

40 

24 

36 

. 6393 

. 3607 

.5062 

. 8784 

.1263 

.3372 

. 5220 

. 4780 

24 

30 

28 

37 

. 6414 

. 3586 

.5057 

. 8836 

.1257 

.3376 

. 5239 

. 4760 

23 

32 

32 

38 

. 6436 

. 3564 

.5052 

. 8888 

.1250 

.3379 

. 5259 

. 4741 

22 

28 

36 

39 

. 6458 

. 3542 

.5047 

. 8940 

.1243 

.3383 

. 5278 

. 4722 

21 

24 

40 

40 

.66479 

.53520 

1.5042 

.88992 

1.1237 

1.3386 

.25297 

.74702 

20 

20 

44 

41 

. 6501 

. 3499 

.5037 

. 9044 

.1230 

.3390 

. 5317 

. 4683 

19 

16 

48 

42 

. 6523 

. 3477 

.5032 

. 9097 

.1224 

.3393 

. 5336 

. 4664 

18 

12 

62 

43 

. 6545 

. 3455 

.6027 

. 9149 

.1217 

3397 

. 5355 

. 46.44 

17 

8 

56 

44 

. 6566 

. 3133 

.5022 

. 9201 

.1211 

3400 

. 5375 

. 4625 

16 

4 

47 

45 

.66588 

.33112 

1.5018 

.89253 

1.1204 

1.3404 

.25394 

.74606 

15 

i:i 

4 

46 

. 6610 

. 3890 

.5013 

. 9306 

.1197 

.3407 

. 5414 

. 4586 

14 

56 

8 

47 

. 6631 

. 3868 

.5008 

. 9358 

.1191 

.3411 

. 5433 

. 4567 

13 

52 

12 

48 

. 6663 

. 3347 

.5003 

. 9410 

.1184 

.3414 

. 5452 

. 4548 

12 

48 

16 

49 

. 6675 

. 3325 

.4998 

. 9463 

.1178 

.3418 

. 5472 

4528 

11 

44 

20 

50 

.66697 

.33303 

1.4993 

.89515 

1.1171 

1.3421 

.25491 

.74509 

10 

40 

24 

51 

. 6718 

. 3282 

.4988 

. 9567 

.1165 

.3425 

. 5510 

. 4489 

9 

36 

28 

52 

. 6740 

. 3260 

.4983 

. 9620 

.1158 

.3428 

. 5530 

. 4470 

8 

32 

32 

53 

. 6762 

. 3238 

.4979 

. 9672 

.1152 

.3132 

. 5549 

. 4450 

7 

2 S 

36 

54 

. 6783 

. 3217 

.4974 

. 9725 

.1145 

.3435 

.. 5509 

. 4431 

6 

24 

40 

55 

.66805 

.33195 

1.4969 

.89777 

1.1139 

1.3439 

.25588 

.74412 

5 

20 

44 

56 

. 6826 

. 8173 

.4964 

. 9830 

.1132 

.3442 

. 5608 

. 4392 

4 

16 

48 

57 

. 6848 

. 3152 

.4959 

. 9882 

.1126 

.3446 

. 5627 

. 4373 

3 

12 

52 

58 

. 6870 

. 3130 

.4954 

. 9935 

.1119 

.3419 

. 5647 

. 4353 

2 

8 

56 

59 

. 6891 

. 3108 

.4949 

. 9988 

.1113 

.3453 

. 5666 

. 4334 

1 

4 

48 

60 

. 6913 

. 3087 

.4945 

.90040 

.1106 

.8456 

. 5685 

. 4314 

0 


M.S. 

M 

Cosine. 

Vis.Sin. 

Sec&ute. 

Ootnng. 

Tangent. 

Cosecul 

Vis. Cos 

Sine. 

M 

M.S. 

8 h 

131 

J 



Natural. 




48° 

3 h 





























287 


Natural Lines. 


Oh 

Li 

42 

o 

Natural Trigonometrical Functions 

137° 

9 h 

M.S 

M 

Sine. 

Vrs. Cos 

Cosec'nte 

Tang. 

Cotang. 

Secaute. 

Vrs.Sin 

Cosine. 

M 

M.S. 

48 

0 

.66913 

.33087 

1.4945 

.90040 

1.1106 

123456 

.25685 

.74314 

60 

12 

4 

1 

. 6935 

. 3065 

.4940 

. 0093 

.IKK) 

23160 

. 5705 

. 4295 

59 

56 

8 

2 

. 6956 

. 3044 

.4935 

. 0146 

.1093 

23463 

. 5724 

. 4275 

58 

52 

12 

l 3 

. 6978 

. 3022 

.4930 

. 0198 

.1986 

23167 

. 5744 

. 4256 

57 

4s 

16 

4 

. 6999 

. 3000 

.4925 

. 0251 

.1080 

.3470 

. 5763 

. 4236 

56 

44 

20 

5 

.67021 

.32979 

1.4921 

.90304 

1.1074 

1.3174 

.25783 

.74217 

56 

46 

24 

6 

. 7043 

. 2957 

.4916 

. 0357 

.1U67 

.3477 

. 5802 

. 4197 

54 

36 

28 

7 

. 7064 

. 2936 

.4911 

. 0410 

.1061 

21481 

. 5822 

. 4178 

53 

32 

32 

8 

. 7086 

. 29 L4 

.4906 

. 0463 

.1054 

23485 

. 5841 

. 4158 

52 

28 

36 

9 

• 7107 

. 2S93 

.4901 

. 0515 

.1048 

.3488 

. 5861 

. 4139 

51 

24 

40 

10 

.67129 

.32871 

1.4897 

.90568 

1.1041 

1.3492 

.25880 

.74119 

50 

20 

44 

11 

. 7150 

. 2849 

.4892 

. 0621 

.10:35 

.3495 

. 5900 

. 4100 

49 

16 

48 

12 

. 7172 

. 2828 

.4887 

. 0674 

.1028 

23499 

. 5919 

. 4080 

48 

12 

52 

13 

. 7194 

. 2806 

.4882 

. 0727 

.1022 

23502 

. 5939 

. 4061 

47 

8 

56 

14 

; 7215 

. 2785 

.4877 

. 0780 

.1015 

.3506 

. 5959 

. 4041 

46 

4 

49 

15 

.67237 

.32763 

1.4873 

.90834 

1.1009 

1.3509 

.25978 

.74022 

45 

11 

4 

16 

. 7258 

. 2742 

.4868 

. 0887 

.1003 

23513 

. 5998 

. 4002 

44 

56 

8 

17 

. 7280 

. 2720 

.4863 

. 0940 

.0996 

.3517 

. 6017 

. 3983 

43 

52 

12 

IS 

. 7301 

. 2699 

.4858 

. 0993 

.0990 

.3520 

. 6037 

. 3963 

42 

48 

1G 

19 

. 7323 

. 2677 

.4854 

. 1046 

.0983 

23524 

. 6056 

. 3943 

41 

44 

20 

20 

.67344 

.32656 

1.4849 

.91099 

1.0977 

1.3527 

.26076 

.73924 

40 

40 

24 

21 

. 7366 

. 2634 

.4844 

. 1153 

.0971 

.353 L 

. 0096 

. 3901 

39 

36 

28 

22 

. 7387 

. 2613 

.4839 

. 1206 

.0964 

.35:34 

. 6115 

. 3885 

88 

32 

32 

23 

. 7409 

. 2591 

.4835 

• 1259 

.0958 

.3538 

. 6135 

. 3865 

37 

28 

3G 

24 

. 7430 

. 2570 

.4830 

• 1312 

.0951 

23512 

. 6151 

. 3845 

36 

24 

40 

25 

.67452 

.32548 

1.4825 

.91366 

1.0945 

1.3545 

.26174 

.73826 

35 

20 

44 

26 

. 7473 

. 2527 

.4821 

• 1419 

.0939 

23549 

. 6194 

. 3806 

34 

16 

48 

27 

. 7495 

. 2505 

.4816 

• 1473 

.0932 

.3552 

. 6213 

• 3/ 8 < 

83 

12 

52 

28 

. 7516 

. 2484 

.4811 

. 1526 

.0926 

23556 

. 6233 

. 3767 

32 

8 

5G 

29 

. 7537 

. 2462 

.4806 

• 1580 

.0919 

.3560 

. 6253 

. 3747 

31 

4 

50 

30 

•67559 

.32411 

1.4802 

-91633 

1.0913 

1.3563 

.26272 

.73728 

30 

10 

4 

31 

. 7580 

. 2419 

.4797 

. 1687 

.0907 

.3567 

. 6292 

. 37uS 

29 

56 

8 

32 

. 7002 

. 2398 

.4792 

• 1740 

.0900 

.3571 

. 6311 

. 3688 

28 

62 

12 

33 

. 7623 

. 2377 

.4788 

. 1794 

.0894 

.3574 

. 6331 

. 3669 

27 

48 

16 

34 

. 7645 

. 2355 

.4783 

. 1847 

.0888 

.3578 

. 6351 

. 3649 

26 

44 

20 

35 

.67066 

.32334 

1.4778 

.91901 

1.0881 

1.3581 

.26371 

.73629 

25 

40 

24 

36 

. 76SS 

. 2312 

.4774 

• 1955 

.0875 

.3585 

. 6390 

. 3610 

24 

'if> 

28 

37 

. 7709 

. 2291 

.4769 

• 2008 

.0S68 

.3589 

0110 

. &590 

23 

32 

32 

38 

. 7730 

. 2269 

.4764 

• 2062 

.0862 

.3592 

. 6430 

. 3570 

22 

28 

136 

39 

• 7752 

. 2218 

.4760 

. 2116 

.0856 

.3596 

. (5449 

. 3551 

21 

24 

40 

40 

.67773 

.32227 

1.4755 

•92170 

1.0849 

1.3600 

.26469 

.73531 

20 

20 

44 

41 

. 7794 

. 2205 

.4750 

. 2223 

.0813 

.3603 

. 6489 

. 3511 

19 

16 

48 

42 

. 7816 

. 2184 

.4746 

. 2277 

.0837 

.3607 

. 6508 

. 3491 

18 

12 

52 

43 

. 7837 

. 2163 

.4741 

. 2331 

.0830 

.3611 

. 6528 

. 3472 

17 

8 

56 

44 

. 7859 

. 2141 

.4736 

. 2385 

.0824 

.3614 

. 6548 

. 3452 

16 

4 

51 

45 

.67880 

.32120 

1.4732 

.92139 

1.0818 

1.3618 

.26568 

.73432 

15 

9 

4 

40 

. 7901 

. 2098 

.4727 

. 2493 

.0812 

23622 

. 6587 

. 3412 

14 

56 

8 

47 

. 7923 

. 2077 

.4723 

. 2547 

.0805 

.3625 

. 6607 

. 3393 

13 

52 

12 

48 

. 7944 

. 2056 

.4718 

. 2601 

.0799 

23629 

. 6627 

. 3373 

12 

48 

16 

49 

. 7965 

. 2034 

.4713 

. 2655 

.0793 

.3633 

. 6647 

. 3353 

11 

44 

20 

50 

.67987 

.32013 

1.4709 

.92709 

1.0786 

1.3636 

.26666 

.73333 

10 

40 

24 

51 

. 8008 

. 1992 

.4704 

. 2763 

.0780 

.3640 

. 6686 

. 3314 

9 

36 

28 

52 

. 8029 

. 1970 

.4099 

. 2817 

.0774 

.3644 

. 6706 

. 3294 

8 

32 

32 

53 

. 8051 

. 1949 

.4695 

. 2871 

.0767 

.3047 

. 6726 

. 3274 

7 

28 

3G 

54 

. 8072 

. 1928 

.4690 

. 2926 

.0761 

.3651 

. 6746 

. 3254 

6 

24 

40 

55 

.68093 

.31907 

1.4680 

.92980 

1.0755 

1.3655 

.26765 

.73231 

5 

20 

44 

56 

. 8115 

. 1885 

.4681 

. 3034 

.0749 

.3658 

. 6785 

. 3215 

4 

16 

48 

57 

. 8136 

. 1864 

.4676 

. 3088 

.0742 

.3662 

. 6805 

. 3195 

3 

12 

52 

58 

. 8157 

. 1843 

.4672 

. 3143 

.0736 

.3666 

. 0825 

. 3175 

2 

8 

56 

59 

. 8178 

. 1821 

.4667 

. 3197 

.0730 

.3669 

. 6845 

. 3155 

1 

4 

5r<i 

GO 

. 8200 

. 1800 

.4663 

. 3251 

.0724 

.3673 

. 6865 

. 3135 

0 

8 

M.S. 

M 

Cosine. 

Vrs. Sin. 

Secaute. 

Cotang 

rangcut. 

losce'nt.J 

Vrs. Cos 

Sine. 

M 

vi s 

8 “ 

QOO 



Natural. 




470 

3** 








































288 


Natural Lines. 


2 h 

43° Natural Trigonometrical Functions. 

136° 

M.S. 

M 

Sine. 

Vrs.Cos. 

Cosec'nte 

Tang. 

Cotang. 

Secante. 

Vrs.Siu 

Cosine. 

M 

54 

0 

.68200 

.3180(1 

1.4663 

.93251 

1.0724 

1.307-3 

.20805 

.73135 

60 

4 

1 

. 8221 

. 1779 

.4058 

. 3306 

.0717 

.3077 

. 0884 

. 3115 

59 

8 

2 

. 8242 

. 1758 

.4654 

. 3360 

.0711 

.3081 

. 0904 

. 3096 

58 

12 

3 

. 8204 

. 1730 

.4049 

. 3415 

.07**5 

.3684 

. 6924 

. 3070 

57 

1G 

4 

. 8285 

. 1715 

.4044 

. 3469 

.0099 

.3088 

. 0944 

. 3050 

50 

20 

6 

.68306 

.31694 

1.4640 

.93524 

1.0092 

1.3092 

.20904 

.73030 

55 

24 

6 

. 8327 

. 1073 

.4*435 

. 3578 

.0086 

.3095 

. 0984 

. 3010 

54 

28 

7 

. 8349 

. 1051 

.4631 

. 3633 

.0680 

.3099 

. 7004 

. 2996 

53 

32 

8 

. 8370 

. 1630 

.4620 

. 3087 

.0074 

.3703 

. 7023 

. 2970 

52 

3G 

9 

. 8891 

. 1009 

.4022 

. 3742 

.0007 

.3707 

. 7**43 

. 2956 

51 

40 

10 

.68412 

.3158S 

1.4617 

.93797 

1.000 L 

1.3710 

.27003 

.72937 

50 

44 

11 

. S433 

. 1506 

.4613 

. 3*51 

.0055 

.3714 

. 7083 

. 2917 

49 

48 

12 

. 8455 

. 1545 

.4608 

. 3906 

.0049 

.3718 

. 71**3 

. 2897 

48 

62 

13 

. 8476 

. 1624 

.4004 

. 3901 

.0013 

.3722 

. 7123 

. 2877 

47 

60 

14 

. 8497 

. 1503 

.4599 

. 4**16 

.0G3G 

.3725 

. 7143 

. 2857 

46 

53 

16 

.68518 

.31482 

1.4595 

.94071 

1.0030 

1.3729 

.27163 

.72S37 

45 

4 

10 

. 8539 

. 1460 

.4590 

. 4125 

.0024 

.3733 

. 7183 

. 2S17 

44 

8 

17 

. 8561 

. 1439 

.4586 

. 4180 

.0618 

.3737 

. 7203 

. 2797 

43 

12 

18 

. 8582 

. 1418 

.4581 

. 4235 

.0012 

.3740 

. 7223 

. 2777 

42 

1G 

19 

. 8603 

. 1397 

.4577 

. 4290 

.0005 

.3744 

. 7243 

. 2757 

41 

20 

20 

.G8624 

.31370 

1.4572 

.94345 

1.0599 

1.3748 

.27203 

.72737 

40 

24 

21 

. 8615 

. 1355 

.4568 

. 4-100 

.0593 

.3752 

. 7 2 S3 

. 2717 

39 

28 

22 

. 8666 

. 1333 

.4503 

. 4455 

.0587 

.3750 

. 7302 

. 2697 

38 

32 

23 

. 8688 

. 1312 

.4559 

. 4510 

.05S1 

.3759 

. 7322 

. 2077 

37 

30 

24 

. 8709 

. 1291 

.4554 

. 4505 

.0575 

.3763 

. 7342 

. 2057 

36 

40 

25 

.687 30 

.31271* 

1.4550 

.94020 

1.0508 

1.3767 

.27302 

.72637 

35 

44 

26 

. 8751 

. 1249 

.4545 

. 4075 

.0562 

.3771 

. 73S2 

. 2617 

34 

48 

27 

. 8772 

. 1228 

.4541 

. 4731 

.0556 

.3774 

. 74**2 

. 2597 

33 

62 

28 

. 8793 

. 1207 

.4530 

. 47*0 

.0550 

.3778 

. 7422 

. 2577 

32 

60 

29 

. 8814 

. 1186 

.4532 

. 4S41 

.0544 

.3782 

. 7442 

. 2557 

31 

54 

30 

.68835 

.31164 

1.4527 

.94890 

1.0538 

1.3786 

.27162 

.72537 

30 

4 

31 

. 8856 

. 1143 

.4523 

. 4952 

.0532 

.3790 

. 7482 

. 2517 

29 

8 

32 

. 8878 

. 1122 

.4518 

. 5007 

.0525 

.379 4 

. 7503 

. 2497 

28 

12 

33 

. 8809 

. 1101 

.4514 

. 5002 

.0519 

.3797 

. 7523 

. 2477 

27 

10 

34 

. 8920 

. 1080 

.4510 

. 5118 

.0513 

.3801 

. 7543 

. 2457 

26 

20 

35 

.6S941 

.31059 

1.4505 

.95173 

1.0507 

1.3805 

.27563 

.72437 

25 

24 

36 

. 8962 

. 1038 

.4501 

. 5229 

.0501 

.3869 

. 7583 

. 2417 

24 

28 

37 

. 8983 

. 1017 

.449*3 

. 5284 

.0495 

.3813 

. 7603 

. 2397 

23 

32 

.‘48 

. 9004 

. 0996 

.4492 

. 5340 

.0489 

.3810 

. 7023 

. 2377 

22 

30 

3!) 

. 9025 

. 0975 

.4487 

. 5395 

.0483 

.3820 

. 7043 

. 2357 

21 

40 

40 

.6904G 

.30954 

1.4483 

.95451 

1.0410 

1.3824 

.27003 

.72337 

20 

44 

41 

. 9007 

. 0933 

.4479 

. 5506. 

.0470 

.3828 

. 7083 

. 2317 

19 

48 

42 

. 9088 

. 0912 

.4474 

. 5502 

.0404 

.3832 

. 7703 

. 2297 

18 

62 

43 

. 9109 

. 0891 

.4470 

. 5018 

.0458 

.3860 

. 7723 

. 2277 

17 

60 

44 

. 9130 

. 0870 

.4465 

. 5073 

.0452 

.3839 

. 7743 

. 2256 

16 

55 

45 

.09151 

.30849 

1.4461 

.95729 

1.6440 

1.3843 

.27704 

.72236 

15 

4 

46 

. 9172 

. 0S‘2S 

.4157 

. 5785 

.0440 

.3 V 47 

. 7784 

. 2216 

14 

8 

47 

. 9193 

. 0807 

.4452 

. 5841 

.0434 

.3851 

. 7804 

. 2196 

13 

12 

48 

. 9214 

. 0786 

.4448 

. 5890 

.(*428 

.3855 

. 7824 

. 2176 

12 

10 

49 

. 9235 

. 0765 

.4443 

. 5952 

.0422 

.3859 

. 7844 

. 2156 

11 

20 

60 

.69256 

.30744 

1.4439 

.96008 

1.0410 

1.3803 

.27804 

.72130 

10 

24 

51 

. 9277 

. 0723 

.4435 

. 6064 

.0410 

.3807 

. 7884 

. 2115 

9 

28 

52 

. 9298 

. 0702 

.4430 

. 0120 

.0404 

.3870 

. 7904 

. 2095 

8 

32 

53 

. 9319 

. 0681 

.4426 

. 6170 

.0397 

.3874 

. 7925 

. 2075 

7 

36 

54 

. 9340 

. 0600 

.4422 

. 6232 

.0391 

.3878 

. 7945 

. 2055 

6 

40 

55 

.69361 

.30039 

1.4417 

.96288 

1.0385 

1.3S82 

.27965 

.72035 

5 

41 

56 

. 9382 

. 0618 

.4413 

. 6344 

.0379 

.3886 

. 7985 

. 2015 

4 

48 

57 

. 9403 

. 0597 

.4408 

. 6400 

.0373 

.3890 

. 8005 

. 1994 

3 

62 

58 

. 9424 

. 0570 

.4404 

. 6456 

.0307 

.3894 

. 8*126 

. 1974 

2 

60 

59 

. 9415 

. 0555 

.4400 

. 0513 

.0301 

.3898 

. 8046 

. 1954 

1 

5C> 

60 

. 9400 

0 05134 

.4395 

. 0509 

.0355 

.3902 

. 8000 

. 1934 

0 

M. S. 

M 

Cosine. 

Vrs.Siu. 

Sccuuie. 

Colallg. 

Tangent. 

Cosec’ut 

Vis, Cos 

Sine. 

M 

8 h 

133° 



Natural. 




46° 


i 

































Natural Lines 


289 


2 h 

44 

0 

Natural Trigonometrical Functions 

135° 

9 h 

M.S 

M 

Sine. 

Yrs.Cos 

Cosec'nte 

Tang. 

Cotang. 

Seen rite. 

Vrs.Sin 

Cosine. 

M 

M.S 

5G 

0 

•69466 

.30534 

1.4395 

.96569 

1.0355 

1 3902 

.28066 

.71934 

60 

4 

4 

1 

. 9487 

. 0513 

.43!) 1 

. 6625 

.0349 

.3905 

. 8086 

1914 

59 

56 

8 

2 

. 9508 

. 0492 

.4387 

. 6681 

.0343 

.3 109 

. 8106 

. 1893 

58 

62 

12 

3 

. 9528 

. 0471 

.4382 

. 6738 

.0337 

.3913 

. 8127 

. 1873 

57 

48 

16 

4 

. 9549 

. 0450 

.4378 

. 6794 

.0331 

.3917 

. 8147 

. 1853 

56 

44 

20 

5 

.69570 

.30430 

1.43 ( '4 

.96850 

1.0325 

1.3921 

.28167 

.71833 

55 

40 

24 

6 

. 9591 

. 0409 

.4370 

. 6907 

.0319 

.3925 

. 8187 

. 1813 

54 

36 

28 

7 

. 9612 

. 03S8 

.4365 

. 6963 

.0313 

.3929 

. 8208 

. 1792 

53 

32 

32 

8 

. 9633 

. 0367 

.4361 

. 7020 

.0307 

.3933 

. 8228 

. 1772 

52 

28 

30 

9 

. 9654 

. 0346 

.4357 

. 7076 

.0301 

.3937 

. 8248 

. 1752 

51 

24 

40 

10 

•69675 

.30325 

1.4352 

.97133 

1.0295 

1.3941 

.28268 

.71732 

50 

20 

44 

11 

. 9696 

. 0304 

.4348 

. 7189 

.0289 

.3945 

. 8289 

. 1711 

49 

16 

48 

12 

. 9716 

. 0283 

.4344 

. 7246 

.02«3 

.3949 

. 8309 

. 1691 

48 

12 

52 

13 

. 9737 

. 0263 

- .4339 

. 7302 

.0277 

.3953 

. 8329 

. 1671 

47 

8 

66 

14 

. 9758 

. 0242 

.4335 

• 7359 

.0271 

.3957 

. 8349 

. 1650 

46 

4 

57 

15 

.69779 

.30221 

1.4331 

.97416 

1.0265 

1.3960 

.28370 

.71630 

45 

3 

4 

16 

. 98oO 

. 0200 

.4327 

• 7472 

.0259 

.3964 

. 8390 

. 1610 

44 

56 

8 

17 

. 9821 

. 0179 

.4322 

• 7529 

.0253 

.3968 

. 8410 

. 1589 

43 

52 

12 

18 

. 9841 

. 0158 

.4318 

• 7586 

.0247 

.3972 

. 8431 

. 1569 

42 

48 

16 

19 

. 9862 

. 0138 

.4314 

• 7643 

.0241 

.3976 

. 8451 

. 1549 

41 

44 

20 

20 

.69SS3 

.30117 

1.4310 

•97699 

1.0235 

1.3980 

.28471 

.71529 

40 

40 

24 

21 

. 9904 

. 0096 

.4305 

• 7756 

.0229 

.3984 

. 8492 

. 1608 

39 

36 

28 

22 

. 9925 

. 0075 

.4301 

. 7813 

.0223 

.3988 

. 8512 

. 148S 

38 

32 

32 

23 

. 9945 

. 0054 

.4297 

• 7870 

.Ol'IS 

.3992 

. 8532 

. 1468 

37 

28 

36 

24 

. 9966 

. 0034 

.4292 

• 7927 

.0212 

.3996 

. 8553 

. 1147 

36 

24 

40 

25 

.69987 

.30013 

1.4288 

•97984 

1.0206 

1.4000 

.28573 

.71427 

35 

20 

44 

26 

•70068 

.23992 

.4284 

• 8041 

.0200 

.4004 

. 8593 

. 1406 

34 

16 

48 

27 

. 0029 

. 9971 

.4280 

. 8098 

.0194 

.4008 

. 8614 

. 1386 

o3 

12 

52 

28 

• 0049 

. 9950 

.4276 

. 8155 

.0188 

.4012 

. 8634 

. 1366 

32 

8 

56 

29 

• 0070 

. 9930 

.4271 

• 8212 

.0182 

.4016 

. 8654 

. 1345 

31 

4 

58 

30 

.70091 

.29909 

1.4267 

•9S270 

1.0176 

1.4020 

.28675 

.71325 

30 

a 

4 

31 

. 0112 

. 9888 

.4263 

• 8327 

.0170 

.4024 

. 8695 

. 1305 

29 

56 

8 

32 

• 0132 

. 9867 

.4259 

• 8384 

-0164 

.4028 

. 8716 

. 1284 

28 

52 

12 

33 

. 0153 

. 9847 

•4254 

. S441 

.0158 

.4032 

. 8736 

. 1264 

27 

48 

16 

34 

• 0174 

. 9826 

.4250 

. 8499 

.0152 

.4036 

. 8756 

. 1243 

26 

41 

20 

35 

•70194 

.29805 

1.4246 

•98556 

1.0146 

1.4040 

.28777 

.71223 

25 

40 

24 

36 

• 0215 

. 9786 

.4242 

• 8613 

.0141 

.4044 

. 8797 

. 1203 

24 

36 

28 

37 

. 0236 

. 9764 

.4238 

• 8671 

.0135 

.4048 

8818 

. 1182 

23 

32 

32 

38 

. 0257 

. 9743 

.4233 

. 8728 

.0129 

.4052 

. 8S38 

. 1162 

22 

28 

36 

39 

. 0277 

. 9722 

.4229 

. 8786 

.0123 

1.4056 

. 8859 

. 1141 

21 

24 

40 

40 

•70298 

.29702 

1.4225 

•98843 

1.0117' 

.4060 

.2S879 

.71121 

20 

20 

44 

41 

. 0319 

. 9681 

.4221 

. 8901 

.0111 

.4065 

. 8899 

. 1100 

19 

16 

48 

42 

. 0339 

. 9660 

.4217 

• 8958 

.0105 

.4069 

. 8920 

. 10S0 

18 

12 

52 

43 

. 0360 

. 9640 

.4212 

• 9016 

.00. 9 

.4073 

. 8940 

. 1059 

17 

8 

66 

44 

. 0381 

. 9619 

.4208 

. 9073 

.0093 

.4077 

. 8961 

. 1039 

16 

4 

59 

45 

.70401 

.29598 

1.4204 

•99131 

1.0088 

1.4031 

.28981 

.71018 

15 

1 

4 

46 

. 0422 

. 9578 

.4200 

. 9189 

.0082 

.4085 

. 9002 

. 0998 

14 

56 

8 

47 

. 0443 

. 9557 

.4196 

. 9216 

.0(76 

.4089 

. 9022 

. 0977 

13 

52 

12 

48 

. 0463 

. 9536 

.4192 

. 9304 

.0070 

.4093 

. 9043 

. 0957 

12 

4S 

16 

49 

. 0484 

. 9516 

.4188 

. 9362 

.0064 

.4097 

. 9063 

. 0936 

11 

44 

20 

60 

.70505 

.29495 

1.4183 

.99420 

1.0058 

1.4101 

.29084 

.70916 

10 

40 

24 

51 

. 0525 

. 9475 

.4179 

. 9478 

.0052 

.4105 

. 9104 

. 0895 

9 

36 

28 

52 

. 0546 

. 9454 

.4175 

. 9556 

.0047 

.4109 

. 9125 

. 0S75 

8 

32 

32 

53 

. 0566 

. 9433 

.4171 

. 9593 

.0041 

.4113 

. 9115 

. 0854 

7 

28 

36 

64 

. 0587 

. 9413 

.4167 

. 9651 

.0035 

.4117 

. 9166 

. 0834 

6 

24 

40 

55 

.70608 

.29392 

1.4163 

.99709 

1.0029 

1.4122 

.29186 

.70813 

6 

20 

44 

56 

. 0628 

. 9372 

.4169 

. 9767 

.0023 

.4126 

. 9207 

. 0793 

4 

16 

48 

57 

. 0649 

. 9351 

.4154 

. 9826 

.(017 

.4130 

. 9223 

. 0772 

3 

12 

62 

58 

. 0669 

. 9330 

.4150 

. 9884 

.0012 

.4134 

. 9248 

. 0752 

2 

8 

66 

59 

. 0690 

. 9310 

.4146 

. 9942 

.0006 

.4138 

. 9269 

. 0731 

1 

4 

GO 

60 

. 0711 

. 9289 

.4142 

1.0000 

1.0000 

.4142 

. 9289 

. 0711 

0 

O 

M.S. 

M 

Coslue. 

Vrs.Sin. 

Secante. 

Cotaug. 

Tangent. 

Coscc'nt.J 

Vrs.Cos 

Siue. 

M 

M S. 

8 h 

134° 



Natural. 



45° 



19 




























290 


Table of Hyperbolic Logarithms. 


N. 

Logarithm. 

nr. 

Logarithm. 

K. 

Logarithm, 

ir. 

Logarithm. 

1.01 

.0099503 

1.65 

J5 007752 

2,29 

.8285518 

2.93 

1.0750024 

1.02 

.0198020 

1.66 

.5068175 

2.30 

.8329091 

2.94 

1.0784095 

L03 

.0295588 

1.67 

43128236 

2.31 

,8372475 

2.95 

1,0818051 

1.04 

.0392207 

1.68 

/> 187937 

2212 

,8415671 

2.96 

1.0851892 

1.05 

.0487902 

1419 

4324 7285 

2213 

.8458682 

2.97 

1,0885619 

LOO 

.0582089 

1.70 

,5306282 

2.34 

,8501509 

2.98 

1.0919233 

1.07 

.0070586 

1.71 

.5304933 

2,35 

.8544153 

2.99 

1.0952733 

1.08 

.0769010 

1.72 

.5423242 

2.36 

.8586616 

3.00 

1.0986123 

1.09 

.0861777 

1,73 

.5481214 

2217 

,8628899 

3.01 

1.10194 (X) 

1.10 

.0953102 

1.74 

45538851 

2218 

.8671004 

3.02 

1.1052568 

1.11 

.1043000 

1.75 

.559(5157 

2219 

.8712933 

3.03 

1.1085626 

LI 2 

.1133287 

1.76 

43653138 

2.40 

.8754687 

3.04 

1,1118575 

1.13 

.1222170 

1.77 

.5709795 

2.41 

,8796267 

3.05 

1.1151415 

1.14 

.1310283 

1.78 

45766133 

2.42 

,8837675 

3.06 

1.1184149 

1.15 

.1397619 

1.79 

43822156 

2.43 

.8878912 

3.07 

1.1216775 

1.10 

.1484200 

1.80 

45877866 

2.44 

,8919980 

3.08 

1.1249295 

1.17 

.1570037 

1.81 

45933268 

2.45 

.8960880 

3.09 

1.1281710 

1.18 

.1655144 

1.82 

45988365 

2.46 

.9001613 

3.10 

1.1314021 

1.19 

.1739533 

1.83 

.6043159 

2.47 

.9012181 

3.11 

1.1346227 

1.20 

.1823215 

1.84 

,6097655 

2.48 

.9082585 

3.12 

1.1378330 

1.21 

.1906203 

1.85 

.6151856 

2.49 

.9122826 

3.13 

1.1410330 

1.22 

.1988508 

1,86 

.6205764 

2.50 

.9162907 

3.14 

1.1442227 

1.23 

.2070141 

1.87 

.6259384 

2.51 

.9202827 

3.15 

1.1474024 

1.24 

.2151113 

1.88 

.6312717 

2.52 

.9242589 

3,16 

1.1505720 

1.25 

.2231436 

1.89 

.6365768 

2.53 

.9282193 

3.17 

1.1537315 

1.20 

.2311117 

1.90 

.6418538 

2.54 

.9321640 

3.18 

1.1568811 

1.27 

.23901G9 

1.91 

.6471032 

22)5 

.9360933 

3.19 

1.1600209 

1.28 

.2468600 

1.92 

.6523251 

22>G 

.9400072 

3.20 

1.1631508 

1.29 

.2540422 

1.93 

.6575200 

2.57 

.9439058 

3.21 

1.1662709 

1.30 

.2623042 

1.94 

.6626879 

2.58 

.9477893 

3.22 

1.1693813 

1.31 

.2700271 

1.95 

.6678293 

2.59 

.9516578 

3.23 

LI 724821 

1.32 

.2776317 

1.96 

.6729444 

2.60 

.9555114 

3.24 

1.1755733 

1.33 

.2851789 

1.97 

.6780335 

2.61 

.9593502 

3.25 

1.1786549 

1.34 

.2926696 

1.98 

.6830968 

2.62 

.9631743 

3.26 

1.1817271 

1.35 

.3001045 

1.99 

.6881346 

2.63 

.9669838 

3.27 

1.1847899 

1.30 

.3074846 

2.00 

.6931472 

2.64 

.9707789 

3.28 

1.1878434 

1.37 

.3148107 

2.01 

.6981347 

2.65 

.9745596 

3.29 

1.1908875 

1.38 

.3220834 

2.02 

.7030974 

2.6G 

.9783261 

3.30 

1.1939224 

1.39 

.3293037 

2.03 

.7080357 

2.67 

.9820784 

3.31 

1.1969481 

L40 

.3364722 

2.04 

.7129497 

2.68 

.9858167 

3.32 

1.1999647 

1.41 

.34:15897 

2.05 

.7178397 

2.69 

.9895411 

3.33 

1.2029722 

1.42 

.3500568 

2.06 

.7227059 

2.70 

.9932517 

3.34 

1.2059707 

1.43 

.3576744 

2.07 

.7275485 

2.71 

.9969486 

3.35 

1.2089603 

1.44 

.3646431 

2.08 

.7323678 

2.72 

1.0006318 

3.36 

1.2119409 

1.45 

.37156:15 

2.09 

.7371640 

2.73 

1.0043015 

3.37 

1.2149127 

1.40 

.3784364 

2.10 

.7419373 

2.74 

1.0079579 

3.38 

1.2178757 

1.47 

.3852024 

2.11 

.7466879 

2.75 

1.0116008 

3.39 

1.2208299 

1.48 

.3920420 

2.12 

.7514160 

2.76 

1.0152306 

3.40 

1.2237754 

1.49 

.3987701 

2.13 

.7561219 

2.77 

1.0188473 

3.41 

1.2267122 

1.50 

.4054051 

2.14 

.7608058 

2.78 

1.0224509 

3.42 

1.2296405 

1.51 

.4121090 

2.15 

.7654678 

2.79 

1.0260415 

3.43 

1.2325605 

1.52 

.4187103 

2.16 

.7701082 

2.80 

1.0296194 

3.44 

1.2354714 

1.53 

.4252677 

2.17 

.7747271 

2.81 

1.0331844 

3.45 

1.2383742 

1.54 

.4317824 

2.18 

.7793248 

2.82 

1.0367368 

3.46 

1.2412685 

1.55 

.4382549 

2.19 

.7839015 

2.83 

1.0402766 

3.47 

1.2441545 

1.50 

.4440858 

2.20 

.7884573 

2.84 

1.0438040 

3.48 

1.2470322 

1.57 

.4510756 

2.21 

.7929925 

2.85 

1.0473189 

3.49 

1.2499017 

1.58 

.4574248 

2.22 

.7975071 

2.86 

1.050S216 

3.50 

1.2527629 

1.59 

.4037:140 

2.23 

.8020015 

2.87 

1.0543120 

3.51 

1.2556160 

1.60 

.4700030 

2.24 

.8064758 

2.88 

1.0577902 

3.52 

1.2584609 

1.01 

.4762341 

2.25 

.8109302 

2.89 

1.0612564 

3.53 

1.2612978 

1.02 

.4824261 

2.26 

.8153648 

2.90 

1.0647107 

3.54 

1.2641266 

1.63 

.4885800 

2.27 

.8197798 

2.91 

1.0681530 

3.55 

1.2669475 

1.64 

.4946962 

2.28 

.8241754 

2.92 

1.0715836 

3.56 

1.2697605 

























Table of Hyperbolic Logarithms. 


291 


N. 

Logarithm. 

N. 

Logarithm, 

N. 

Logarithm. 

N. 

Logarithm. 

3.57 

1.2725655 

4.21 

1.4374626 

4.85 

1.5789787 

5.49 

1.7029282 

3.58 

1.2753627 

4.22 

1.4398351 

4.86 

1.5810384 

5.50 

1.7047481 

3.59 

1.2781521 

4.23 

1.4422020 

4.87 

1.5830939 

5.51 

1.7065646 

3.GO 

1.2809338 

4.24 

1.4445632 

4.88 

1.5851452 

5.52 

1.7083778 

3.G1 

1.2837077 

4.25 

1.4469189 

4.89 

1.5871923 

5.53 

1.7101878 

3.G2 

1.2864740 

4.26 

1.4492691 

4.90 

1.5S92352 

5.54 

1.7119944 

3.63 

1.2892326 

4.27 

1.4516138 

4.91 

1.5912739 

5.55 

1.7137979 

3.G4 

1.2919836 

4.28 

1.4539530 

4.92 

1.5933085 

5.66 

1.7155981 

3.G5 

1.2947271 

4.29 

1.4562867 

4.93 

1.5953389 

5.57 

1.7173950 

3.66 

1.2974631 

4.30 

1.4586149 

4.94 

1.5973653 

5.58 

1.7191887 

3.67 

1.3001916 

4.31 

1.4609379 

4.95 

1.5993875 

5.59 

1.7209792 

3.68 

1.3029127 

4.32 

1.4632553 

4.96 

1.6014057 

5.60 

1.7227666 

3.69 

1.3056264 

4.33 

1.4655675 

4.97 

1.6034198 

5.61 

1.7245507 

3.70 

1.3083328 

4.34 

1.4678743 

4.98 

1.6054298 

5.62 

1.7263316 

3.71 

1.3110318 

4.35 

1.4701758 

4.99 

1.6074358 

5.63 

1.7281094 

3.72 

1.3137236 

4.36 

1.4724720 

5.00 

1.6094379 

5.64 

1.7298840 

3.73 

1.3164082 

4.37 

1.4747630 

5.01 

1.6114359 

5.65 

1,7316555 

3.74 

1.3190856 

4.38 

1.4770487 

5.02 

1.6134300 

5.66 

1.7334238 

3.75 

1.3217558 

4.39 

1.4793292 

5.03 

1.6154200 

5.67 

1.7351891 

3.76 

1.3244189 

4.40 

1.4816045 

5.04 

1.6174060 

5.68 

1.7369512 

3.77 

1.3270749 

4.41 

1.4838746 

5.05 

1.6193882 

5.69 

1.7387102 

3.78 

1.3297240 

4.42 

1.48^1396 

5.06 

1.6213664 

5.70 

1.7404661 

3.79 

1.3323660 

4.43 

1.4883995 

5.07 

1.6233408 

5.71 

1.7422189 

3.80 

1.3350010 

4.44 

1.4906543 

5.08 

1.6253112 

5.72 

1.7439687 

3.81 

1.3376291 

4.45 

1.4929040 

5.09 

1.6272778 

5.73 

1.7457155 

3.82 

1.3402504 

4.46 

1.4951487 

5.10 

1.6292405 

5.74 

1.7474591 

3.83 

1.3428648 

4.47 

1.4973883 

5.11 

1.6311994 

5.75 

1.7491998 

3.84 

1.3454723 

4.48 

1.4996230 

5.12 

1.6331544 

5.76 

1.7509374 

3.85 

1.3480731 

4.49 

1.5018527 

5.13 

1.6351056 

5.77 

1.7526720 

3.86 

1.3506671 

4.50 

1.5040774 

5.14 

1.6370530 

5.78 

1.7544036 

3.87 

1.3532544 

4.51 

1.5062971 

5.15 

1.6389967 

5.79 

1.7561323 

3.88 

1.3558351 

4.52 

1.5085119 

5.16 

1.6409365 

5.80 

1.7578579 

3.89 

1.3584091 

4.53 

1.5107219 

5.17 

1.6428726 

5.81 

1.7595805 

3.90 

1.3609765 

4.54 

1.5129269 

5.18 

1.6448050 

5.82 

1.7613002 

3.91 

1.3635373 

4.55 

1.5151272 

5.19 

1.6467336 

5.83 

1.7630170 

3.92 

1.3660916 

4.56 

1.5173226 

5.20 

1.6486586 

5.84 

1.7647308 

3.93 

1.3686394 

4.57 

1.5195132 

5.21 

1.6505798 

5.85 

1.7664410 

3.94 

1.3711807 

4.58 

1.5216990 

5.22 

1.6524974 

5.S6 

1.7681496 

3.95 

1.3737156 

4.59 

1.5238800 

5.23 

1.6544112 

5.87 

1.7698546 

3.96 

1.3762440 

4.60 

1.5260563 

5.24 

1.6563214 

5.88 

1.7715567 

3.97 

1.3787661 

4.61 

1.5282278 

5.25 

1.6582280 

5.89 

1.7732559 

3.98 

1.3812818 

4.62 

1.5303947 

5.26 

1.6601310 

5.90 

1.7749523 

3.99 

1.3837912 

4.63 

1.5325568 

5.27 

1.6620303 

5.91 

1.7766458 

4.00 

1.3862943 

4.64 

1.5347143 

5.28 

1.6639260 

5.92 

1.7783364 

4.01 

1.3887912 

4.65 

1.5368672 

5.29 

1.6658182 

5.93 

1.7800242 

4.02 

1.3912818 

4.66 

1.5390154 

5.30 

1.6677068 

5.94 

1.7817091 

4.03 

1.3937663 

4.67 

1.5411590 

5.31 

1.6695918 

5.95 

1.7833912* 

4.04 

1.3962446 

4.68 

1.5432981 

5.32 

1.6714733 

5.96 

1.7850704 

4.05 

1.3987168 

4.69 

1.5454325 

5.33 

1.6733512 

5.97 

1.7867469 

4.06 

1.4011829 

4.70 

1.5475625 

5.34 

1.6752256 

5.98 

1.7884205 

4.07 

1.4036429 

4.71 

1.5496879 

5.35 

1.6770965 

5.99 

1.7900914 

4.08 

1.4060969 

4.72 

1.5518087 

5.36 

1.6789639 

6.00 

1.7917594 

4.09 

1.4085449 

4.73 

1.5539252 

5.37 

1.6808278 

6.01 

1.7934247 

4.10 

1.4109869 

4.74 

1.5760371 

5.38 

1.6826882 

6.02 

1.7950872 

4.11 

1.4134230 

4.75 

1.5581446 

5.39 

1.6845453 

6.03 

1.7967470 

4.12 

1.4158531 

4.76 

1.5602476 

5.40 

1.6863989 

6.04 

1.7984040 

4.13 

1.4182774 

4.77 

1.5623462 

5.41 

1.6882491 

6.05 

1.8000582 

4.14 

1.4206957 

4.78 

1.5644405 

5.42 

1.6900958 

6.06 

1 8017098 

4.15 

1.4231083 

4.79 

1.5665304 

5.43 

1.6919391 

6.07 

1.8033586 

4.16 

1.4255150 

4.80 

1.5686159 

5.44 

1.6937790 

6.08 

1.8050047 

4.17 

1.4279160 

4.81 

1.5706971 

5.45 

1.6956155 

6.09 

1.8066481 

4.18 

1.4303112 

4.82 

1.5727739 

5.46 

1.6974487 

6.10 

1.8082887 

4.19 

1.4327007 

4.83 

1.5748464 

5.47 

1.6992786 

6.11 

1.8099267 

4.20 

1.4350845 

4.84 

1.5769147 

5.48 

1.7011051 

6.12 

1.8115621 




























292 Table of Hyperbolic Logarithms. 


N. 

Logarithm. 

N. 

Logarithm. 

N. 

Logarithm. 

N. 

Logarithm. 

6.13 

1.8131947 

6.77 

1.9125011 

7.41 

2.0028305 

8.05 

2.0856720 

6.14 

1.8148247 

6.78 

1.9139771 

7.42 

2.0041790 

8.06 

2.0869135 

6.15 

1.8164520 

6.79 

1.9154509 

7.43 

2.0055258 

8.07 

2.0881534 

6.16 

1.8180767 

6.80 

1.9169226 

7.44 

2.0068708 

8.08 

2.0893918 

6.17 

1.8196988 

6.81 

1.9183921 

7.45 

2.0082140 

8.09 

2.0906287 

6.18 

1.8213182 

6.82 

1.9198594 

7.46 

2.0095553 

8.10 

2.0918640 

6.19 

1.8229351 

6.83 

1.9213247 

7.47 

2.0108949 

8.11 

2.0930984 

6.20 

1.8245493 

6.84 

1.9227877 

7.48 

2.0122327 

8.12 

2.0943306 

6 21 

1.8261608 

6.85 

1.9242486 

7.49 

2.0135687 

8.13 

2.0955613 

6.22 

1.8277699 

6.86 

1.9257074 

7.50 

2.0149030 

8.14 

2.0967905 

6.23 

1.8293763 

6.87 

1.9271641 

7.51 

2.0162354 

8.15 

2.0980182 

6.24 

1.8309801 

6.88 

1.9286186 

7.52 

2.0175661 

8.16 

2.0992444 

6.25 

1.8325814 

6.89 

1.9300710 

7.53 

2.0188950 

8.17 

2.1004691 

6.26 

1.8341801 

6.90 

1.9315214 

7.54 

2.0202221 

8.18 

2.1016923 

6.27 

1.8357763 

6.91 

1.9329696 

7.55 

2.9215475 

8.19 

2.1029140 

6.28 

1.8373699 

6.92 

1.9344157 

7-56 

2.0228711 

8.20 

2.1041341 

6.29 

1.8389610 

6.93 

1.9358598 

7.57 

2.0241929 

8.21 

2.1053529 

6.30 

1.8405496 

6.94 

1.9373017 

7.58 

2.0255131 

8.22 

2.1065702 

6.31 

1.8421356 

6.95 

1.9387416 

7.59 

2.0268315 

8.23 

2.1077861 

6.32 

1.8437191 

6.96 

1.9401794 

7.60 

2.0281482 

8.24 

2.1089998 

6.33 

1.8453002 

6.97 

1.9416152 

7.61 

2.0294631 

8.25 

2.1102128 

6.34 

1.8468787 

6.98 

1.9430489 

7.62 

2.0307763 

8.26 

2.1114243 

6.35 

1.8484547 

6.99 

1.9444805 

7.63 

2.0320878 

8.27 

2.1126343 

6.36 

1.8500283 

7.00 

1.9459101 

7.64 

2.0333976 

8.28 

2.1138428 

6.37 

1.8515994 

7.01 

1.9473376 

7.65 

2.0347056 

8.29 

2.1150499 

6.38 

1.8531680 

7.02 

1.9487632 

7.66 

2.0360119 

8.30 

2.1162555 

6.39 

1.8547342 

7.03 

1.9501866 

7.67 

2.0373166 

8.31 

2.1174596 

6.40 

1.8562979 

7.04 

1.9516080 

7.68 

2.0386195 

8.32 

2.1186622 

6.41 

1.8578592 

7.05 

1.9530275 

7.69 

2.0399207 

8.33 

2.1198634 

6.42 

1.8594181 

7.06 

1.9544449 

7.70 

2.0412203 

8.34 

2.1210632 

6.43 

1.8609745 

7.07 

1.9558604 

7.71 

2.0425181 

8.35 

2.1222615 

6.44 

1.8625285 

7.08 

1.9572739 

7.72 

2.0438143 

8.36 

2.1234584 

6.45 

1.8640801 

7.09 

1.9586853 

7.73 

2.0451088 

8.37 

2.1246539 

6.46 

1.8656293 

7.10 

1.9600947 

7.74 

2.0464016 

8.38 

2.1258479 

6.47 

1.8671761 

7.11 

1.9615022 

7.75 

2.0476928 

8.39 

2.1270405 

6.48 

1.8687205 

7.12 

1.9629077 

7.76 

2.0489823 

8.40 

2.1282317 

6.49 

1.8702625 

7.13 

1.9643112 

7.77 

2.0502701 

8.41 

2.1294214 

6.50 

1.8718021 

7.14 

1.9657127 

7.78 

2.0515563 

8.42 

2.1306098 

6.51 

1.8733394 

7.15 

1.9671123 

7.79 

2.0528408 

8.43 

2.1317967 

6.52 

1.8748743 

7.16 

1.9685099 

7.80 

2.0541237 

8.44 

2.1329822 

6.53 

1.8764069 

7.17 

1.9699056 

7.81 

2.0554049 

8.45 

2.1341664 

6.54 

1.8779371 

7,18 

1.9712993 

7.82 

2.0566845 

8.46 

2.1353491 

6.55 

1.8794650 

7.19 

1.9726911 

7.83 

2.0579624 

8.47 

2.1365304 

6.56 

1.8809906 

7.20 

1.9740810 

7.84 

2.0592388 

8.48 

2.1377104 

6.57 

1.8825138 

7.21 

1.9754689 

7.85 

2.0605135 

8.49 

2.1388889 

6.58 

1.8840347 

7.22 

1.9568549 

7.86 

2.0317866 

8.50 

2.1400661 

6.59 

1.8855533 

7.23 

1.9782390 

7.87 

2.0630580 

8.51 

2.1412419 

6.60 

1.8870696 

7.24 

1.9796212 

7.88 

2.0643278 

8.52 

2.1424163 

6.61 

1.8885837 

7.25 

1.9810014 

7.89 

2.0655961 

8.53 

2.1435893 

6.62 

1.8900954 

7.26 

1.9823798 

7.90 

2.0668627 

8.54 

2.1447609 

6.63 

1.8916048 

7.27 

1.9837562 

7.91 

2.0681277 

8.55 

2.1459312 

6.64 

1.8931119 

7.28 

1.9851308 

7.92 

2.0693911 

8.56 

2.1471001 

6.65 

1.8946168 

"7.29 

1.9865035 

7.93 

2.0706530 

8.57 

2.1482676 

6.66 

1.8961194 

7.30 

1.9878743 

7.94 

2.0719132 

8.58 

2.1494339 

6.67 

1.8976198 

7.31 

1.9S92432 

7.95 

2.0731719 

8.59 

2.1505987 

6.68 

1.8991179 

7.32 

1.9906103 

7.96 

2.0744290 

8.60 

2.1517622 

6.69 

1.9006138 

7.33 

1.9919754 

7.97 

2.0756845 

8.61 

2.1529243 

6.70 

1.9021075 

7.34 

1.9933387 

7.98 

2.0769384 

8.62 

2.1540851 

6.71 

1.9035989 

7.35 

1.9947002 

7.99 

2.0781907 

8.63 

2.1552445 

6.72 

1.9050881 

7.36 

1.9960599 

8.00 

2.0794415 

8.64 

2.1564026 

6.73 

1.9065751 

7.37 

1.9974177 

8.01 

2 0806907 

8.65 

2.1575593 - 

6.74 

1.9080600 

7.38 

1.9987736 

8.02 

2.0819384 

8.66 

2.1587147 

6.75 

1.9095425 

7.39 

2.0001278 

8.03 

2.0831845 

8.67 

2.1598687 

6.76 

1.9110228 

7.40 

2.0014800 

8.04 

2.0844290 

8.68 

2.1610215 





































Table op Hyperbolic Logarithms. 


293 


N. 

Logarithm. 

N. 

Logarithm. 

N. 

Logarithm. 

N. 

Logarithm. 

8.G9 

2.1621729 

9.33 

2.2332350 

9.97 

2.2995806 

71 

4.2626799 

8.70 

2.1633230 

9.34 

2.2343062 

9.98 

2.3005831 

72 

4.2766661 

8.71 

2.1644718 

9.35 

2.2353763 

9.99 

2.3015846 

73 

4.2904594 

8.72 

2.1656192 

9.36 

2.2364452 

10 

2.3025851 

74 

4.3040651 

8.73 

2.1667653 

9.37 

2.2375130 

11 

2.3978953 

75 

4.3174881 

8.74 

2.1679101 

9.38 

2.2385797 

12 

2.4849067 

76 

4.3307333 

8.75 

2.1690536 

9.39 

2.2:196452 

13 

2.5649494 

77 

4.3438054 

8.76 

2.1701959 

9.40 

2.2407096 

14 

2.6390573 

78 

4.3567088 

8.77 

2.1713367 

9.41 

2.2417729 

15 

2.7080502 

79 

4.3694479 

8.78 

2.1724763 

9.42 

2.2428350 

16 

2.7725887 

80 

4.3820266 

8.79 

2.1736146 

9.43 

2.2438960 

17 

2.8332133 

81 

4.3944492 

8.80 

2.1747517 

9.44 

2.2449559 

18 

2.8903718 

82 

4.4067193 

8.81 

2.1758874 

9.45 

2.2460147 

19 

2.9444390 

83 

4.4188406 

8.82 

2.1770218 

9.46 

2.2470723 

20 

2.9957323 

84 

4.4308168 

8.83 

2.1781550 

9.47 

2.248128S 

21 

3.0445224 

85 

4.4426513 

8.84 

2.1792868 

9.48 

2.2491843 

22 

3.0910425 

86 

4.4543473 

8.85 

2.1804174 

9.49 

2.2502386 

23 

3.1351942 

87 

4.4659081 

8.86 

2.1815467 

9.50 

2.2512917 

24 

3.1780538 

88 

4.4773368 

8.87 

2.1826747 

9.51 

2.2523438 

25 

3.2188758 

89 

4.4886364 

8.88 

2.1838015 

9.52 

2.2533948 

26 

3.2580965 

90 

4.4998097 

8.89 

2.1819270 

9.53 

2.2544446 

27 

3.2958369 

91 

4.5108595 

8.90 

2.1860512 

9.54 

2.2554934 

28 

3.3322045 

92 

4.5217886 

8.91 

2.1871742 

9.55 

2.2565411 

29 

3.367*2958 

93 

4.5325995 

8.92 

2.1882959 

9.56 

2.2575877 

30 

3.4011974 

94 

4.5432948 

8.93 

2.1894163 

9.57 

2.2586332 

31 

3.4339872 

95 

4.5538769 

8.94 

2.1905355 

9.58 

2.2596776 

32 

3.4657359 

96 

4.5643482 

8.95 

2.1916535 

9.59 

2.2607209 

33 

3.4965076 

97 

4.5747110 

8.96 

2.1927702 

9.60 

2.2617631 

34 

3.5263605 

98 

4.5849675 

8.97 

2.1938856 

941 

2.2628042 

35 

3.5553481 

99 

4.5951199 

8.98 

2.1949998 

9.62 

2.2638442 

36 

3.5835189 

100 

4.6051702 

8.99 

2.1961128 

9.63 

2.2648832 

37 

3.6109179 

101 

4.6151205 

9.00 

2.1972245 

9.64 

2.2659211 

38 

3.6375862 

102 

4.6249728 

9.01 

2.1983350 

9.65 

2.2669579 

39 

3.6635617 

103 

4.6347290 

9.02 

2.1994443 

9.66 

2.2679936 

40 

3.6888795 

104 

4.6443909 

9.03 

2.2005523 

9.67 

2.2690282 

41 

3.7135721 

105 

4.6539604 

9.04 

2.2016591 

9.68 

2.2700618 

42 

3.7376696 

106 

4.6634391 

9.05 

2.2027647 

9.69 

2.2710944 

43 

3.7612001 

107 

4.67282S8 

9.06 

2.2038691 

9.70 

2.2721258 

44 

3.7841896 

108 

4.6821312 

9.07 

2.2049722 

9.71 

2.2731562 

45 

3.8066525 

109 

4.6913479 

9.08 

2.2060741 

9.72 

2.2741856 

46 

3.8286414 

110 

4.7004804 

9.09 

2.2071748 

9.73 

2.2752138 

47 

3.8501476 

111 

4.7095302 

9.10 

2.2082744 

9.74 

2.2762411 

48 

3.8712010 

112 

4.7184989 

9.11 

2.2093727 

9.75 

2.2772673 

49 

3.8918203 

113 

4.7273878 

9.12 

2.2104697 

9.76 

2.2782924 

50 

3.9120230 

114 

4.7361985 

9.13 

2.2115656 

9.77 

2.2793165 

51 

3.9318256 

115 

4.7449321 

9.14 

2.2126603 

9.78 

2.2803395 • 

52 

3.9512437 

116 

4.7535902 

9.15 

2.2137538 

9.79 

2.2813614 

53 

3.9702919 

117 

4.7621739 

9.16 

2.2148461 

9.80 

2.2823823 

54 

3.98S9841 

118 

4.7706846 

9.17 

2.2159372 

9.81 

2.2834022 

55 

4.0073332 

119 

4.7791235 

9.18 

2.2170272 

9.82 

2.2844211 

56 

4.0253517 

120 

4.7874917 

9.19 

2.2181160 

9.83 

2.2854389 

57 

4.0430513 

121 

4.7957906 

9.20 

2.2192034 

9.84 

2.2864556 

58 

4.0604430 

122 

4.8040210 

9.21 

2.2202898 

9.85 

2.2874714 

59 

4.0775374 

123 

4.8121844 

9 99 

2.2213750 

9.86 

2.2884861 

60 

4.0943446 

124 

4.8202816 

9.23 

2.2224590 

9.87 

2.2894998 

61 

4.1108739 

125 

4.S283137 

9.24 

2.2235418 

9.88 

2.2905124 

62 

4.1271344 

126 

4.8362819 

9.25 

2.2246235 

9.89 

2.2915241 

63 

4.1431347 

127 

4.8441871 

9.26 

2.2257040 

9.90 

2.2925347 

64 

• 4.1588839 

128 

4.8520303 

9.27 

2.2267833 

9.91 

2.2935443 

65 

4.1743873 

129 

4.8598124 

9.28 

2.2278615 

9.92 

2.2945529 

66 

4.1896547 

130 

4.8675345 

9.29 

2.2289385 

9.93 

2.2955604 

67 

4.2046926 

131 

4.8751973 

9.30 

2.2300144 

9.94 

2.2965670 

68 

4.2195077 

132 

4.8828019 

9.31 

2.2310890 

9.95 

2.2975725 

69 

4.2341065 

133 

4.8903491 

9.32 

2.2321626 

9.96 

2.2985770 i 

70 

4.2484952 

134 

4.897S398 
















































294 


Triangulatiox. 



206. 

To find the distance 6 from the point C to an 

inaccessible house. 

Measure a base line a and the angles B and C. 
Then a: ft = sin..4 :sin.R. 

b = —f-—Suppose a = 160 feet, C = 87° 45', 
sm. A 

B = 65° 35'. Required the distance ft? 

A = 180—(2? + C)=180—(87° 45'-i- 65° 35')=26° 40'. 

ft = = 325.56 feet. 


siu.26° 40' 


0.4488 



207. 

To find the distance a over a lake. 

Measure the angle A and t he two sides ft and c. 
Say A = 34° 25', ft — 1684 feet, and c — 1310 feet. 
Required the distance o? 

. „ c . . 1310 

C ° ' ft sin.^4 C ° ' 1684 X sin.34° 25' 

— cot.34° 25' =t 0.083 = cot. 94° 45'. 

a : ft = sin.34° 25': sin.94° 15'. 

W84 X S l"- 34 ° 25 ' = 955. 1 f«,t. 


a ■- 


s in. 94° 15 



To measure the height of a lighthouse. 
Measure the distance ft and angle A ; then the 
height a will be a = b tan.^4. Suppose ft = 134 
feet and the angle A =<56° 34'. Required the 
height «? 

a = 134 X tan.56° 34' = 134 X 1.5147 = 202.97 feet. 

The height of the line ft above the ground ean 
be measured separately and added. 



209. 

To find the height of an object from a height. 
The distance c and the angles A and B are 
given. 

a4-ft = c sin.(.4 + B). Suppose ^4 = 36° 42', 
B ~2\° 22', and the side c = 115 feet. Required 
the height a? 

a + ft =115 sin.(36 0 42'+ 21° 22') = 115 sin.58° 4' 
= 115 X 0.84866 = 97.216 feet. 

The horizontal distance d = c cos.58° 4' = 115 
m X 0-52893 = 60.827 feet. 

///y ft = 60.827 X tan.21° 22' = 60.827 X 0.36433 = 22.162 
feet. 

The height a = 97.216 — 22.162 = 75.054 feet. 



To find the height of an object from a point 
below the base. 

Measure the angles A and B and the distance 
c. ft = c sin.R, and d — c cos.B. Suppose c — 86 
feet, A = 42° 20', B = 10° 15'. Required the hori- 
zontal distance d and the height ft? 

ft = 86 sin.10° 15'= 15.3028 feet. 
d = 86 cos. 10° 15' = 84.627. 

(a + ft) = d tan.(.4 + B) = 84.627 X tan.52° 35' 

= 110.616 feet. 

a = 110.616 —15.3021 = 95.313 feet. 







































Triangulation. 


295 


211 . 



To find the height of a distant mountain. 

The distance from the instrument to the 
mountain is approximately known by a map 
to be say 12 miles, and the angle v measured 
from the top of the mountain to the horizontal 
line is 1°44'. Required the height of the moun¬ 


tain? 12 X 5280 = 68300 feet. Feet. 

Heightofmountain// = 63360 tan.l°44' = 1917.27 
For curvature and refraction add 82.32 

Height of instrument 4. 

The required height 2003.59 


212 . 



To find the height h of a mountain, and hori¬ 
zontal distance a from the instrument to under 
the highest point. 

Measure the augle H and distance d. 
h=dsin.Il, and a = dcos.II. 

Supposed = 1560feet.and the angle//= 42° 54'. 
Required the height h and distance a ? 

h = 1560 X sin.42° 54' = 1061.9 feet. 
a = 1560 X cos.42° 54/ = 1142.7 feet. 



213. 


To measure the distance d from o to a tree on 
the other side of the stream. 

Draw from o the line « -f 6 at right angles to 
d. Draw c at right angles to a + 6. From a 
point p sight the tree, which cuts the line a -f b, 


so that a:b = d:c. 


Distance, d = 



214. 


To find the distance d from o to a tree on the 
other side of a stream. 

Make a at right angles to d, and / at right 
angles to e ; then a:d = c\a. 



215. 



To find the height h of an inaccessible object, 
also its horizontal distance from the instrument. 
Measure the base line a and the angles B t C, 


a:c = sin.^4:sin.(7. c- 

a: e = sin.^4': sin./?. 

h:c= sin .II: sin.C. 
g = c cos.(c g). 


a ^.A=K0-(D + C). 
a sin.2? 


sin 

e 


sin.d' 

^ _ c sin.// 

sin.C 

h' = e sin.(eg). 































296 


Triangulation. 



« sin.(C+ d) 
sin.4 


>1 

J 

>! 

l ) 

: 


216. 

To find the distance b between a tree and a 
house on the other side of the river. 

, . .. . ^ , rtsin.Z) 

a : a = sin.4 sin.i). u =— ; —-y . 

sin.4' 

4' = 180— {B + C+D). 

«:« = sin ..4 :si■>.(*- + D). ?- sin (B ' + 


cot.D' 


e 


dsin.B 
b:d = sin.i?: sin.i)'. 


cot .B. 


b = 


sin.4 


d sin.i? 
sin. iy' 


217. 

To find the distance c' between a house and 
a tree. 

Put the instrument at B , in line with the 
tree and house; measure the angle i?, and lay 
out the base line a; measure the angles Cand d. 

.. . _ asin.C 

ci:c = sin.4 .sin.C. c 


A'=m — {B + C). 

a:c+c*'=sin.4 :sin.(C-fc'). c+c' = 

a sin.C a / . . „ . \ 

■ ——j = — —- I sin.tC + d) —sin.C J . 
sin.4 sin.4 \ ' ' / 


sin.4' * 
a sin.(C-f- d) 


sin.4 










































Mechanics. 


297 


ELEMENTS OF MECHANICS. 

Mechanics is that branch of natural philosophy which treats of forces 
in equilibrium and in motion ; it is divided into two distinct parts, namely— 

STATICS AND DYNAMICS. 

Statics is the science of forces in equilibrium or at rest; it is subdivided 
into various branches bearing upon solids, liquids, gases, and imponderable 
fluids. 

Statics, strictly speaking, refers to forces in equilibrium in regard to 
solids. 

Hydrostatics treats of forces in equilibrium of liquids. 

Aerostatics treats of pressure and equilibrium of air or other gases. 

Electrostatics refers to static electricity held or stored in a body. 

Thermostatics refers to stationary temperatures in bodies. 

The scieuce of statics is very well developed in text-books, and particularly 
so in works on roofs, girders, bridges, and ou strength of materials ; but such 
cannot be said about dynamics, which is yet in a confused condition. 

Dynamics is the science of forces in motion, producing power and work, 
and is also subdivided into various branches like statics, bearing upon solids, 
liquids, gases, and imponderable fluids. 

Dynamics , strictly speaking, refers to power and work of solids. 

Hydrodynamics, or Hydraulics, treats of motion, power, and work of 
liquids. 

Aerodynamics treats of power and work of air and other gases. 

Electrodynamics treats of power and work of electricity and magnetism. 

Thermodynamics treats of power and work produced directly by heat like 
that of steam. 

Imponderable Matter. 

The term imponderable implies that the body has no weight, but is, never¬ 
theless, a material substance which is in such condition as to constitute the 
connection of attraction or repulsion between ponderable bodies. We know 
that imponderable substances are matter by their manifestation of possess¬ 
ing inertia , and, whatever aggregate form a body may be converted into, its 
inertia can never be destroyed or interfered with in the least. 

All matter can be resolved into four aggregate forms—namely, solid, liquid, 
gaseous, and imponderable; but there is no sharp line of distinction between 
these four forms—that is, a body may be in a semifluid state, like glaciers, 
which flow in a solid state; and many soft substances are neither solid nor 
liquid, but may be called plastic. 

The compositions of imponderable fluids are as varied as those of ponder¬ 
able matter, and the probability is that they are all binary compounds of the 
same matter which we know when in a ponderable state. Each kind of im¬ 
ponderable sqj^stance manifests a nature peculiar to itself, and some kinds 
cannot be displaced by ponderable bodies passing through them, as is the 
case with that imponderable fluid which forms the connection of universal 
attraction, and also that constituting magnetic attraction and repulsion, 
which fluid passes through the ponderable like light passes through a per¬ 
fectly transparent substance. The spectrum is a fair representation of 
known ponderable matter in imponderable slates. It is also manifest by 
the spectrum that some imponderables are not refracted, but pass straight 
through as if the prism were not there, whilst other imponderables do not 
pass through the prism, but are reflected therefrom. 

This is purely speculative on the part of Mr. Nystrom, and is not to be ac¬ 
cepted as established fact.—W. E. M. 





298 


Statics. 


STATICS. FORCES IN EQUILIBRIUM. 

Ik mechanics, the term force means any action which can he expressed 
simply by weight, and which can tie realized only by an equal amount of 
reaction. Forces are derived from a great variety of sources; but whenever 
it is simply/orce, it can invariably be expressed by weight, without regard to 
motion, time, power, or work. 

The magnitude and direction of a force can be represented by a straight 
line, but no force can be realized without an equal amount of reaction in the 
opposite direction, which can likewise be represented by a straight line. 




F 


-&■ 


1. Action and Reaction. 

Let the line F represent the magnitude and 
direction of a force acting on the point c at rest 
or in motion, and let the line R represent an 
equal amount of resistance acting in the oppo¬ 
site direction ; then the force F and resistance 
R are said to be in equilibrium. R is called re¬ 
sultant. 


4k 

,k 


a 


c 


R 


F 


5j 

~'V 


ft 


2. Resultant of two Parallel Forces. 

W is a weight whose magnitude and direction 
are represented by the arrow, and F is a force 
acting parallel to )V, both acting on the inflex¬ 
ible bar ab. Make dd' equal to F’and ee' equal 
to IF; join d’ e, and the crossing c is the fulcrum 
for the resultant R. Make R equal to I VF 
and draw it from c parallel to IF and F. 


Static Moment and Lever. 

F : W= a: b. Static moments Fb = Wa. 

a and b are called levers; the force or weight multiplied by the lever acted 
upon is called static moment. 

A lever is the rectangular distance from the direction of a force to the 
fulcrum. 

Fulcrum is the point on which a lever turns. 



3. Forces Acting Obliquely. 

Fand IF represent the magnitude and direc¬ 
tion of two forces acting at the points d. and e. 
Continue the direction of the two forces until 
they meet at d'. From d' set off the F and W 
equal to F and W respectively j complete the 
parallelogram, and the diagonal R' represents 
the magnitude and direction of the resultant R 
acting at the fulcrum c. The levers for these 
forces are the rectangular distances a and b. 

Static moments Fb = Wa. 





4. Forces Acting Against One Another. 

Prolong the forces F and IF until they meet 
at d\ from which set off the forces IF' and F 
equal to IF and F respectively. Complete the 
parallelogram, and c'd' is the magnitude and 
direction of the resultant R , which, acting in 
the fulcrum c, will balance the two forces. The 
levers a and b are drawn from the fulcrum c at 
right angles to the direction of the respective 
forces. 

Static moments F b — IF a. 






















Moment of Stability. 


299 



5. 

Two Forces Acting Against One Another. 

Prolong F and ]V until they meet at (V ; set 
off W' and F equal to W and F respectively 
and complete the parallelogram ; then c'rf' rep¬ 
resents the direction and magnitude of the re¬ 
sultant R acting at the fulcrum c. The levers 
a and b are drawn from the fulcrum c at right 
angles to the respective forces. 

Static moments Fb = TV a. 



6 . 


Form of Bar Independent of Levers. 

The bar upon which the forces act can be of 
any desired form, such as eerf, but the actual 
levers will still be as before defined—namely, the 
rectangular distances from the fulcrum to the 
direction of the respective forces. 

Static moments Fb = Wa. 



7. 


Combination of Levers. 

The static moments in this combination will be 
W a a' a" — wb b’ b". 


W = 


wbb’ b" 
a a'a" 


w = 


W a a' a" 
bb’b" ’ 



Moment of Stability. 

8 . 

A body W is acted upon by a force F to turn 
the body over the fulcrum A. .Required its mo¬ 
ment of stability. 

Moment of stability Wa = Fb static moment. 
The weight of the body is considered concen¬ 
trated in its centre of gravity W, and the lever 
a is the rectangular distance from the fulcrum 
c to the direction of gravity. 




a = 


Fb 
W ' 


and b = 


W a 
F • 


9. 

Formulas are the same as above. 

_ Example. —The weight of the body is W— 30,390 
pounds, the lever a = 6 feet, and the lever 
b — 18 feet. Required the force F to turn the 
body over the fulcrum B. 

„ Wa 30390 X 6 iO . 0n . 

F~ -r— =- 1Q -- = 12,130 pounds, 

b lo 

the answer. 





























300 


Retaining Wales for Water. 


10 . 



wmmtMmsmmm 


Retaining Walls for Water. 

The moment of stability of a retaining wall should 
in practice he at least four times the static moment 
of the water-pressure. Let on represent tlie water- 
pressure per unit of surface at the bottom ; then the 
triangle onm represents the total water-pressure on 
the retaining wall. When the linear dimensions are 
expressed in feet, we can express the moments per 
foot of length of wall. One cubic foot of fresh water 
at 60° Fahr. weighs G2.33 pounds. The centre of pres¬ 
sure of the water is at the height of the centre of 
gravity of the triangle o nm, or oue-third of the depth 
d. The total water-pressure or force F on the wall 
will then be F= | X 62.33d 2 = 31.16 d 2 . The lever of the force F is b — \d. The 
static moment of the force F will then be Fb = jFd; but F = 31.16d 2 ,when 
the static moment will be i X 31.16 d 3 = 10.3866 d 3 . 

The weight w per cubic foot of hydraulic masonry is about 90 pounds for 
brick and 140 pounds for granite. 
h = height and t — thickness of the retaining wall. 

Weight of the wall W — hi w. 
a = lever for the weight IF of the wall. 

The moment of stability of the wall without water-pressure will then be 
IF a = h tw a. 

The real moment of stability of the wall with water is 
IF a — Fb = h l w a — 10.3866 d 3 . 

In practice, make Wa = 4Fb. 


11 . 



When the cross-section of the retaining wall 
is in the form of a triangle of the same area as 
the rectangular one, it will be 2$ stronger in 
moment, of stability, on account of the lever a 
being that much increased. The static moment 
of the water-pressure will be the same as in the 
foregoing case. 




12 . 


When the section of the retaining wall is in 
form of a triangle, but the inclined side is on 
the water side, the moment of stability and 
static moment are both diminished, but that 
of stability suffers most. 


C 



13. 

Mt. Stb. IFa = ^ h t w X i t = £ h fi w. 
E — hypothenuse A C. 

Force F= 


h (F -62.33. 
n 


Fb = 


2 h 


62.33. 











































Natural Slope and Weight of Granular Substances. 


301 


When walls are used to retain earth-work, etc., it is impossible to say what 
the action of the force may be. For practical purposes its action is assumed 
similar to water, with Certain modifications given below. 



14 . 

Whatever may be the shape and position of the 
retaining wall, its moment of stability is always 
Wa, and the static pressure of the water acts at 
right angles to the surface of the wall. The force 
of tlie water-pressure in pounds is equal to the area 
of the pressed surface in square feet multiplied by 
half the depth in feet, and the product by 62.33. 
The centre of pressure is at one-third of the depth 
from the bottom. 


Retaining’ Walls Tor Earthwork. 



The action of earth or other granular substances, 
like sand, gravel, grain, etc., on retaining walls is 
similar to that described for water, and the mo¬ 
ments are calculated by the same formulas, with 
the only exception that the natural slope of the 
granular materials diminishes the force F as the 
cosine for that slope. 

The natural slope of a granular substance is the 
greatest angle with the horizon at which it will re¬ 
pose in a heap. Let s denote the angle of natural 
slope and XV = weight per cubic foot of the material 
retained by the wall; which values for some sub¬ 
stances are contained in the following table: 


Natural Slope and Weight of Granular Substances. 


Granular Substances Loosely Heaped. 

Slo 

s. 

pe. 

cos.s. 

Weight, 

w. 

Lime (powder). 

Saw-dust, wheat-flour. 

Broken stone or coal. 

Malt-flou r. 

45 

44 

43 

40 

39 

38 

37 

36 

35 

35 to 40 

0.70711 

0.71934 

0.73135 

0.76604 

0.77715 

0.78801 

0.79863 

0.80902 

0.81915 

0.8 to 0.75 

95 

94 

47 

45 

48 

80 to 100 

Sand (moist)... 

Sand (dry)... 

Malt-corn. 

Wheat, rye, and corn. 

Peas. 

Gravel and earth.. 


The force F per foot of length of wall will be F—kwd cos .s. For safe 
calculation, we may assume w = 100 pounds per cubic foot of earth or gravel 
pressing against the wall, and s = 32° 51', safety angle of natural slope, which 
cosine is 0.84; then the force F will be F — £ X 100 X 0.84 d = 42 d. 

Static moment Fb — j Fd , but F= 42 d. Fb — 14 d 2 . 

Moment of stability Wa = h(wa = i hw fi. 

W — weight per cubic foot of materials in the wall, which is 90 for brick, 
120 for rubble concrete, and 140 for granite. 

The real stability of the wall will be 

Wa — Fb = hhwP — 14 t* 2 . 

In practice, make Wa = 4 Fb or h w ( l — 28 d s . 




























302 


Cranes. 


16. 




HOISTING-CRANES. 



The ordinary foundry crane consists of the post 
P, resting in a shoe a, and cap b , jib G , and stay 8. 

The weight JFacts on tlie lever l, which is reacted 
in the supports a and & on the lever h. Consider 
the support b as the fulcrum, and the force F act¬ 
ing on the lever h in the shoe a. Static moments 
Fh = Wl. 


F= 


Wl 


h 


W= 


Fh 


l 


h- 


Wl 


F 


l = 


Fh 

~W' 


S= strain of compression on the stay S, and L =» length of the stay. 

WL WIL ... „ Wl 

Force ol tension of the jib, & = 




h 


m n 


n 


The horizontal strain in the cap b is equal to that in tlio shoe «. The 
vertical pressure in the shoe a is equal to the weight IF and weight of the 
crane added together. In foundry cranes the block c is moved in or out to 
suit the location of the weight to be lifted; and the jib and stay are botli 
made double, so that the chain can pass bet ween the parts. 

The chain in cranes should be kept constantly moist with grease for two 
good reasons—namely: 

1st. Lubricat ion decreases the friction and wear of the chain. 

2d. It prevents, to a certain extent, the crystallization of the chain and the 
hook. For a foundry crane in constant use, the chain should be greased with 
lubricating oil about once a month. It has been found advantageous to an¬ 
neal the chain about once a year for preventing crystallization, but the an¬ 
nealing itself injures the fibres of the iron after having been repeated a few 
times, so that greasing answers a better purpose. 

Supposing the crane is strong enough, the weight IF which can be lifted 
wit h it depends upon the proportion of gearings in the windlass and pulleys. 

A man working a crank can exert a force of about 40 pounds moving with 
a velocity of 2 feet per second, which is 80 effects. If the gearing in the crane 
is say 100 to 1, the one man can lift 800 pounds. The gearing in large foundry 
cranes varies between 200 and 300 to 1. 

The loss of power by friction is about 10 + \/n per cent, w hen n is the 
proportion of gearing. 

Wharf Cranes. 



This kind of crane is often used for loading and 
unloading vessels. Its static moments are the same 
as for the foundry crane—namely, Fh — IF l. 


F= 


Wl 
h ' 


TF= 


Fh 


l 


h = 


Wl 
F' 


l = 


Fh 


IF* 


ost, 


The lateral strength of the curved part of the post 
must compensate the stay in the foundry crane. 


Shop Cranes. 


L— length of the tension-rod. 

T= force of tension. 

F— pressure ih each of the post journals. 
f= force of compression of the jib. 

1 = distance between the centres of the block 
the post. 

,K=f. h = wl 


and 


T= 


h 

WLl 


m n 


/--= 


Wl 

n 


F= 


F ‘ 
fn 
h ' 


1 = 


Fh 


1F= 


IF 

fn 
l ' 


All forces, weight, and measures are in pounds and 
feet. 































Stability of a Round Tower.' 


803 


Stability of Towers to the Force of Wind. 


The moment of stability of a tower or other structure exposed to the 
wind should be at least four times the static moment, of the greatest storm to 
which the object is exposed. The moment of stability is like that in retain¬ 
ing walls, but the static moment depends upon the velocity of the wind. 

Let A denote the area in square feet of the structure exposed to the wind, 
the greatest force of which will be 50.4 ; b = height of the centre of gravity 
of the exposed surface above the ground. The greatest static moment of 
the wind will then be 50 A b, and the minimum stability of the structure 
should be limited to 


4 X 50 A b = 200 A b. 

Stability, Wa = Fb static moment. 





Practically, the stability Wa ought to be four times the 
static moment Fb. 

W a = 4 Fb, or W a = 200 A b, 

, 200 A b 

and W =-, 

a 

in which TF=the whole weight of the tower in pounds, 
^ A — area of one side of the tower facing the wind. 


Example. —Suppose the tower to be 4 feet square and 20 feet high, from 
which a = 2 and b = 10 feet. The area of the side will be A = 4 X 20 — 80 
square feet. W T hat weight of the tower is required to maintain it stable to 
a tornado ? 


___ 200 A b 

W — - 

a 


200 X 80 X 10 
2 


— 80,000 pounds. 


It is supposed that the wind acts at right angles on one side of the tower ; 
but if acting in the direction of the diagonal of the square section, a greater 
surface will be exposed, but at such angle to the wind that the acting force 
will be the same as when blowing directly on only one side. 


Stability of a Round Tower. 

On a round tower of diameter equal to the side of the square the force of 
the wind is only one-half of that on the square tower. 

The illustration represents a tower or chimney of the form of a conic 
frustum of diameters d at the top and D at the base; h = height of the tower; 
all in feet. 

The height of the centre of gravity b is calculated from 
20. the following formula: 



h h l D — d \ 

2 6 \ D + d /* 


Example. —The height of the tower h — 260 feet, diameter 
at the top d = 10 feet, and 2? = 25 feet at the base. Re¬ 
quired the height of the centre of gravity of the surface 
of the tower ? 


6 = 


260 

2 ~ 


260 

6 


( 


25 — 10\ 
25 + 10/ 


— 111.43 feet. 


The projecting area to the wind is 

A ( 2 ) + d) = Ai. (25 + 10) — 4550 square feet. 

2 2 

Of this area only one-half is effective to the force of the wind, or A =2275 
square feet. The static moment of a tornado will then be 

Fb = 50 A b = 50 X 2275 X 111.43 — 12675162.5 foot-pounds. 


















304 


Levers. 


If the material in the tower and that in the base it stands on were hard 
enough to stand the crushing force at the fulcrum in upsetting the tower, 
the lever of the moment of stability would be half the diameter D of the 
base; but as such is not the case, in practice a deduction from the leverage 
must be made to allow for the softness of the materials acted upon at the 
fulcrum. 

When the base of the tower is square, and the force of the wind acts at 
right angles to one of its sides, the fulcrum will be a line; whilst on a cir¬ 
cular base the fulcrum will be a point in which the whole weight of the tower 
acts to crush the materials. 

In ordinary good brickwork the lever of the moment of stability may be 
taken at 0.7 of the radius of the circular base. 

The weight of the tower in the preceding example should then be 


W- 


4Fb 4 X 12675162.5 


a 


0.7 X 12.5 


5794368 pounds. 




A small deduction ought also to be made from the lever in a square base, 
where it may be taken at 0.9 of half the side of the square. 

The cohesive force of the materials increases the stability of the structure 
when the masonry is perfectly solid at the base. 


Levers. 

Levers are of three distinct kinds, with reference to the relative positions 
of the Force F, Weight W, and Fulcrum C. 

1st. Fulcrum C is between the force F and the weight W. 

2d. Weight IT is between the fulcrum C and the force F. 

3d. Force F is between the fulcrum C and the weight IT. 

Example 1. Figure 21. The weight W — 68 pounds, the lever l = 3.86 feet, 
and L = 10 feet 6 inches. 

Required the force F= ? 

,, , , „ Wl- 68X3.86 

Formula 1. F= = -——— = 2o pounds nearly. 

Jj 1U.0 

a = distance between the force F’fmd the weight W. 

The formulas 3, 4, 7, 8, 11, 12 are for finding the fulcrum C when the force 
F,\ weight IT, and the distance a are given. 

Example 2. Fig. 22. The force F= 360 pounds, IT = 1870, and a = 8 feet 
4 inches. 

Required the position of the fulcrum c? 


Formula 7. I = 


Fa 


360 X 8.333 2999.988 


= 19.86 feet. 


IT— F 1870 — 360 1510 

L = 8.333 + 19.86 = 28.193 feet., the answer. 

Example 3. Fig. 26. The weight of the lever is Q= 18 pounds. The centre 
of gravity is x = 2.25 feet from the fulcrum. IT = 299 pounds, l = 5.5 feet 
and L = 11.95. 

Required the force F = 1 in pounds. 


F= 


Wl — Qx 299X5.5 — 18X 2.25 
L ~ 11.95 


Inclined Plane. 


= 134.25 pounds. 


Example 4. Fig. 45. A load IT = 3466 pounds is to be drawn up an inclined 
plane l = 638 feet long and h = 86 feet high. 

What force is required to keep the load on the inclined plane? 

h IT 86 X 3466 


F — 


638 


= 467.2 pounds. 




















Mechanics.—Statics. 


305 


Example 4. Fig. 4S. A Cylinder of cast iron, weighing W = 5245 pounds, la to 
be rolled up an inclined plane; the angles v — 18° 20' and v' = 8° 10' 

What force is required to keep the cylinder on the plane? 

F = W. sin.(»+tO =5245Xsin.(18° 20'+S° 10') = 2340 pounds. 
Example 5. Fig. 47. An iron hall which weighs 398 pounds, is tied to an In¬ 
clined plane with a rope; the angle of the rope and the inclined plane is 
v' — 16° 40', and v — 14° 30'. What force is acting on the rope ? 


„ FFsin.a 
* ' » 

cos.tr 


398Xsin.14° 30' 
cos.l6° 40' 


= 104 pounds. 


Example 0. Fig. 35. What force F is required to raise a weight W = 8469 
pounds, by a double moveable pulley ? 

F = i W = ±XS4G9 = 2117-25 pounds. 

Example 7. Fig.38. How much weight can a force F— 269 pounds lift by 
three compound moveable pulleys ? 

W= ‘2 U F — 2^X269 = 2152 pounds, the answer. 

Screw. 

Example 8. Fig. 54. What force is required to lift a weight W = 16785 pounds, 
by a screw, with a pitch P = 0-125 feet, the lever being r — 5 feet. 4 inches ? 


F= 


WP 167S5X0-125 


= 62-62 pounds, the answer. 


2*-r 2X3-14X5-333 

Including friction the force will be 

F W{P + fdrt ) 

‘Inr ’ 

Find the friction f on page 267. d diameter of the screw in feet. 

Wedge. 

Example 9. Fig. 51. The head of the wedge a = 3 inches, and length 
l -= 16£ inches; the resistance to be separated is It = 4846 pounds. Required 
the force F = ? (Friction omitted.) 


4^40X3 = 8gl dg . 

16'5 


Including friction the force F will be, 


148 


^[f + A 2+ -f,)] 


in which the friction / is to be found on page 267. 

Catenaria. 

Example 10. An iron chain 256 feet long, weighing 1560 pounds, is to be sus¬ 
pended between two points in the same horizontal line, but 196 feet apart. 

How deep will the chain hang under the line of suspension, and with what 
farce will the chain act at the points of suspeusion? 

Fiaure and Formula 43. we have given, 

\V = ^Xd5 60 = 78 0 pounds, l = £X256 == 128 feet, and a = £X196 =98 feet. 
j l== o.f -j/ 128 2 — 98 2 = 61.7 feet, the required depth under the horizontal line. 

2X61.7 = 1>259> QTV== 38 o 27' } and 2v = 76° 54'. 


cot.r = 


98 


The required force will be, 


F — 


780Xsin-38 o 27' 
sin.76° 54' 


: 507 pounds. 


20 



















306 


Lever and Static Momentum, 



























































Lever and Static Momentum. 


307 


27 F:W=r:R. FR = Wr, 

W w 


1 / \ \ 

W = ^ r _^ 
r» r “-W 


28 

T/I 7 ".. ./ p n D' 


7! ' ' i ) Tjr I It It 

F ~RR’> W rr' ’ 

Tl = number of revolutions of the wheels, 


n : n' — R' : r, v : v' = rr': RR', 

V = velocity of \V 7 v' — velocity of F . 

rv_ r 2Q 

-r-r r t t I Tl T-\ TV / TV / f 


-=f—'— 7 — s' 

ri hV r r r T ir r K K K 

F RRR" y ™ rr'r" ’ 



l PxJ / x/ r\ V \ r 

Hb. V 

n : «" = r'r" : R R\ v: v' = rr'r" : 
RR'R". 

r r'r'' &C. = radii of the pinions. 

R R'R"&C. = radii of the wheels. 


30 

Let P and Q represent the magnitudes and direc¬ 
tions of two forces which act to move the body B. 
By completing the parallelogram, there will be ob¬ 
tained a diagonal force F, whose magnitude and di¬ 
rection is equal to the resultant of P and Q. F is 


called the resultant of P and Q. 

\J§5^ A 

31 

If three or more forces act in different directions 
to move a body P, find the resultant of any two of 
them, and consider it as a single force. Between 
this and the next force find a second resultant, 
thus : P. Q , and A’ are magnitudes and directions of 
the forces. P+ Q = r, r+ R F— P+ Q+fi, or F 
is the magnitude and direction of the three forces, 

P, Q, and B. 

^x-° 

*<f\ 

V 

32 

A force Q acting (alone) on the body B, can move 
it to a in a unit of time, another force Pis able to 
move it to b in the same time ; now if the two 
forces act at the same time, they will move the 
body to c. c is the resultant of a and b. 












































308 


POLLFYS. 













































































■Funicular and Catenarian. 


309 
























































310 


Inclined Plane. 




45 - W sin.®, 

I 

w FI F 

W = —- = —-y 

h sin.v 

W = EA = Wcos.v. 

L 

w = normal component. 


F= w sin.y 


cos.u' 


W 


F cos.v' 


sin.t; 

w = W (cos.® — sin.v tan.t/). 
r W sin.u 

r - 7 > 

cos.tr 

w= F cos,u> 

sin.v * 

w = W (cos.v-fsin.v. tarn/). 


49 

W = weight of the body, and direc¬ 
tion of the force of gravity ; to be drawn 
at right-angles to the base b, and F par¬ 
allel to F. 

By calculation see Formula, Fig. 46. 


\ x 


£'-^*ar 


50 

w = the force with which the body 
presses against the plane, to be drawn 
at right-angles to the plane l; then the 
parallelogram is completed. 

By calculation see Formula, Fig. 47. 


To solve an Inclined Plane by diagrams. 


F — magnitude and direction of the 
force, which is obtained by completing 
i the parallelogram. 

By calculation see Formula, Fig. 45. 














































Wedge and Screws 


311 



'51 


Wedge. 


F = 


R a 


R = 


FI 


l 7 a 

F = force required to drive the wedge. 



52 

i 

Let the line F represent the magnitude and di¬ 
rection of a force acting to move the body B on the 
line CD ; then the line a represents a part of F 
which presses the body B against CD, and the line 
b represents the magnitude of the force which 
actually moves the body B. 

b = \/ F~ — b = Fcos.v. 


( 1 

- -»T7T 





53 


F: W = k : b = sin.v : cos.v = tan.u. 

j? Wh 

F - nr 


Trr Fb 

W= — = 

h tan.u 


= W tan.v. F — F. 
F 


F cot.u. 



54 Force by a Screw. 

P = Pitch of the screw, 

r = radius on which the force F acts. 

F : W= P : 2 nr. 

F2j?r 


F 2tzF 





55 Force by Compound Screws. 

P = Pitch of the large screw, 
p = Pitch of the endless screw. 

R = radius of spur-wheel for the endless 
screw. 

W: F = 4tt 2 R r : P p. 

f=-F- p ‘‘, w-™f Rr 

47P R r 


PP 9 

On the spur-wheel Is a cylinder by which 
the weight W is wound up, the formula will 
be, _ radius of the cylinder,) and 

F : W =p r' : 2n R r. 

F w pi w ,* r . 

F ~ 2n Ft’ P r 











































































312 


Strength of Materials 


STRENGTH OF MATERIALS. 

• 

Table I., shows the weight a column can hear with safety; when the weight 
presses through the length of the columu. The tabular number is the weight 
in pounds or tons per square inch on the transverse section of a column 
a length less than 12 times its smallest thickness. 

Talilc !• 


RESISTANCE EOlt COMPRESSION. 56 


Kind of Materials . 



rounds. 

Tons. 

i \ 

Oak, of good quality, 

• 

m 

432 

0-1885 


W \ 

Oak, common, - 

m 

m 

2S0 

0-125 

\ 

Spruce, red (Sapin rouge), 

m 

m 

540 

0-241 

* 



“ white, (Sapin blanc), 

m 

m 

140 

0-6256 




Iron, wrought, ... 

m 

• 

14400 

6-43 




Iron, cast, ... - 

m 

• 

28750 

12-85 




Basalt, - 

• 

m 

2875 

1-285 




Granite, hard, • » 

• 

m 

1000 

0-446 




“ common, * - • 

m 

m 

575 

0-256 

Jl 



Marble, bard, - 

m 

m 

1435 

0-640 




“ common, * 

• 

m 

431 

0-192 




Sandstone, hard, . • • 

• 

m 

1295 

0-577 




“ loose, - - - 

• 

• 

5-6 

0-0025 




Brick, good quality, • • 

• 

m 

175 

0-078 




“ common, ... 

• 

m 

58 

0-0259 




Lime-stone, of hardest kind,- 

• 

m 

720 

0-321 




“ common, • 

• 

m 

432 

0193 




Plaster-Paris, - 

• 

• 

86 

0-0384 




Mortar, good quality, and 18 months old, 

58 

00259 

SL, 



Do. common, ... 

• 

• 

36 

0-016 




When the length or height of the column is more than 12 times its smallest 
thickness, divide the tabular weight by the corresponding number in this 
Table. 


LengthXthickness 

12 

18 

24 

30 

36 

42 

4S 

54 

60 

Divide by 

1-2 

1-6 

2 

2-8 

4 

5 

6 

8 

12 


Example. A building which is to weigh 2000 tons is to be supported by piles 
of Sapin rouge Spruce 18 feet in length, and 12 inches diameter. How many piles 
are required to support the building? 


and 


12 2 XO'”85X0'241 

1-6 


=■ 17 tons, the weight which 


each pile can bear, 


2000 

17 


118 piles. 


Professor llodgkinson’s Formulae for Crushing Strength of 

Cast Iron Pillars. 


The ends of the pillars should be perfectly flat and square, and the load to 
hear even on the whole surface. 

T=crushing weight in tons. 

H=outside and d inside diameters in inches. 

I =length or height of pillar in feet. 


r=4665 



.55 _^ 3.(9 

l 1,7 


) 









































Diameters in Inches. For Tubes subtract the weight due to the bore, 


Cast Iron Pillars, 


313 

! 


4> 

a 

vi 

A 


a 

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't 

CO 

b 

l'- 

o 

Hi 

05 

0 

03 

05 

0 




co 

H 

CO 

co 

03 

rH 

rH 


05 

b 

cO 

to 

0 


co 

co 

co 

CM 

CM 

CM 



; o h< 

rfOrf 

h* 0 b- 

CQ N fO 


CO CO rH 
r—‘ CO 0 
CO *0 rH 
03 rH H 


HI 03 CO 
05 rH C/D 
CO N O 


I s * Tfl N 
CO H 0 
H* Hi CO 


W lO o 

rH 0 

CO 03 03 


CO CO H* CO O 

H 05 N CO O 

03 rH *-H r-H rH 



to* 

• 

0 

0 

CO 












• 

rH 

0 

CM OO 

0 CO 

CO 

0 

o 

CO 0 05 

CO o 

0 co 

CO 


o 

Hi 

co 

rH 

CO o 

0 rH 

O 

CO 

CO 

CO O b- 

to Hi 

03 

rH 

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H* 

rH 

Hi 

03 

rH 05 

0 0 

H> 

CO 

03 

03 03 i-H 

rH rH 

rH 

r— 1 

H 05 00 



03 

CO 

CO 














CO 

03 

a> ^ 

0 O 03 Hi 

0 

eo co 





0 


03 

03 

rH 

CO b* 

Hi 0 

1-H 

b- H< 

03 O CO 03 CO 

0 

CO 

CO 05 0 



i> 

03 

rH 

CO 'd! 

CO 03 

03 

rH 

rH 

rH r-H 05 

CO b. 

0 

0 

>0 ■cT 'S* 



0 

O 














0 

o 

O 

CO 1-H 

0 05 











03 

o 

O 

O rH 

0 rH 

0 

co 

0 

0 00 03 

CO 

CO 

CO CM O 



CO 

rH 

0 

CO 03 

rH r-H 

b» 0 

lO 'ct< -Si 

co CO 

CM 

03 03 03 03 


ft 

« 

» 

V 

rH 

A 

e» 

H 


CO 

** < 0 

h t o *-0 co o . .,. 

g _ o --H >0 OCMCOOOCO ONK 5 WH o 7 ? ® 7 

* u^TjiOOCMCM CM ri i— 1 i— 1 i—l r -1 Ol 00 N 1 > 

d 

£ „ i (5 iO >0 ^ CO 

•* 3 CM CO CM rH P-ICJI- 50 U 5 r^OOcOCOCM CM CM CM rH H 

rH 

of i—( iC 5 CO-HCOCMCO»OCM®OOCOtJicC) 

B ^ 00 i (5 OCOCOO-cT CpCCCMCMCM CM ^ 1 

•ScMCOCOiCIrH 2 hoooo ooooo ooioo 

H 

iHNMH »/5 C 0 b- 00 C 5 O rtOjeorO tDi'-OOCS© 

iH rlr-liH—liH ririrl rtf) 


r 



These values are about half of that by Prof. Hodgkinson’s formula. The points after the numbers mean ciphers. 






































































314 


{Strength of Materials. 


Table II* 

COHESIVE STRENGTH PER SQ. INCH OF CROSS-SECTION 


Just tear asunder. 

With safety. 

Pounds. 

Tons. 

Pounds. 

Tons 

134250 

59-93 

33600 

14-98 

133152 

59-43 

33300 

14-86 

128632 

56-97 

32160 

14-24 

65000 

29-2 

16260 

7-3 

59470 

26-7 

14900 

6-7 

56000 

25-0 

14000 

6-2o 

36000 

16-00 

9000 

40 

40000 

17-85 

10000 

4-46 

34400 

15-35 

8600 

3-84 

45000 

20-05 

11250 

5-00 

18000 

8-03 

4500 

2-0 

32500 

14-37 

8130 

3-6 

61200 

27-2 

15300 

6-8 

5000 

2-23 

12500 

0*56 

8S0 

0-356 

220 

0-09 

3320 

1-48 

830 

0-37 

53000 

23-6 

13250 

5-9 

45000 

2005 

11250 

5-0 

16000 

7-14 

4000 

1-87 

11500 

5-13 

2875 

1-28 

20000 

8-93 

5000 

2-23 

11400 

6-09 

2850 

1-27 

21000 

9-38 

5250 

2-34 

12000 

5-36 

3000 

1-34 

11500 

5-13 

2875 

1-28 

10000 

4-46 

2500 

1-11 

13600 

6-07 

3400 

1-52 

12000 

6-35 

3000 

1-34 

13000 

6-8 

3250 

1-45 

7800 

3-48 

1950 

0-87 

7600 

3-40 

1900 

0-85 

6400 

2-86 

2130 

0-95 

3200 

1-43 

1100 

0-49 

38000 

17 

12600 

5-36 

65000 

29 

21600 

9-38 

90000 

40 

30000 

13-4 


Kind of Material t. 

Cast Steel, 

Blistered Steel, 

Steel, Shear, - 
Iron, Swedish bar, 

Russian, 

English, 

common, over 2 in. sq. 
sheet, parallel rolling, 
at right angles to roll, 
Cast iron, good quality, 

“ inferior, 

Copper, cast, - 
“ rolled, 

Tin, cast, - 
Lead, cast, 

“ rolled, * 

Platinum, wire, 

Brass, common, '» 


Wood. 

Ash, 

Beach, 

Box, - 
Cedar, 

Mahogany, 

“ Spanish, 
Oak, American white, 
“ English “ 

“ seasoned, * 

Pine, pitch, 

“ Norway, 
Walnut, - 
Whalebone, 

Hemp ropes, good, 
Manilla ropes, 

Wire ropes, 

Iron chain, 

“ with cross pieces, 


To Find tlxe Cohesive Strength* 

Rule. —Multiply the cross-section of the materials in square inches by the j 
tabular number in Table II., and the product is the cohesive strength. 

Example. An iron-bar has a cross-section of 2 - 27 sq. in. How many tens are 
required to tear it asunder, and how many pounds can it bear with safety ? 

English iron 2’27X25 = 56-75 tons, which will tear it asunder, tmd it will bear 
with safety 

2-27X14000 = 31780 pounds. 






































Chains, Hemp and Wire Ropes. 315 


Safety 

Inches and 16ths. 

Wlit. per fathom. 

Price per fathom. 

(Ultimate 

proof. 

Chain. 

iHerap. 

1 Wire. 

Chain. 

Hemp. 

1 Wire. 

Chain. 

Hemp. 

Wire. 

1 

.Strain. 

Cwt. 

Diam. 

Circ’m. 

Ciro'm. 

Pounds Pounds Pounds 

$ cts. 

$ cts. 

$ cts. 

Cwt ~ 

1.3 

1 

o-io 

0-4 

0-23 

0-08 

0-06 

0-15 

0-06 

0-08 

2-6 

4-5 

2 

1.6 

OS 

0-93 

0-47 

0*24 

0-25 

0-12 

0*15 

9 

10 

3 

2-1 

0-12 

2-11 

1-06 

0-54 

0-36 

0-17 

0*22 

20 

18 

4 

2*12 

11 

3-75 

1-89 

1-10 

0-48 

0-25 

0-32 

35 

28 

5 

3*7 

1*6 

5-86 

2-94 1*83 

0-60 

0-33 

0*43 

55 

40 

6 

4-2 

1-10 

8-45 

4*52 

2-56 

0-96 

0-42 

0-54 

80 

55 

7 

4-15 

1-14 

11-5 

6.09 

9-42 

1-25 

0*48 

0*62 

109 

69 

8 

5-8 

2-2 

15-0 

7-55 

4*39 

1-44 

0-60 

0-78 

138 

SO 

9 

6-3 

2-6 

18-8 

9-56 

5-48 

1-60 

0-76 

0-90 

160 

94 

10 

6*14 

2-11 

23-0 

11-8 

7*00 

1*86 

0-95 

1-20 

218 

109 

11 

7*9 

2-15 

27*7 

14-3 

8-38 

2-16 

1-14 

1-50 

187 

127 

12 

8-4 

3-3 

33-0 

171 

9.90 

2-43 

1-37 

1-80 

254 

147 

13 

8-15 

3-8 

38-5 

19 9 

11-9 

2-70 

1-60 

2-10 

293 

168 

14 

9-10 

3-12 

44*7 

23-1 

13-6 

3-06 

1-85 

2*28 

335 

199 

15 

10-5 

4*1 

51*1 

26-3 

16-0 

3-70 

2-10 

2*45 

397 

220 

1 in. 

11* 

4-6 

58-0 

30-2 

18-6 

4-33 

2-42 

2-73 

440 

246 

1*1 

11-11 

4-11 

65-6 

34-1 

21-3 

4-68 

2-73 

3-10 

492 

278 

1*2 

12-6 

5 in. 

73-7 

38-2 

24-2 

5-58 

3-06 

3*50 

545 

302 

1*3 

13-1 

5-5 

82-1 

42-6 

27-4 

5-86 

3-40 

3-91 

604 

332 

1*4 

13*12 

5-10 

91-0 

47-1 

30*7 

6-42 

3-77 

4-35 

663 

365 

1-5 

14-7 

6 in 

100 

52-0 

35* 

7.08 

4-16 

4-89 

730 

399 

1*6. 

15*2 

6-5 

110 

57-1 

38*7 

7-75 

4*57 

5'35 

798 

435 

1*7 

15*15 

6-10 

120 

63-4 

42-6 

8-42 

5*07 

5-86 

869 

472 

1-8 

16-8 

6-15 

131 

67-9 

46*7 

915 

5-44 

6-35 

944 

553 

1-10 

17*14 

7-10 

154 

79-8 

56-4 

10-07 

6-38 

7-63 

1105 

638 

1-12 

19-4 

8*4 

178 

92-6 

66-0 

12-38 

7-40 

8-83 

1275 

729 

1*14 

20-10 

8-14 

205 

106 

76-5 

14-15 

8-48 

10-00 

1457 

825 

2 i a 

22* 

9.8 

232 

121 

88-0 

16-00 

9-70 

11-50 

1650 

1072 

2*4 

24-12 

10-12 

293 

153 

112 

20-75 

10-25 

14-60 

2141 

1288 

2*8 

27-8 

12 in. 

363 

189 

140 

25* 

15-10 

18-00 

2575 

1559 

2*12 

30-4 

134 

438 

229 

172 

30-25 

18-30 

21*80 

3117 

1854 

3 in 

33- 

14-8 

522 

272 

205 

36-00 

21-80 

25*90 

3708 


The prices of the chains are taken from that in England and added 50 
per cent. Price of hemp ropes from Weaver, Fitler & Co., Rope manu¬ 
facturers, Philadelphia. The prices of Wire ropes are deduced from 
the price list of John A. Roebling, Patent Wire Rope Manufacturer, 
Trenton, N. J. 

The safe strength is here taken one half of the ultimate strength which 
may he trusted for new ropes, but when much in use only one quarter or less 
should be used. 






































316 


Strength of Materials. 


LATERAL STRENGTH 




OP MATERIALS. 




The formulas for lateral strength are here reduced to the simplest pos¬ 
sible form, and are in consequence subject to conditions which must be 
particularly attended to. In calculating the strength ot beams of ir¬ 
regular sections, as shown by the figures 68 to 75 on page 318, it is necessary 
to maintain ilie proportions marked on the figures and the calculation will 
be correct. For the sections 206 to 209 any proportion will answer in the 
formulas. The weight of the beam itself lias not here been taken into con¬ 
sideration, for which allowance must be made if considerable. See pages 
317 and 318. 

Notation. 


I — length of beam in feet. See figures 59 to 64. 

h = height, 6 = breadth or thickness in inches of the beam, where the 
strain is acting. 

k = coefficient for the different materials and sections of beams, to be 
found in the tables. 

x = modulus of elasticity of materials. See Table. 

/= elastic deflection in inches. 

weight in pounds which the beam can bear with safety, being 
about one quarter of the ultimate strain at which the beam 
would break. 

Example 1. Fig. 58. A rectangular beam of oak fastened in a wall 
projects out 1=6 feet 4 inches, 6 = 8 inches, and 6=6 inches. Required 
what weight it can bear on the end 1V=*] 


W_ 30X6X81 _ 16og p 0un( ] 3j w ith perfect safety. 

Example 2. Fig. 59. A beam of section fig. 69, with thickness 6=1*26 
inches, height 6=22-5 inches, supported at the two ends in a length J=25 
feet. Required what weight it can bear in the middle. For cast 
iron coefficient k= 260. 


w = 4X 260XP25><22-6» 
25 


20325 lbs.=11*8 tons nearly. 


Example 3. Required the elastic deflection for the same beam and con¬ 
dition as in the foregoing example l See Table, modulus of elasticity 
#=2285 for cast iron. See page 320. 


/= 


26325X25* 

16X2285X1 ‘25X22-5* 


= 0*80 inches, nearly. 


Example 4. Fig. 62. A wrought iron girder of section fig. 75. consist¬ 
ing of four angle irons of a =3-5X0'5X 2 ><4=14 square inches, the plate 
being 0-6:l-35=0 - 37 inches thick,.and 6=18 inches deep by J=22 feet. Re¬ 
quired how much weight evenly distributed the girder can bear with 
safety 1 


ir=3 8X800>Of>O8 1133 .= 32*75 tons. 

22 

If plates being riveted to the angle iron at top and bottom, add that 
area to a. 

Example 5. Fig. 80. The crank R= 3*5 feet, force F=3860 lbs., length 
of the shaft 1=61 feet, diameter D=6-25 inches. Required the twisting 
in degrees. The shaft being of wrought iron for which #=4110. Page 320. 

Degree. 

& 4110X626 4 










Strength and Elasticity of Materials. 317 































































































































318 


Different Forms of Beams.. 


64 


70 



i 




Coefficient k. 

Cast iron, 150 
Wro’t iron, 120 
Wood, 30 


65 



Coefficient k . 

Cast iron, 236 
Wro’t iron, 189 


71 



Coefficient k. 
Cast iron, 150 
Wro’t iron, 120 
Wood, 30 


b 


1 > 



Coefficient k. 

Cast iron, 250 
Wro’t iron, 200 




66 

Coefficient 
Cast iron, 
Wro’t iron, 
Wood, 

b h'-LP. 



72 

Coefficient k. 

Cast iron, 700 
Wro’t iron, 560 


67 

Coefficient k. 
Cast iron, 88 
Wro’t iron, 70 
Wood, 18 

b }?=&—(?. 



73 

Coefficient k. 
Cast iron. 900 


68 


74 


V 

fc I 

— A- - 

Coefficient k. 

Wro’t iron, 700 

b l?=.a h. 


69 




Coefficient k. 

V if 1 


_i_ fl._J8.l_ 

Cast iron, 260 

yj| 

Wro’t iron, 208 





< 10 » 




k= 800, 
b fi 2 =a h, 
a<= area of all 
the four angle 
irons in square 
inches. 



















































Strength of Materials. 


319 



A "beam fixed in one end and loaded 
at the other, should have the form of a 
Parabola, in which l = abscissa and h = 
ordinate. y= depth, x— length from W. 


y=h 




77 


W= 


kb h° kbh* 


l cos.v 


b> 


w 


~r a —i i 


J!L 


i ' i T 


h 


* s/m /f 




78 


W— 


36 kb h* 


l 


Divide the length into 24 equal parts, 
place 14 in the middle and 5 at each end. 



79 To cut oat the stoutest rectangular 
beam from a log. 

1st case, divide the diameter in 3 equal 
parts, and draw lines at right-angles as 
represented. 

2d, divide the diameter in 4 equal parts. 

1, 6=1*414 b, non-elastic. 

2, A=l*73 b, elastic beams. 



w~ 

„ A 3 IF R _ 3 / H 

Z)=4a /■- =80a - , 

V x v x n 


Twisting in degrees ^ 


425 FR l 



! 81 


D : d=z-$/R : ^/r } 

3 / H 
D= 80 a /—, 

\ x n 


Twisting in degrees = 


2233000 


E l ^ 

HJ 






























































































320 


Strength op Materials. 


Absolute and Ultimate Strength, of Materials* 


Kind of Materials. 


Coefficient k. 


Elasticity. 


Safety. 

Inter. 

Pr. cir. 

Ultimate. 

X 

Wrought iron, .... 

120 

162 

240 

488 

4110 

Cast iron, .... 

150 

200 

300 

600 

2285 

Cast steel, soft,.... 

385 

519 

170 

1540 

4300 

Cast steel, hardened, . 

1050 

1100 

2100 

4200 

6000 

Blistered steel, soft, . 

175 

233 

350 

700 

4200 

Brass,. 

58 

75 

113 

226 

1280 

Copper,. 

53 

71 

106 

212 

2160 

Z n ic y • • • • • 

15 

20 

30 

61 

2360 

Tin, • • • • • • 

17 

23 

34 

69 

• • • 

Lead, ..... 

4 

6 

9 

18 

100 

Ash, . • * • • • 

45 

56 

85 

170 

221 

Hickory, .... 

67 

90 

135 

270 

• • • 

Chestnut, sweet, . . 

42 

56 

85 

170 

• • • 

Oak, white, .... 

50 

66 

100 

200 

300 . 

Oak, English, .... 

25 

33 

50 

100 

248 

Canadian oak, . . . 

37 

49 

73 

147 

283 

Pine, white, . 

34 

45 

67 

135 

• • • 

Yellow pine, 

38 * 

50 

75 

150 

268 

Teak,. 

51 

68 

102 

205 

316 


The safe stress is here taken one-quarter of the ultimate breaking weight, but 
when the weight is acting at short intervals one-third might be relied upon, or in 
pressing circumstances one-lialf, when the materials in the beams are known to be 
of good quality; but the latter should never be exceeded. 

Pi-operties of some South American "Woods, 

Taken from the borders of the rivers Perene and Madre de Dios, and expert- 
merited upon by the author of this Pocket-Book. 


Peruvian Names of the 
Woods. 

Color. 

Specific 

gravity. 

Wt. per 
cub. foot, 
lbs. 

Hard¬ 

ness. 

H 

Ultimate 

strength. 

k 

Elas¬ 

ticity^ 

X 

Chonta (Palm), . . . 

Black, . . . 

1.564 

96.75 

28 

450 

649 

Balsamo.. 

Brown, . . 
Brown stripes, 

1.207 

75.25 

22 

422 

492 

Shacaranda, .... 

0.991 

61.75 

18 

343 

322 

J ebe (Ind.-rubber tree)* 

Light yellow, 

0.797 

49.65 

15 

351 

305 

Amarillo,. 

Yellow, . . 

0.734 

45.75 

13 

334 

300 

Caoba,. 

Light brown, 

0.613 

38.20 

11 

128 


Iluachapeli, .... 

Oillv, « • • 

0.566 

35.25 

10 

134 

180 

Nogal, •••••• 

Dark brown, 

0.551 

34.35 

10 

131 

158 

Jebe(best Ind.-rubber)* 

White, . . . 

0.527 

32.85 

9 

- 162;. 
62 1 

262 

M. Barigon, .... 

Wliite, . . 

0.282 

17.58 

6 

92 


* There are different kinds of trees which give India-rubber, but of different 
quantity and quality.- \ 

The woods wjsre perfectly dry. Four experiments on each were made. 

The hardness, H, is compared with that of substances on page 333. 

The coefficient, k, is the ultimate lateral strength of the woods. 
x = modulus of elasticity determined near the ultimate strength. 


k = 


Wl 


and 


x 


WP 


46 A 2 " 16/6 A 3 

Meaulng of letters is the same as that on page 272. 


Fig. 59, p. 317. 




































Dimensions, Weight, and Price or Rolled Iron. 


321 


Angle 

Iron. 

! 

Dimensions. 

Per Foot. 
Weight. Price. 

Inches. 

lbs. 

cents. 

A(** +1 *) 

1*77 

5*13 1 

i (li+U) 

2-32 

6-74 2 


2-09 

6*07 

c 

T 8 e ( Ii+1 h 

3-49 

101 

h ( 2 + 2 > 

317 

8*60 4 

t ( 2+2 ) 

4-59 

13*3 i 

A( 2i+2i > 

4.97 

14-4 

( 

r 7 s( 2i+2i ) 

6.84 

19-9 

f( 3 + 3 ) 

713 

20-7 

i ( 3+3 ) 

9*32 

27*1 * 

1 ( s ‘ +3i ) 

8-40 

24*4 ji 

r ^(3i+3i) 

12-2 

34*9 

r 7 s( 4+4 ) 

11-2 

32-5 

I ( 4+4 ) 

155 

45-0 

ShipFrames. 



i-'i+lV 21 

2.5 

7-95 

rV 2 + V 3 

4-36 

138 

r 2 i+f\' 3i 

6.68 

21-2 

4 3 i+tV 4i 

8.85 

28T 


11-0 

35-0 

f- 4+ ,v 6 

16*4 

51 '° 


Variety of Forms. 


Section. 


Price 


Figure. 


12 


IV 


Area. 

Wt.p.ft. 

Wt. p. Mile, | 

Per Ft. 

Sq. In. 

lbs. 

Tons 

$ cts. 

7-4 

25 

117 86 

0-56 

5-9 

20 

92-7 

0-45 

71 

24 

113.14 

0-54 

• 




1-95 

6*6 

31*45 

0-16 

5*41 

18-3 

86-43 

0-45 

4.44 

15 

70*71 

0-37 

4-22 

14-3 

67-75 

0-35 

7-00 

23-6 

111-57 

0-58 

5-32 

18 

Chair. 

0-72 

965 

32.6 

Channel. 

1-16 

5-41 

183 

Channel. 

0-65 

2-66 

9 

Purlin. 

0-35 

2-66 

9 

T iron. 

0-32 

0-65 

2-2 

Window- 

12 

0-50 

1-7 

Sashes. 

12 

0-89 

3 

Sash bar. 

12 

2-07 

7 

Shoe. 

0-25 

6-66 

22-5 

Girder. 

0-80 


1 0 


21 


This is the beam for which 
the formulas and table are set 
up. Top and bottom are alike. 


This compound Girder is 
made to order of any size, for 
about 6 cents per pound. 


• 20 intermediate tUea, 


21 






















































322 


Bridges. 


FORCES IN STRUCTURES, 

[Important structures should not he undertaken without consulting the works of 
Stoney, Du, Bois or Greene , or others of equal authority .] 

A vertical pole, Fig. 1, presses on its support with a force equal to its 
weight; but when the pole*has an incline, Fig. 2, resting with its upper end 

on a second support at F, the action is divided 
into two equal static moments—namely, 


Fig. 1. 


Fig. 2. 



F:W 
IV a 


a: b, 
IF = 


W a = Fb. 


Fb 


a 


w 


IF e 


IF = weight of the pole, supposed to act in 
its centre of gravity. 

a = lever of the weight TF, drawn hori¬ 
zontally from the fulcrum c to the vertical 
direction of the centre of gravity of the pole. 
F= force holding the pole at F 
b = lever of the force F. 
h = vertical height of the centre of gravity of the pole above the fulcrum c. 
e = distance from the centre of gravity to the fulcrum. 
w= pressure at c in the direction of the pole. 

These notations of letters will be the same in the following figures. 


Fig. 3. 


A Beam Resting Against a Vertical Wall. 

F : IF = a : b, Fb = IF a. 

1 Va Fb Fb 

— W =-, a • 

b a ' 

IF a We 



b = 


F 


IF » 


The horizontal pressure at the fulcrum c is 
equal to the force F. 

The vertical pressure at the fulcrum c is 
equal to TF. 

The diagonal R shows the direction of the 
resultant of the two forces F and TF. 


Force of a Half-Circle Arc Resting on Two Walls. 

Fig- 4. It appears in this case that the arc presses 

vertically down upon the two walls, but such 
is not the case. The formulas are the same 
as those lor Fig. 3. If the walls are not stable 
enough to stand with safety the force F then 
a tie-rod t must be inserted. If the walls are 
sufficiently stable without the tie t, then they 
should not be level on the top, but inclined 
like cd, so as to be at right angles to the di¬ 
rection of the resultant R. 

It is supposed that there is no lateral strength 
in the arc, but that the centre of gravity of 
half the,arc acts like that in Fig. 3. 




f — k - L— - 


r — 


rt-Xr: 






















































Bridges. 


323 


5=E 


A Weight Suspended on a Pair of Shears. 

Fig. 5. The -weight P is hung from the angle of 

two spars. 

Pa 

F= omitting the weight of the spars. 

F = (P + 2 IF), when IF = weight of one 

spar. 

p 

The vertical pressure at c will be — -f IF. 

A 

Thrust on spar = ^ (W + P). 

Truss-Bridge with Two Rafters and a King-Rod. 

Fig. 6. The bridge consists of two rafters IFf, a tie- 

beam cc, king-rod P. The span S is divided 
FA F , into two parts by the king-rod, which bears 

one-half of the load uniformly distributed on 
the bridge. The tie-beam must be strong 
enough to bear with safety half the load in 
half the span. 

■ x P— weight uniformly distributed on the 
bridge, including the weights of the tire-beam 





a —« 


Thrust on each rafter w ■■ 


K w+ i 

Timss-Bridge with two Queen-Rods 


”f(f+4 

( w+ iy 


Fig. 7. 



The truss-bridge consists of a tie-beam cc, 
two rafters IF and l, two queen-rods, and a 
horizontal straining-beam IF'. 

The span S is divided into three equal parts 
by*the queen-rods. Bridges have usually two 
trusses, and therefore the stresses must be 
halved. 

F = ||(p + 2 IF + IF'). 

Thrust on rafter w = — (P + 2 IF+ IF'). 

A 0 


The force F and thrust w are divided on each side of the bridge. 

The tension on the tie-beam is equal to F. 

Truss-Bridge with one King- and two Queen-Rods. 


Fig. 8. 



The span S is divided into four equal parts 
by the king- and queen-rods, making the load 
on each of the rods j P. The weight on the 
king-rod is supported by the diagonal trusses 
tt, by which it is transmitted to the two queen- 
rods—that is, £ P on each—making £+£ = f P 
on each queen-rod. 


F ~i{ p + w )- 


W = weight of all the trusses and horizontal 
straining-beam. 




























































324 


Bridges. 


p = the whole load on the bridge, uniformly distributed, and including 
that of tie-beaiu and flooring. 

t P 

Thrust on t and t, w' — —. 

o 0 

3 ip 

Thrust on rafters w = ' 


8 b ' 


The tension on the tie-beam is equal to F. 


Truss-Bridge of Many Panels. 


The span S is divided into so many parts as to make the lateral strength 
of the tie-beam or lower chord in each division sufficiently strong to bear 
the load with safety. The rectangular space of each division, such as 
B 1 E F, is called a panel. 

The heavy-drawn lines represent compression members, except the lower 
chord A C, and the dotted lines are tension-rods. 

p= whole load uniformly distributed on 
the bridge, including the weight of the bridge 
unloaded. 

p = load and weight on each panel. 
n — number of panels in the span S. 
t = the strain on each tensiou-rod, which 
P 

will be p = —-H x. 


Fig. 9. 



S 


n 


on the upper chord in the same panel. 


It 


The compression of the diagonal struts = -y-. 


Assume the span to be S — 96 feet, which, divided into eight parts, makes 
a = 12 feet; and if the angle of the struts are 60°, then 

b — a tan.60° = 12 X 1.732 = 20.784 feet. 

The length of the struts l = 12 X sec.60° = 12 X 2 = 24 feet. 


Table of Calculation of Compression and Tension. 


a = -, the width of each division or panel. 

n 

b — height of the bridge or panel. I = length of each strut. 

x = the additional weight thrown upon each tie-rod by its neighbor toward 
the middle i?/ of the bridge, that is, half the tension on B D is thrown on 
G H\ the whole tension on G H is thrown on 1J\ the whole tension on IJ 
is thrown on KL\ and lastly, the bearing of the bridge on the pier at A is 
the sum of all the tensions -f ip, which will be half the weight and load of 
the bridge. 

The tension on the lower chord in each panel is equal to the compression 


Tension t on Rods, 
t = p + x. 

B I=p -b o — p. 

G II=p -b ip = lip. 
IJ = p + lip = 2ip. 
K L —p + 2ip = 3ip. 
A l A = ip + 3ip = 4p. 


Compression and Tension. 
A 3i p X 12 


= 2.019 p. 


20.784 


K1= L J ~ (3i + 2i) p 
12 


X 2077-84 = 3 ’ 462 ^ 


[ G — .TII — (3i -b 2j -f li)p 
X-^ = 4.327 p. 


Struts, compression. 

U 

b‘ 


T n _24 X p , 

° - 20 . 784 * ‘ 5 ‘ 7p " 


JII -- 


24 X 1 i p 


20.784 


= 1.732 p. 


20.784 


G B — H1= (3i + 2i fli 
+ »)P X ^7 , -z i = 4.616 p. 


T 24 X 21 p „„„„ 

J x=-ro:iiir= 2M7p - 


A L — 


24 X 31 p 


20.784 


20.784 


: 4.042 p. 


The tension and compression of the chords at mid-span can be checked by 
multiplying the quarter spau into the half load on bridge and dividing by 
height of bridge. 


























Strength of Timber. 


325 


Bowstring Bridge. 


Fig. 10. This is a strong and cheap bridge suitable 

for small spans; it is, in fact, an arched girder, 
generally built of timber, but is also made of 
iron. 

The arch should be made in the form of a 
parabola; the tie-beam must be made strong 
enough to bear the maximum load on each 
panel. 

The bracing between the arch and tie-beam may be all ties, in which case 
the compression and tension will be uniform and alike in the arch and tie- 



beam namely, 5 that is, the maximum load P on the bridge, includ¬ 

ing the weight of the bridge, multiplied by the span S, and the product 
divided by eight times the height h of the girder, is the tension on the tie- 
beam or compression of the arch. 

When the bracing consists of ties and struts, then the tension compression 
is calculated like that for fig. 9. 


STRENGTH OF TIMBER. 


(From Mr. Lasletl’s experiments.) 


Names op Different Kinds op Woods. 

(The strengths mean per square inch of cross-section.) 

Trans¬ 

verse 

breaking 

weight. 

Tensile 

strength. 

Compres¬ 

sive 

strength. 

English Oak (Mean). 

Lbs. 

687 

Tons. 

2.546 

Tons. 

3.337 

T ron wood. 

1273 

4.311 

5.208 

Chow. 

975 

3.214 

5.621 

T ron Ra rk. 

1407 

3.740 

4.601 

Ttlne. Gum. 

712 

2.700 

3.078 

C!nnn.Hifl.n Ash... 

638 

2.453 

2.453 

Average of the. above hardwoods. 

949 

3.161 

3.615 

Baltinfore Oak, African Teak, Green Heart, Sabicu, 
American and Eucalyptus Mahogany, and Eng¬ 
lish Ash Average hardwoods... 

967 

2.120 

3.493 

Dantzic Fir, Riga Fir, Spruce Fir, Larch, Cedar, 
Red Pine, Yellow Pine, Pitch Pine, Kauri Pine. 

683 

1.597 

2.486 

Average fwcptv-nine hard and soft woods. 

830 

2.416 

3.168 



The strength of wood or timber is so varied that it is impossible to give 
correct data, for even in the same set of experiments on the same kind of 
wood, taken from the same tree, the variation reaches 50 per cent, or more. 
Green timber is about half as strong as sun-dried timber. The compressive 
strength per unit of cross-section of the same kind of wood is 10 to 15 per 
cent, greater than the tensile strength. 









































326 


Suspension Bridges. 



SUSPENSION BRIDGES. 


When a chain or flexible cord is suspended between two supports, the form 
of the curve taken by the chain is called a catenary, which resembles very 
much that of the parabola of the conic sections. 

For the same height II and span 5 the catenary has a slightly larger radius 
of curvature at the lowest point than has the parabola; but when a bridge 
is suspended on the catenary, the weight per unit of length of the chain is 
greater at T in the middle of the span S than it is at and near the points of 
suspension. This greater weight depresses the catenary in the middle, so as 
to practically make it the form of a parabola. 

The formulas for the catenary are very difficult to manage, whilst those 
for a parabola are very simple, and will here be used. 


Xoiation. 


IF= total load on the bridge. 

T— tension on thechain in t hecentre. 
t = tension at any point of distance 
D from the centre. 
t' — tension at abutments. 

S= span. 

L — length of chain between the 
piers. 

The formulas will answer for 


H— versed sine, or height of abut¬ 
ments above ceutre of chain. 

D and d = co-ordinates for any point 
of the chain. 

t> = augle of suspension at piers. 
(The angle of the counter-chain 
ought to be equal to that of sus¬ 
pension.) 


any system of weights and measures. 


n S Id 

D = \u 


d = 

tan.u = 
T= 
T = 


II 
4 I f-II 

£2 • * 

<S_ 

4 II’ 

£ IF tan.v. 

JF£ 

8 H’ ’ 
IF S' 


11 8 T ’ ' 

H = generally. 


. 1 . 

. 2 . 

. 3. 
. 4. 
. 5. 

• 6 . 

. 7. 


L = 2 '^■7 + ^ H% > a PP rox * • 8 - 


IF 

t' = — seo.u 


TF= 2 1' cos.v. 

TF 


• • 


cos.v ■ 


2 r 


• • • • 


. 9. 


. 10 . 


. 11 . 


sin.v = — .12. 


A bridge is generally suspended on two or more chains, and the tensions 
2\ t\ and t mean that on all the chains. 

On account of the tension on the backstays, a, b is equal to that on the 
chains or cables at the top of the piers; the vertical pressure on each pier 
will be equal to the weight IF of the whole bridge, to which must be added - 
the weight of the pier itself to get the total pressure on the base. 

The anchorage must be firm enough to hold with safety the tension t'. It 
is best to curve the backstay into the anchorage, as shown at a, and the 
anchors should be set in stone masonry laid in cement. 

























327 


Suspension Bridges. 


Example. The chains or cables for a suspension bridge of S = 400 feet span 
are to hang H— 32 feet below the supports on the piers, and to support a 
weight JF= 600,000 pounds, uniformly distributed. Required the length of 
the cables betweeu the piers, the angle v, and tlio strain l' on the cables at 
the top of the piers? 

The length of the cable will be found by Formula 8. 


- 2 V 


160000 16 

—7— + 7T X 1024 = 409 feet. 
4 9 


The angle v will be found by Formula 3. 

S' 400 „ _ 

ta™ 3.1111. 

This tangent answers to the angle t> = 72° 11'. The tension l' at the piers 
will then be, Formula 9: 

t — ~ sec.v = —X 3.2G83 = 9S0,490 pounds. 

If the bridge is to be suspended on say four cables that is, two on each 
side of the roadway, then each cable must be strong enough to bear with 
safety 980490 : 4 = 245142.5 pounds. 

The tension T in the middle of the span is found by Formula 4, namely: 

T=i W tan.v = £ X 600000 X 3.1111 = 933,333 pounds. 

The ultimate strength of the bridge, so far as the strength of the cables is 
concerned, will be found by Formula 10, in which case t' means the ultimate 
strength of the cables, sav 980490 X 6 = 5,882,940. 

1F= 2 X 5882940 X 0.30597 = 4,532,300 pounds. 

The ordinates D for the curve can be calculated for assumed values of the 
abscissa d by Formula 1. 

S 2 

The parameter of the parabola is p = . 

£2 

The focus of the parabola is d = —. 

In H 

d = height of focus above the lowest part of the cable. 


Dimensions and Cost of Large Bridges. 


Name of Bridge. 

Maxi 

App. 

Height. 

mum. 

Span. 

Length. 

Reputec 

Totul 

Amount. 

Cost. 
Per ft. 
run. 

Nature of Bridge. 

Britannia. 

Ft. 

125 

Ft. 

460 

Ft. 

1511 

$ 

3,009,325 

& 

1990 

Two lines R.R. Tubular. 

Charing Cross.. 

50 

154 

1365 

900,000 

655 

Four lines R. R. Double 

Boyne. 

90 

264 

550 

700,000 

1270? 

Warren. 

Four lines R. R. Lattice. 

Crurulin. 

200 

150 

1800 

195,000 

105 

Two lines R. R. Lattice 

Craige Machie. 

20 

200 

413 

61,000 

145.5 

on open-work piers. 
One line R. R. Lattice 

Grand River ( 

130 

12 

620 

150,000 

250 

and jrlate girders. 

One line R. R. Plate 

(Mauritius) J 

Deepdale. 

150? 

60 

740 

101,330 

135 

girders. 

Two lines R. R. Lattice 

Westminster.... 

20 

120 

1160 

1,175,000 

1010 

on open-work piers. 
Road. Cast and wrought 

Frpeburg. 

167 


808 

120,000 

400,000 

145 

iron arch, 83 feet wide. 
Wire-rope suspension 
bridge. Road only. 
Wire-rope suspension 

Niagara. 

245 

821 

800 

500 

Landore. 

75 

110 

1760 

143,600 

16.3 

bridge. Road and R. R. 
Wooden trusses. 

East River. 

154 

1596 

10043 

9,000,000 

900 

Wire-rope suspension. 





































328 


Catenary. 


THE CATENARY. 



i 


The curve formed by a flexible rope or chain 
suspended from two points is called a catenary 
or chain-line. 

Let v denote the angle of the curve with the 
vertical in any point P whose abscissa is x 
and ordinate jr, / = length of the curve OP; 
1F = weight of the whole chain ; P= force of 
tension at the angle v. The formulas for the 
catenary will then be 


IsinJh) , , ., 


& — s i n 2 V hypJog.cot.?o. 

X — 

1 sin.^v , 

x— . . ( cosec.v — 1 ). 

stn.2v 

1= 

x hyp.log.cot.jiV 


( cosec.v — 1 ) 

Jr =■ 


y( cosec.v — 1) 

hyp. log.cot.?v ’ 
x sin.2v 

sin.v( 1 — sin.v) * 
IF sin.v 
sin. 2v 




The formulas for the catenary are very difBcult to manage, because the 
angle v must be given; but by the aid of the following table the solution 
becomes very simple: 


Table Tor tbe Catenary Curve. 


Angle 

Abscissa 

Ordinate 

Curve 

1 

l_ 

V. 

Z . 

y • 

l. 

X' 

V 

30 

1.00000 

1.31690 

1.73210 

1.3169 

1.3153 

40 

0.55573 

1.01068 

1.19175 

1.8186 

1.1792 

45 

0.41421 

0.88137 

1.00000 

2.1278 

1.1346 

50 

0.30540 

0.76291 

0.83910 

2.4981 

1.1000 

54 

0.22078 

0.65284 

0.70021 

2.9570 

1.0725 

60 

0.15470 

0.54930 

0.57735 

3.5507 

1.0511 

62 

0.13257 

0.50940 

0.53171 

3.8425 

1.04:38 

64 

0.11260 

0.47021 

0.48773 

4.1759 

1.0372 

66 

0.09484 

0.43169 

0.44523 

4.5518 

1.0314 

68 

0.07853 

0.39376 

0.40403 

5.0141 

1.0261 

70 

0.06418 

0.35637 

0.36397 

5.5527 

1.0213 

71 

0.05762 

0.33786 

0.34433 

5.8636 

1.0192 

72 

0.05146 

0.31946 

0.32492 

6.2079 

1.0171 

73 

0.04569 

0.30116 

0.30573 

6.5914 

1.0152 

74 

0.04030 

0.28296 

0.28675 

7.0213 

1.0134 

75 

0.03528 

0.26484 

0.26795 

7.5068 

1.0117 

76 

0.03061 

0.24681 

0.24933 

8.0631 

1.0102 

77 

0.02630 

0.22887 

0.23087 

8.7023 

1.0088 

78 

0.02234 

0.21099 

0.21256 

9.4445 

1.0073 

79 

0.01872 

0.19318 

0.19438 

10.820 

1.0062 

80 

0.01543 

0.17542 

0.17633 

11.372 

1.0052 

81 

0.01247 

0.15773 

0.15838 

12.654 

1.0041 

82 

0.00983 

0.14008 

0.14054 

14.254 

1.0033 

83 

0.00751 

0.12248 

0.12278 

16.309 

1.0025 

84 

0.00551 

0.10491 

0.10510 

19.046 

1.0018 

85 

0.00382 

0.08738 

0.08749 

22.874 

1.0013 

86 

0.00244 

0.06987 

0.06993 

28.613 

1.0008 

87 

0.00137 

0.05238 

0.05241 

38.171 

1.0005 

88 

0.00061 

0.03491 

0.03492 

57.279 

1.0002 

89 

0.00015 

0.01745 

0.01745 

114.586 

1.0000 
































Catenary. 


329 


Application of the Catenary Table. 

The chain for a suspension bridge of 300 feet, span is to hang 60 feet below 
its supports on the piers. The chain is to support a weight of 52,000 pounds, 
uniformly distributed in its length. Required the length of the chain and 
the angle v and strain at the supports? Half the span or y — 150 feet, for 
which x = 60 feet. 




which corresponds nearly with an angle of i> = 50° in the table, and the 
length of half the chain will be / = 150 X 1.1 = 165 feet. 

The strain at the supports will be 


)V s-in.v 
sin. 2,v 


52000 X sin.5 0° 
H».100° 


= 40449 pounds. 


The ordinates x and y and the length l for any angle v in the table are 
found as follows: 

When v = 50° at the support, find x and y where v = 70° ? 

0.30540 : 0.06418 = 60: x. x = ° = 12.609 feet. 

O.oOolO 

0.76291:0.35637 = 150:2/. y = °‘ 35 . 6 ??* 3 — = 70.068 feet. 

u./ozyi 


Length l = 70.068 X 1-0213 = 71.56 feet. 

The ordinates can thus be calculated for a sufficient number of points in 
the catenary to define the course of the curve. 

The strain at the lowest point or centre of the catenary will be 

vj tan.v = 26000 X ton.50° = 30984 pounds ; 

when v = angle at the piers, and w= half the weight on the whole chain. 

The catenary is not a line of the conic sections; its figure has the appear¬ 
ance of a parabola, but is a little fuller at the vertex. 

All the curves of the conic sections are of the second order, or of the ex¬ 
ponent n = 2; whilst the exponent of the catenary is nearly n = 2.3. 

«/- 

The formula for any parabola is y = y p x, 

2 3 - 

when that of the catenary will approach y = y px. 

Length of the curve O P or l = |(2 y + 1 /y£ -V 9x2). 












830 


Stone Bridges. 


STONE BRIDGES. 



The abore illustration represents an ordinary stone bridge with one 
elliptic and one circular arch. The construction of the elliptic arch can be 
made by the methods explained in geometry or in conic sections, or by ordi¬ 
nates, as shown on pages 150 and 151. When the rise h is £th or moreof the 
span (S', circle arcs maybe resorted to in constructing the ellipse; but when 
A< £th S, the ellipse should be very accurately constructed by Formula 188 
or 189, page 179. The radius of curvature of the intrados of the ellipse at the 

key-stone is R = —, and the smallest radius of curvature at the sides is 


Ufl 


Ah' 


r = ——. The depth of arc at the crown or key-stone should be 0.35 \/ R 

when the bridge has only one span, and for several spans the depth of key¬ 
stone should be at least 0.4 \/R, according to average practice. 

The involute I of the ellipse should be drawn as shown by the dotted lines, 
in order to know the proper direction of the seams in the arch-stones, which 
should tangent the involute. 

Having given the span S and rise A of a circular span, the radius of curv- 

h 

ature will be R — -f —. 

o h Z 

Bridge Glossary. 


Abutment is the stone-work against 
which ilie arches of a bridge abut. 
(See Piers.) 

Arch-stones, the stone ‘blocks of 
which the arch is built. 

Back, the upper or outside surface 
of the arch. 

Crown, the vertex of the arch. 

Extrados, the same as back or the 
outside surface of the arch. 

Faces, the two projecting areas of 
the arch. 

Haunches, the extrados from the 
crown to abutment. 

Intrados, the inside surface of the 
arch. 

Key-stone, the centre stone at the 
crown of the arch. 

Piers, the stone-work upon which 
a bridge rests. When the bridge is 


in the form of a girder, like Figs. 00, 
00, which does not press sideways on 
the piers, the piers are not called 
abutments. 

Rise; the vertical height h of the 
intrados above the abutments. 

Skew-backs, the seats for the arch. 

Soffit, the same as intrados. 

Span, the horizontal distance S be¬ 
tween the piers. 

Spandrels, the filled-in space above 
the piers. 

Spandrel-fillings, the materials filled 
in the spandrels. 

Springing line, the inner junction 
between the arch and the pier. 

Springs, the same asspringing lines. 

Springers, the foot-soles of the 
arch. 

Voussoirs, the same as arch-stones. 





































































331 


Roofs of Wood and Iron. 


ROOFS OF WOOD AND IRON. 



The Figs. 1 and 2 illustrate the common form of wooden roofs, as constructed 
over spans ot from 30 to 80 feet. When the span exceeds 60 feet, a proportionate 
number of struts and tie-rods must be inserted, as shown by the dotted lines, or as 
illustrated for iron roofs. 


Table of Timber Dimensions, In Indies, for Roofs over Spans 

from 30 to 80 feet. 


Name of timbers. 

30 

35 

40 

45 

Span 

50 

in feet. 

55 

60 

70 

80 

Tie-beams, 

a 

5X0 

6X7 

0X8 

7X8 

8X9 

8 X12 

9X11 

10X11 

10X12 

Truss rafter, 

b 

5X5 

5X6 

6X7 

7X7 

8 X8 

8 X 9 

9 X 

9 

9X10 

10X11 

Collar-beams 

c 

5X5 

5X6 

6X7 

7X7 

8 X8 

8 X 9 

9 X 

9 

9X10 

10X11 

Com. rafter, 

d 

2X5 

2X5 

2X6 

2X6 

2X6 

2 X 7 

2 X 

7 

2iX 8 

3 X 9 

Purlins, 

e 

5X6 

5X6 

5X7 

6X7 

6 X8 

6 X 8 

6 X 

9 

6 X 9 

6 X 9 

Struts, . . 

f 

3X4 

3X5 

3X6 

4X7 

4X8 

5 X 8 

5 X 

9 

6 X 9 

6 X 9 

King’s rod, 

h 

1 

1 

1 

n 

u 

H 

H 


12 

2 

Bolts, . . 


2 

2 

2 

2 

£ 

l 

n 


H 

11 


Lioad on Roofs in Pounds per Square Foot, exclusive of 

Framing. 


Lead covering. 

Pounds. 

. 8 

Tiles, .... 

Pounds. 

9 to 16 

Zinc covering, . 

2 

Boarding, f thick, 

3 

Corrugated Iron,. 

. 3.5 

Boarding, 1£ thick, 

6 

Slates, 

. . 10 

Pressure of wind, 

40 


In high latitudes the roofs may be covered with snow, which makes a pressure 
of 10 pounds per square foot per foot of depth of the snow. 

Empirical statements of sizes of timber are valuable only as a cheek. Each 
truss should be graphically solved as set forth by Du Bois, On Roof and Bridge 
Construction , or some other writer of equal authority. 

Engineers cannot be too careful in the calculations for structures upon 
whose stability life depends. 

In addition to the graphical calculation of the framed structure, careful cal¬ 
culation of the strength of each member should be made. The rolling or 
partial loads should also be carefully calculated and tabulated, to avoid pos¬ 
sibility of weak points. The computation and erection of bridges require 
broad and thorough knowledge of statical theories, and the engineer under¬ 
taking such work without adequate knowledge should be held as a murderer 
should life be lost by his negligence or ignorance. 















































332 


Strains on Roofs. 



STRAINS ON ROOFS. 

The above figures illustrate four different kinds of pointed iron roofs, of which 
figure 3 is most in use. 


Maxwell’s Graphical Method (Du Bois). 


Solution of Fig. 3. 



Determine the whole load on each truss. Take the load as concentrated at 
the apexes, The reaction at each pier is £ of 7 h = W,. To any convenient 
scale laydown the line 3 \h. Draw lines parallel to” C and T from the ex¬ 
tremities of 3 hh. They will intersect and give the stresses to the same scale 
for T and C. Take the first upper vertex; its polygon of forces is Cpcs in 
the diagram. Take the first lower vertex ; its polygon of forces is Ts Q't. 

In this manner treat each apex, measuring the stresses on each member. 

For a thorough exposition of this method and others used in framed struc¬ 
tures, suspension bridges, stone arches, etc., etc., see “The Strains in Framed 
Structures” (Du Bois). 






















Steam Hammer. 


333 



BOLLMAN’S AMERICAN TRUSS BRIDGES. 

Notation of Letters. 


\V = total load uniformly distributed on 
the bridge. 

w = load on each point of suspension. 

S= span. 

D and d = distances from abutments A 
and B to point of suspension. 

A and a = cross areas of the tension 
and counter-tension rods in 
square inches. 


R and r = lengths of tension and coun¬ 
ter-tension rods. 

H = depth of truss, which is usually 
one-seventh of the span. 

N — number of points of suspension. 

2'and t = tensions on the rods R and r 
respectively. 

C = compression on the top at centre. 


These formulas will answer for any system of weights and measures. 


— TF ji w J) R 

~(N+ 1)’ SH ' 


wdr 

SH' _ 82f’ 


When T and t are tons, A and a = square inches, Z>, d, S, H, R and r = feet, 
then 


w D R _ wdr 

5 SH’ 5 SH 


STEAM HAMMER, 

A heavy steam hammer with short fall produces a better forging than a 
light hammer with a high fail, although the dynamical work may be the same 
in both cases. This is accounted for by the inertia of the ingot forged. 

The effect of blows of a heavy hammer and short fall will penetrate through 
the metal, and nearly with the same effect on the anvil side, while a light 
hammer and high fall will affect the metal on or near the surface of the blow, 
because most of the work is in the latter case discharged in the inertia of the 
ingot forged. In forging a large shaft, it is generally piled up with iron bars 
sometimes rolled into a segment form to suit the pile. When placed under 
the hammer in a welding heat, very light and gentle blows are first given. The 
work of a light hammer will be discharged in the bars nearest to the blow, 
while a heavy hammer will squeeze the whole mass together throughout, and 
a sound welding will be produced. 

The additional expense of a heavy hammer is fully compensated by the waste 
of labor and materials under a small one. I have often seen, in broken shafts, 
the bars in the centre as clear and unwelded as when first piled, which is a 
sure indication that the shaft has been forged by a too light hammer. In 
crank-shafts for propeller engines forged under a light hammer, when brought 
to the machine-shop the best part of the metal is worked away by planing 
and turning,and the poorest left for the engine; but if forged under a heavy 
hammer, the difference in quality of metal will not be so great. 

Weiglit of Steam Hammers. 

The weight of a steam hammer in pounds should be at least eighty times 
the square of the diameter of the shaft in inches. 


























334 


Bridges. 





Fine Lines in Tension and Thick Lines in Compression. 

The Warren girder consists of equilateral triangles formed by the trusses 
and ties, which make divisions or bays in the span. 

The depth of the girder is 0.10825 of the span. 

For a uniformly distributed load Stoney’s method may be used. For a roll¬ 
ing load, such as a train of cars, a table will have to be made. See Du Bois, 
Graphical Statics The theory is too elaborate to be discussed in a general 
Pocket-Book, 


Weight of" one pair of Warren’s Girders In tons, 

far a single track of railway on the top or on the bottom (approximate). 

Span of the girder in feet. 


the 

50 

60 

70 

80 

90 

100 

110 

120 

130 

140 

150 

160 

Top, 

11 

15 

18 

23 

27 

32 

38 

44 

51 

58 

66 

75 

Bottom, 

15 

19 

24 

29 

35 

41 

48 

56 

64 

72 

80 

89 


Table of Dimensions in inches of Rolled Iron, 

for roofs on spans from, 30 to 80 feet (figures 3 to 6, page 332). 


Name of 
iron. 

30 

s 

40 

pan in feet. 

50 

60 

70 

80 

Rafter T-iron, 

L 

2?X2iX£ 

3£X3X£ 

4X3£Xf 

5X4£X* 

5£X6Xf 

6X6Xf 

Struts T-iron, 

s 

2£X2yXi 

3X2£X£ 

3X3X1 

4X4X1 

4£X4£XI 

5X44Xf 

King bolt, 

K 

1 

H 

n 

H 

If 

If 

Queen bolt, -j 

Q 

1 

T 

? 

1 

n 

H 

H 

l 

O' 

f 

f 

f 

F 

i 

U 

H 

Tie-rod, \ 

T 

H 

if 

H 

ii 

if 

H 

l 

V 

l 

H 

U 

if 

H 

if 

Weight, lbs., 


1500 

3000 

4800 

7000 

9550 

12400 


The last line shows the approximate weight in pounds of each principal when 
the rise of the roof is 0.2 of the span. 




















































Phcenix Beam. 


335 


ELEMENTS OF PHCENIX BEAMS. 

^Rigidity is a different quality from strength. A beam may be quite 
strong enough to carry a given load, but under this load it may deflect more 
than is desirable. About one-thirtieth of an inch per foot of clear span is 
the usual maximum of deflexion that is permissible. Under ordinary loads 
this is attained when the clear span is about 2(5 times the depth of the beam, 
and the dividing lines in the table show where this limit is passed for each 
beam. 

Like the load-factor, the deflexion-factor is dependent upon the depth and 
flange area of the beam to which it is to be applied; the general formula for 
the deflexion of any beam under an equally distributed load being 

.004 \\\I? 

T” + tK 

By inserting the values proper to each beam, the results given in the fol¬ 
lowing tables have been obtained. A close approximation to the actual de¬ 
flexion may be obtained by dividing the square of the length of the span in 
feet by 62 times the depth of the beam in inches. 


Definition of Terms used in Formulae. 

W — Equally distributed load on any beam in net tons. 

L = Length of clear span, expressed in feet. 
a = Area of top, or bottom, flange, in square inches. 

a' = Area of stem of beam, in square inches. 

D — Effective depth of beam, expressed in feet. 

d — Effective depth of beam, expressed in inches. 

S — Safe strain per square inch of effective section 
2000 pounds. 

S = Deflexion in inches at middle for a central load. 

8' — Deflexion in inches at. middle for a uniformly distributed load. 

General formula for any I beam under an equally distributed load : 

8fl(a + 7)<9 

w= —-_—. 

L 

Now, in this formula, it is only necessary to insert the proper values for 
“effective depth” and “effective section,” given in the table for each par¬ 
ticular beam, in order to determine its strength. 

The load-factor for each beam is thus dependent upon its depth and the 
quantity of metal in its flanges. This load-factor, when divided by the clear 
span, gives a quotient that indicates the number of tons that the beam will 
carry with perfect safety. The table gives the safe loads per foot of clear 
span for various-sized beams. 


(»+f) 


in tons of 













336 


Elements op Phcenix Beams. 


TABLE I—ELEMENTS OF PHCENIX BEAMS. 




Dimeusions, Inches. 

Area, Square Inches. 

Beam. 

Width 

Average 

Thickness 


a* of 

Sum of 



of 

Flange. 

Thickuess 
of Flauge. 

of 

Stem. 

Flange. 

Stem. 

, a ' 
ct-f"- 

6 

15" 

200 

5 16 

1.156 

.65 

6.142 

7.715 

7.428 

15" 

150 


.911 

.50 

4.330 

6.340 

5.386 

12" 

170 

5* 

1.050 

.59 

5.777 

5.446 

6.684 

12" 

125 

4 » 

.802 

.49 

3.810 

4.880 

4.623 

10£"135 

5 

.875 

.50 

4.375 

4.750 

5.166 

IQP'105 

44 

.745 

.44 

3.353 

3.793 

3.986 

9" 

150 

6t 

1.039 

.60 

5.586 

3.828 

6.224 

9" 

84 

4 

.700 

.40 

2.800 

2.800 

3.261 

9" 

70 

34 

.680 

.31 

2.381 

2.238 

2.754 

8" 

81 

44 

.625 

.38 

2.812 

2.476 

3.225 

8" 

65 

4 

.527 

.35 

2.109 

2.282 

2.489 

7" 

69 

4 

.625 

.37 

2.500 

1.900 

2.816 

7" 

55 

34 

.507 

.35 

1.775 

1.949" 

2.100 

6" 

50 

34 

.531 

.31 

1.858 

1.284 

2.072 

6" 

40 

2* 

.517 

.25 

1.421 

1.158 

1.614 

5" 

36 

3 

.400 

.30 

1.200 

1.200 

1.400 

5" 

30 

24 

.375 

.25 

1.000 

1.000 

1.166 

4" 

30 

24 

.410 

.25 

1.135 

.730 

1.257 

4" 

18 

2 

.281 

.21 

.562 

.682 

.676 


Beam. 

Effective 

D Feet. 

Depth, 

<Z Inches. 

Load Factor. 

>B ( a + -f) s 

When S— 6 Tons. 

Deflection Factor. 
(•♦-?> 

15" 200 

1.150 

13.S0 

410 

1415 

15" iso; 

1.170 

14.04 

302 

1062 

12" 170 

.910 

10.92 

292 

797 

12" 125 

.930 

11.16 

208 

576 

10i"135 

.800 

9.62 

178 

478 

10i"105 

.812 

9.74 

155 

378 

9" 150 

.658 

7.90 

197 

388 

9" 84 

.691 

8.30 

108 

225 

9" 70 

.698 

8.38 

92 

193 

8" 81 

.610 

7.37 

94 

175 

8" 65 

.618 

7.42 

74 

137 

7" 69 

.530 

6.37 

72 

114 

7" 55 

.537 

6.44 

54 

87 

6" 50 

.456 

5.47 

45 

62 

6" 40 

.458 

5.50 

35 

49 

5" 36 

.383 

4.60 

2-5 

30 

5" 30 

.385 

4.62 

21 

25 

4" 30 

.298 

3.58 

18 

16 

4" 18 

.304 

3.65 

10 

9 


































Elements of Phcenix Beams. 337 


PHCENIX BEAMS. 

Their Adaptation and Duty ns Flooring Joists. 

Clear 

Span. 

8' 

apart 

3 V 

apart 

4' 

apart 

4*' 

apart 

5' 

apart 

5*' 

apart 

6' 

apart 

10 feet. 
Load lbs. 

I 

30 □' 

4,200 

35 a' 

4,900 

6' 

40 □' 
5,600 

r 

45 □' 
6,300 

50 □' 

7,000 

55 □' 

7,700 

7 or 8" 

60 □' 
8,400 

12 feet. 

Load lbs. 

I 

36 □' 

5,040 

6 or 

42 

5,SS0 

7" 

48 

6,720 

54 

7,560 

7" 

60 

8,400 

66 

9,240 

8 

72 

10,080 

" 

14 feet. 

Load lbs. 

I 

42 □' 
5,880 

7 or 

49 

6,860 

8" 

56 

7,S40 

63 

8,820 

8 or 9" 7« 

70 

9,800 

77 

10,780 

9' 

84 

11,760 

'70 

16 feet. 

Load lbs. 

I 

48 □' 
6,720 

8' 

56 

7,840 

64 

8,960 
9" 70 

72 

10,0S0 

9' 

80 

11,200 

84 

88 

12,320 

10* 

96 

13,440 

'105 

18 feet. 

Load lbs. 

I 

54 □' 
7,560 

8 or 9" to 

63 

8,820 

9' 

■72 

10,080 

84 

81 

11,340 

90 

12,600 

10 * 

99 

13,S60 

'105 

108 

15,120 

20 feet. 

Load lbs. 

I 

60 □' 
8,400 

9 84 or 10* 

70 

9,800 

80 

11.200 

10 * 

90 

12,600 

'105 

100 

14,000 

110 

15,400 

12' 

120 

16,800 

125 

22 feet. 

Load lbs. 

I 

66 □' 
9,240 

77 

10,780 

10*" 105 

88 

12,320 

99 

13,860 

110 

15,400 

12" 125 

121 

16,940 

132 

18,480 

12" 170 

24 feet. 

Load lbs. 

I 

72 □' 

10,080 

10* or 1 

84 

11,760 

2" 125 

96 

13,440 

12" 

108 

15,120 

125 

120 

16,800 

12 " 

132 

18,480 

”0 or 15' 

144 

20,160 

150 

26 feet. 

Load lbs. 

I 

78 □' 
10,928 
10* or 12 

91 

12,740 

12 " 

104 

14,560 

125 

117 

16,380 

12" 

130 

18,240 

170 or 15' 

143 

20,020 

160 

156 

21,840 

15" 150 

28 feet. 

Load lbs. 

I 

84 □' 
11,760 

12 " 126 or 

98 

13,720 

15" 150 

112 

15,6S0 

72" no 0 

126 

17,640 

r 15" 150 

140 

19,600 

15" 150 

154 

21,560 

15" 

168 

23,520 

200 

30 feet. 

Load lbs. 

I 

90 0 ' 

12,600 

12 or 15150 

105 

14,700 

12" 170 0 

120 

16,800 

r 15" 1 ®® 

135 

18,900 

15" 160 

150 

21,000 

165 

23,100 

15 " 200 

180 

25,200 

In the above table the load is taken at 140 lbs. per □ foot of floor. 


22 




































































































































338 


Elements of Phcenix Beams. Table II. 



15" 200 Lbs. 

15" 150 Lbs. 

12" 170 Lbs. 

12" 125 Lbs. 


nr 410 

ITT 802 

«r- 292 

208 

IV — - - 



' _ L • 

L 



L ' 



L. ' 

^ar Span, 
in Feet. 

m 
C5 C 

fcr • 

% G 
§.2 

cc £ 

CD & 

;. of Beam, 
in Lbs. 

*© W3 

cc 2 

9 3 

«® - 

_60 _• 

H § 
O-S 

£ " 

U rzz 

;. of Beam, 

in Lbs. 

"D rn 

3 G 
9 © 

—i pH 

to . 
a o 

o.2 

©* 

03 O 
<D 03 

of Beam, 

in Lbs. 

*53 go 

C5 £ 
© O 

.© «r 

c g 
o.2 
e.« 

<rj O 

IV V 

t’g 

t. of Beam, 

in Lbs. 

O 

Zk 

o r\ 

£ 


c M 

£ 



£ 



£ 



// 



n 



// 



tf 


10 

41.0 

.116 

667 

30.2 

.114 

500 

29.2 

.147 

567 

20.8 

.144 

417 

11 

37.2 

.140 

733 

27.4 

.138 

550 

26.6 

.177 

623 

18.8 

.174 

458 

12 

34.2 

.167 

800 

25.2 

.154 

600 

24.3 

.210 

680 

17.3 

.207 

500 

13 

31.6 

.196 

867 

23.2 

.182 

650 

22.4 

.246 

737 

16.0 

.243 

542 

14 

29.3 

.227 

933 

21.6 

.212 

700 

20.9 

.286 

793 

14.9 

.282 

583 

15 

27.4 

.261 

1000 

20.0 

.254 

750 

19.4 

.328 

850 

13.8 

.325 

625 

16 

25.6 

.296 

1067 

18.9 

.289 

800 

18.3 

.374 

907 

13.0 

.360 

667 

17 

24.1 

.334 

1133 

17.8 

.327 

850 

17.2 

.423 

963 

12.2 

.408 

708 

18 

22.8 

.376 

1200 

16.8 

.367 

900 

16.2 

.475 

1020 

11.5 

.459 

750 

19 

21.6 

.419 

1267 

15.9 

.410 

950 

15.4 

.530 

1077 

10.9 

.513 

792 

20 

20.5 

.463 

1333 

15.1 

.455 

1000 

14.6 

.587 

1133 

10.4 

.578 

833 

21 

19.5 

.510 

1400 

14.4 

.502 

1050 

13.9 

.648 

1190 

9.9 

.636 

875 

22 

18.6 

.560 

1467 

13.7 

.551 

1100 

13.3 

.711 

1247 

9.4 

.698 

917 

23 

17.8 

.612 

1533 

13.1 

.602 

1150 

12.7 

.777 

1303 

9.0 

.763 

958 

24 

17.1 

.667 

1600 

12.6 

.656 

1200 

12.2 

.846 

1360 

8.7 

.832 

1000 

25 

16.4 

.725 

1667 

12.1 

.712 

1250 

11.7 

.918 

1417 

8.3 

.903 

1042 

26 

15.8 

.785 

1733 

11.6 

.769 

1300 

11.2 

.992 

1473 

8.0 

.977 

1083 

27 

15.2 

.846 

1S00 

11.2 

.828 

1350 

10.8 

1.068 

1530 

7.7 

1.053 

1125 

28 

14.6 

.906 

1867 

10.8 

.889 

1400 

10.4 

1.147 

1587 

7.4 

1.131 

1167 

29 

14.1 

.972 

1933 

10.4 

.942 

1450 

10.0 

1.230 

1643 

7.1 

1.211 

1208 

30 

13.7 

1.04 

2000 

10.0 

1.017 

1500 

9.7 

1.314 

1700 

6.9 

1.294 

1250 












10J" 135 Lbs. 

I0i 

" 105 Lbs. 

9" 150 Lbs. 

0" 84 Lbs. 


W 178 

nr- 155 

nr 197 

nr 108 


” “ L 


L 



L ' 



L ’ 

10 

17.8 

.149 

450 

15.5 

.164 

350 

19.7 

.203 

500 

10.8 

.192 

280 

11 

16.2 

.180 

495 

14.0 

.197 

385 

17.8 

.243 

550 

9.8 

.231 

308 

12 

14.8 

.214 

540 

12.9 

.236 

420 

16.4 

.296 

600 

9.0 

.276 

336 

13 

13.7 

.251 

585 

11.8 

.278 

455 

15.2 

.347 

650 

8.3 

.324 

364 

14 

12.7 

.291 

630 

11.1 

.322 

490 

,14.1 

.402 

700 

7.7 

.376 

392 

15 

11.8 

.333 

675 

10.2 

.364 

525 

13.2 

.459 

750 

7.2 

.432 

420 

16 

11.1 

.380 

720 

9.7 

.414 

560 

12.3 

.530 

800 

6.7 

.488 

448 

17 

10.5 

.431 

765 

9.1 

.470 

595 

11.6 

.585 

850 

6.3 

.550 

476 

18 

9.9 

.481 

810 

8.6 

.528 

630 

10.9 

.654 

900 

6.0 

.622 

504 

19 

9.3 

.533 

855 

8.1 

.589 

665 

10.3 

.737 

950 

5.7 

.695 

532 

20 

8.9 

.595 

900 

7.7 

.652 

700 

9.8 

.807 

1000 

5.4 

.768 

560 

21 

8.5 

.658 

945 

7.3 

.719 

735 

9.3 

.891 

1050 

5.1 

.839 

588 

22 

8.1 

.721 

990 

7.0 

.788 

770 

8.9 

.980 

1100 

4.9 

.927 

616 

23 

7.7 

.784 

1035 

6.7 

.862 

805 

8.5 

1.07 

1150 

4.7 

1.01 

644 

24 

7.4 

.856 

1080 

6.5 

.941 

840 

8.2 

1.17 

1200 

4.5 

1.10 

672 

25 

7.1 

.928 

1125 

6.2 

1.025 

875 

7.9 

1.27 

1250 

4.3 

1.19 

700 

26 

6.8 

1.00 

1170 

5.9 

1.105 

910 

7.6 

1.38 

1300 

4.1 

1.27 

728 

27 

6.6 

1.08 

1215 

5.7 

1.187 

945 

7.3 

1.48 

1350 

3.9 

1.36 

756 

28 

6.3 

1.16 

1260 

5.5 

1.271 

980 

7.0 

1.59 

1400 

3.8 

1.48 

784 

29 

6.1 

1.24 

1305 

5.3 

1.360 

1015 

6.8 

1.70 

1450 

3.7 

1.60 

812 

30 

5.9 

1.33 

1350 

5.1 

1.455 

1050 

6.6 

1.83 

1500 

3.6 

1.73 

840 
















































































Elements op Phoenix Beams. Table II. 339 




9" 70 Lbs. 

w — 92 . 

8 

W 

" 81 Lbs. 

T _ 94 

1 

S" 65 Lbs. 

v 74 

1 

7" 69 Lbs. 

72 


L 

L 


V ~ L ' 

v — --- 

L ' 

Clear Span, 
in Feet. 

if. 

73 2 

ss 

" K 

Correspou’g 

Deflection. 

Wt. of Beam, 
in Lbs. 

Safe Load, 

Net Tons. 

Correspon’g 

Deflection. 

C2 . 

O a; 

Cl- M 

C ^ 

£ 

Safe Load, 

Net Tons. 

Correspon’g 

Deflection. 

Wt. of Beam, 

in Lbs. 

Safe Load, 

Net Tons. 

Correspon’g 

Deflection. 

Wt. of Beam, 

in Lbs. 



// 



n 



rr 



// 


10 

9.2 

.190 

233 

9.4 

.215 

270 

7.4 

.215 

216 

7.2 

.252 

230 

11 

8.4 

.231 

256 

8.5 

.258 

297 

6.8 

.264 

238 

6.5 

,303 

253 

12 

7.7 

.275 

280 

7.8 

.308 

324 

6.2 

.312 

260 

6.0 

.363 

276 

13 

7.0 

.318 

303 

7.2 

.361 

351 

5.7 

.365 

282 

5.5 

.424 

299 

14 

6.7 

.380 

326 

6.7 

.420 

378 

5.3 

.424 

303 

5.1 

.491 

322 

15 

6.2 

.432 

350 

6.2 

.478 

405 

4.9 

.475 

325 

4.8 

.568 

345 

16 

5.7 

.448 

373 

5.9 

.546 

432 

4.6 

.549 

347 

4.5 

.645 

368 

17 

5.4 

.548 

396 

5.5 

.617 

459 

4.3 

.616 

368 

4.2 

.724 

391 

18 

5.1 

.615 

420 

5.2 

.693 

486 

4.1 

.697 

390 

4.0 

.818 

414 

19 

4.8 

.690 

443 

5.0 

.783 

513 

3.9 

.780 

412 

3.8 

.914 

437 

20 

4.6 

.761 

466 

4.7 

.859 

540 

3.7 

.863 

433 

3.6 

1.01 

460 

21 

4.4 

.842 

490 

4.5 

.952 

567 

3.5 

.946 

455 

3.4 

1.10 

483 

22 

4.2 

.925 

513 

4.2 

1.02 

594 

3.4 

1.05 

477 

3.2 

1.19 

506 

23 

4.0 

1.01 

536 

4.1 

1.14 

621 

3.2 

1.13 

498 

3.1 

1.32 

529 

24 

3.8 

1.08 

560 

3.9 

1.23 

648 

3.1 

1.25 

520 

3.0 

1.45 

552 

25 

3.6 

1.16 

583 

3.7 

1.32 

675 

2.9 

1.32 

642 

2.9 

1.59 

575 

26 

3.5 

1.27 

606 

3.6 

1.44 

702 

2.8 

1.43 

563 

2.8 

1.72 

598 

27 

3.4 

1.38 

630 

3.5 

1.57 

729 

2.7 

1.55 

585 

2.7 

1.86 

621 

28 

3.3 

1.49 

653 

3.3 

1.65 

756 

2.6 

1.66 

607 

2.6 

2.00 

644 

29 

3.2 

1.60 

676 

3.2 

1.78 

783 

2.5 

1.77 

628 

2.5 

2.14 

667 

30 

3.1 

1.73 

700 

3.1 

1.91 

810 

2.4 

1.88 v 

650 

2.4 

2.27 

690 

- j- 


1" 55 Lbs. 

6" 40 Lbs. 

5" 30 Lbs. 

4" 30 Lbs. 



54 


35 


21 


18 


W 


IF — 

# 

IF — 


IF = — 




L 


L 




L 


J 

L 

10 

5.4 

.248 

183 

3.5 

.286 

133 

2.1 

.336 

100 

1.80 

.448 

100 

11 

4.8 

.293 

201 

3.2 

.348 

146 

1.9 

.405 

110 

1.63 

.545 

110 

12 

4.5 

.357 

220 

2.9 

.410 

160 

1.7 

.471 

120 

1.50 

.643 

120 

13 

4.2 

.423 

238 

2.7 

.486 

173 

1.6 

.563 

130 

1.38 

.752 

130 

14 

3.9 

.491 

256 

2.5 

.562 

186 

1.5 

.660 

140 

1.28 

.872 

140 

15 

3.6 

.558 

275 

2.3 

.636 

200 

1.4 

.757 

150 

1.20 

1.00 

150 

16 

3.4 

.651 

293 

2.2 

.738 

213 

1.3 

.854 

160 

1.12 

1.13 

160 

17 

3.2 

.722 

311 

2.0 

.805 

226 

1.2 

.945 

170 

1.06 

1.29 

170 

18 

3.0 

.803 

330 

1.9 

.907 

240 

1.2 

1.12 

180 

1.00 

1.44 

180 

19 

2.8 

.882 

348 

1.8 

1.01 

253 

1.1 

1.21 

190 

.95 

1.62 

190 

20 

2.7 

.992 

366 

1.7 

1.11 

266 

1.0 

1.28 

200 

.90 

1.79 

200 

21 

2.5 

1.06 

385 

1.6 

1.21 

280 

1.0 

1.45 

210 

.85 

1.95 

210 

22 

2.4 

1.17 

403 

1.6 

1.39 

293 

.95 

1.62 

220 

.81 

2.14 

220 

23 

2.3 

1.28 

421 

1.5 

1.49 

306 

.90 

1.75 

230 

.78 

2.35 

230 

24 

2.2 

1.39 

440 

1.5 

1.58 

320 

.85 

1.88 

240 

.75 

2.57 

240 

25 

2.1 

1.50 

458 

1.4 

1.79 

333 

.82 

2.05 

250 

.72 

2.79 

250 

26 

2.1 

1.69 

476 

1.3 

1.87 

346 

.80 

2.25 

260 

.69 

3.01 

260 

27 

2.0 

1.80 

495 

1.3 

2.09 

360 

.77 

2.43 

270 

.66 

3.26 

270 

28 

1.9 

1.90 

513 

1.2 

2.15 

373 

.75 

2.64 

280 

.64 

3.51 

280 

29 

1.8 

2.01 

531 

1.2 

2.39 

386 

.72 

2.81 

290 

.62 

3.77 

290 

30 

1.8 

2.23 

550 

1.1 

2.43 

400 

.70 

3.03 

300 

.60 

4.02 

300 






























































































340 


Floors with Iron Joists. 



FLOORS WITH IRON JOISTS. 

{Phoenix Iron Company , Philadelphia.) 

Having given the weight a floor is to hear per unit of surface, select the 
proper size I girders and joists required lor sustaining the floor with 

safety. , „ 

To find the distance from centre to centre of beams for laying in floors 
when the span and load per square foot are given : The total load for eacli 
beam mav be represented by TF= P IB] in which P— load per square foot; 
L = clear span in feet; B = distance from centre to centre of beams. In 

8 D (a + f') 5 

the equation IF =-- substitute the value of IF, as above. 


Then 


PLB = 


8 D {a + e ) S 


whence B = 


8 •»(» + !> 


L ’.~. PL 2 ’ 

and from this formula B may be determined for any given values of P, L, 
and S. - 

Assuming P at 140 pounds per square foot and S at 6 tons safe strain per 
effective square inch, the proper spaces for each beam may be determined 
from the following table: 

Table III. 


Beam. 

Value of 

8 ®(“+V> 



Clear Span, in 

Feet. 


10 

12 

14 

16 

18 

20 

22 

15" -200 

410 





18.10 

14.64 

12.10 

15" -150 

302 




16.86 

13.32 

10.80 

9.00 

-170 

292 




16.30 

12.87 

10.43 

8.62 

12" -125 

208 



15.09 

11.55 

9.15 

7.39 

6.11 

1 (> A "— 13o 

178 



13.05 

9.93 

7.88 

6 36 

5.22 

10i"-105 




11.28 

8.65 

6.83 

5.54 

4.58 

9" -150 

197 



14.00 

10.70 

8.68 

7.04 

5 81 

9" - 84 

108 


10.71 

7.87 

6.02 

4.76 

3.86 

3.18 

9" - 70 

92 

13.00 

9.02 

6.70 

5.14 

4.05 

3.28 

2.72 

8 " - 81 

94 

13.44 

9.32 

6.85 

5.24 

4.14 

3.36 

2.77 

8 " - 65 

74 

10.60 

7.34 

5.39 

4.13 

3.28 

2.64 


7" - 69 

72 

10.29 

7.14 

5.25 

4.02 

3.13 



7" - 55 

54 

7.70 

5.36 

3.94 

3.01 




6 " - 50 

45 

6.43 

4.45 

3.28 





6 " - 40 

35 

5.00 

3.47 

2.55 





5" - 36 

25 

3.57 

2.48 






5" - 30 

21 

3.00 

2.08 







24 26 


10.17 

7.49 

7.24 

5.13 

4.41 

3.84 

4.88 

2.68 


8.66 

6.40 

6.17 

4.38 

3.76 

3.28 

4.16 



For any given span, and when it is equally convenient to set the beams 
apart at the distances B, B, B, corresponding to their several sizes in the 

above table, the cheapest beam to use is the deepest of the series, even with the extra 

charge for size and length added. 

Example .—For a 26 feet span, what is comparative weight of iron and cost 
per square foot of floor sustained of a 10s"-105, a 12"—125, and a 15"-150 beam 
placed apart at their respective distances, B, in table? 

Iron per Per lb. Cents. Eatio. 

101" 10.57 lbs. (Si = 6 c. =• . 63 « = fi.oo 

12" 9.51 lbs. @ 6 ^ + i = 6 ic. = .6181 = .971 

15" 7.81 lbs. @ 6 i + £ + £ = 7c. = .5467 = .80* 

The rigidity of a floor is greater when a suitable number of deep beams are 
used in preference to a greater number of shallower ones of equal strength. 
For any given span, to find the rigidity of a floor, ascertain from the table 

















































Floors with Iron Joists. 


341 


the bending moments of the several beams under comparison; and from 
table III. their respective distances B for given span, and divide the former 
hy the latter. The quotients will represent their respective ratios of rigidity 
in the floor. 

When Pis any other weight than 140 pounds per square foot of flooring, 
then calculate the distance B from centre to centre of the beams by the 
given formula. 

Under no circumstances should the floor-beams he strained beyond the 
limits of their elasticity, or, in other words, so strained that on the removal 
of the load they will not return to their original condition without set or 
permanent deflection. 

Fig. 82. 


'mmm 

■mtm 

mm* 

mm 

"■mm* 

Ti 

m//m 

W///////M 

V A 

1 



TIES 






\ 

1 



vr* 

GIRDER 



c_ 

o 

H 


! 

r 

1 


JOISTS 




—TAT¬ 

TIES 



I 

4 

- 

1 


mm?, 


mm 


mm. 

mm 

wm< 



If a beam is required to sustain its load at the centre, the figures in the 
table must be divided by 2; if at, any other point, the weight at the centre 
is to the weight at any other point as the product of the segments of the 



span at the given point is to the square of half the span. The coefficient of 
safety or load factor is placed above each beam in the table, and this divided 
by the clear span in feet gives the strength of the beam for a distributed load 
iii net tons of 2000 pounds. 



The deflection for each beam corresponding to this load will be found in 
the next line, and the weight of the beam itself, for the given length, should 
be deducted from the safe load. For any less load uniformly distributed, the 



deflection will be directly proportionate to that given in the table. The de¬ 
flection should not exceed oue-thirtieth of an inch per foot of clear span, and 
the dividing lines in table II. show where this limit is passed for each beam. 

Beams are generally laid in floors as shown in Fig. 82, the joists either rest¬ 
ing on top of the gilders, as in Fig. 83, or bolted to the sides of the girders. 

Fig. 84 shows the detail of connection when the under sides are made flush, 
Fig. 85 the joint when the upper sides are flush, and Fig. 86 shows the form 
























































































342 


Floors with Iron Joists. 


usually adopted when the beams are of the same size or when the centre lines 
are brought together. Arrangements of this kind are also used to connect 
the trimmer-beams of hatchways, jambs, and stairways. Fig. 87 shows the 
end of a double girder resting on a cast shoe plate, the beams being joined 
with a cast separator and bolts. 





The wall end of the joists should also be provided with a shoe or bearing 
plate of iron or stone, as the brickwork is apt to crush under the end of the 
beam unless the load is distributed by this means over a sufficient surface. 
Anchor straps should be bolted to the end of each girder, and to the ivall end of 
every alternate joist, thus binding the walls firmly from falling outwards in 
the event of fire or other accident. 

Several simple modes of anchorage are shown in Figs. 84, 85, and 86. 

When one beam does not 
give sufficient strength for a 
girder, it is customary to bolt 
together two or more with 
cast separators between them. 
(Figs. 87 and 88.) For carry¬ 
ing a wall nine inches thick, 
two beams laid close together, 
of depths proportioned to the 
span, and for thirteen-inch 
walls the same beams, with a 
space between them, make a very good arrangement, care being taken to 
bind the two beams firmly together with separators and bolts every six or 
eight feet. 

When the length of the 
span becomes too great for 
the girder, and posts are in¬ 
troduced for intermediate 
supports, joint boxes of sim¬ 
ple pattern are provided at 
each floor, forming caps and 
Fig. bases for the wrought-iron 

columns, and at the same 
time serving to unite the 
girders continuously through 
the length of the building. 

The cap or base may be of 
any ornamental pattern de¬ 
sired to give a finish to the column. 

Between the joists the spaces are filled up with brick arches, resting on the 
lower flanges against cast-iron or brick skew-backs. 





The bricks should he moulded with a slight taper to suit the arch, and he 
laid in place with as little mortar as possible. Above the arch the space is 
filled with grouting, in which wooden strips 2" X 1" are bedded for nailing 
































































































Plate- and Box-Girders. 


343 



the flooring to. The thrust of the arches is taken up by a series of tie-rods, 
placed in lines from 6 to 8 feet apart, and usually from £ to 1 inch in diam¬ 
eter, [as shown in plan,] that run from beam to beam from one end of the 
building to the other, being anchored into each end wall with stout washers, 
an angle bar or 
channel serving 
as a wall-plate for 
distributing the 
strain produced 
by the thrust of 
the first arch. 

Instead of the 
brick arches cor¬ 
rugated iron is 
sometimes used 
to fill in the 
spaces. It is 
placed on the 
lower flanges of 
the beams and 
filled in above 
with cement in 
place of brick¬ 
work. 

The centres for 
turning the 

arches can be suspended by iron straps hooked on the lower flange, and de¬ 
tachable <rn one side so that the frames can be shifted from point to point 
as the work progresses. If a flush surface is preferred for the ceiling, it may 
be obtained by wedging strips of pine between the beams, and tacking the 
laths diagonally to the under side of these, finishing with a smooth and fair 
surface of plastering, aud thus entirely concealing the iron-work above. 


Fig. 90. 


Fig. 91. 


Fig. 92. 


Plate- and Box-Girders. 

For large halls in which columns cannot be allowed, the floor-beams must 
be made deep and strong enough to bear the floor above. For this purpose, 

plate-girders, as represented by Fig. 91, also 
box-girders , as represented by Fig. ty, are 
made. 

The strength of these girders is calculated 
by the standard formula for I beams. 

a = area of all the four angle-irons and top 
and bottom plates. 

a' — area of the stem or stems, in square 
inches. 

In this kind of girders the iron should not 
be strained to more than £th, or better, ^th, 
of the breaking strength of the iron. 

W = equally distributed load on the girder, 
including its own weight. 

The weight of the girder, including rivets and lap-plates, will be 

w — 3.6 L (a + a'). 

The length of a girder which can bear with safety only its own weight 
will be 



L = 




22 DS (a+ -£-)' 


a + a' 


The same rule will hold good for the different sections of beams shown on 
page 318. 

For lattice-girders the area a' is omitted in the formula for strength. 

W — 8DSa 


Lattice-girder 


L 










































Phcenix Columns. 




Fig. 93. 





•p ,*! 

»| 





PHCENIX COLUMNS. 



One 

One 


One 


Segment. 

Column. 


Segment. 

Mark. 

tn g 

a _ . 

•'-'WO 

a £ 


Mark. 


a • 



~ r o - 

*—* 

^'O 0 


c ■— 



H.S 

Weigh 
Poun 
per Ys 

a ~ 

i° 

^ cr 
ca 

Weigh 
Poun 
per F 


M § 

a 5 

JO _ 

H.S 

tc s ^ 

A 

rs 

94 

3.8 

12.6 

D 

4 

28 

4 Segment. 

4 

T% 

12 

144 

4.8 

5.8 

16.0 

19.3 

5 Segment. 

i 5 g 

8 

32 

36 

3f" diam. 

§ 

17 

6.8 

22.6 


1 7 S 

4 

9 

15 

40 

44 

48 


4 

16 

6.4 

21.3 

94" diam. 

IP 

IS 

3 

8 

194 

7.8 

26.0 





23 

9.2 

30.6 



28 

32 

4 Segment. 

TS 

26J 

10.6 

35.3 


4 


4 

30 

12.0 

40.0 


TB 

4J§" diam. 

A 

5 

¥ 

33J 

37 

13.4 

14.8 

44.6 

49.3 

E 

f 

TB 

4 

36 

40 

44 

• 

4 

184 

7.4 

24.6 

6 Segment. 

9 

TS 

5 

48 

53 

II- 

IS 

224 

9.0 

30.0 


¥ 

44 

4 

43 

4 

1 

58 

4 Segment. 

# 

TB 

264 

304 

10.6 

12.2 

35.3 

40.6 

11" diam. 

63 

68 


4 

344 

13.8 

46.0 


73 

5}§" diam. 


384 

15.4 

51.3 


83 

5 

5 

42| 

17.0 

56.6 



4 

25 

10.0 

33.3 


t 5 b 

30 


TB 

i 

30 

12.0 

40.0 


3 

35 


35 

14.0 

46.0 


t 7 b 

40 

r 


40 

16.0 

53.0 

G 

4 

45 


4 

45 

18.0 

60.0 

IB 

50 

4 Segment. 

5 

S 

48 

53 

19.2 

21.2 

64.0 

70.6 

8 Segment. 

6 

S 

44 

55 

60 


u 

58 

23.2 

77.3 


4 

65 


4 

63 

25.2 

84.0 


43 

70 

7 T V' diam. 

43 

3 

68 

73 

27.2 

29.2 

90.0 

97.3 

1 If" diam. 

7 

s 

1 

75 

85 


l 

83 

33.2 

110.6 


14 

95 



93 

37.2 

124.0 


14 

105 


H 

103 

41.2 

137.3 


13 

115 


One 

Column. 


C 3s 
rt ^ 

q — 

*- *—< 

^ cr 
Cfl 


14. 

16. 

18. 

20 . 

22 . 

24. 


16.8 

19.2 

21.6 

24.0 

26.4 

28.8 

31.8 

38.8 

37.8 

40.8 

43.8 

49.8 


24. 

28. 

32. 

36. 

40. 

41. 
48. 
52. 
56. 
60. 
68 . 
76. 
84. 
92. 


? 

JO C r 

13 > 5 ^ 

Jp *"■ 


46.6 

53.3 

66.0 

66.6 

73.3 
80.0 


56. 

64. 

72. 

80. 

88 . 

96. 

106. 

116. 

126. 

136. 

146. 

166. 


80.0 

93.3 

106.6 

120.0 

133.3 

146.6 
160.0 

173.3 

186.6 
200.0 
226.6 

253.3 
280.0 
306.6 


Notes— 1. Diameters given are for the interior of columns, and are constant 
for all thicknesses of the same column. 2. The weight of rivet-heads adds 
from 2 to 5 per cent, to the weight of finished columns. 











































































Phcenix Columns. 


345 


For wrought-iron columns 


• • • • 


• • • 


For cast-iron 


W = 


1F = 


FA 


1 + ( l) 

FA 


w 


1 + SoO 


ar 


W = Breaking load in lbs; ——— — Safe load for wrought 


w 


and ——- = Safe load for cast iron, 
b 

r._ ( 50,000 lbs for wrought-iron. 

(80,000 lbs. for cast-iron. 

A = Sectional area of metal, in square iuches. 


h = diameter. 


I = length. 
— — length in terms of the diameter. 


In order to find the load which a cast- or wrought-iron column will sustain 
with safety, ascertain first the number of times its diameter will divide into 
its length ; seek for the quotient in column I. of the table, and on the same 
line, in column IV. or V. (according as the material shall be cast or wrought 
iron), the safe load on each square inch of its cross section may be taken; 
multiply by the number of square inches contained in the cross section for 
the total safe load. 


Ratio 

Maximum Load. 

Safe Load. 

of Length to 
Diameter. 

Per Square Inch. 

Per Square Inch. 

1 

Cast. 

Wrought. 

Cast. 

Wrought. 

h 

W — Lbs. 

W= Lbs. 

Factor J = 

Factor \ = -y-. 

I. 

II. 

III. 

IV. 

V. 

8 

68 965 

48 971 

11 494 

12 243 

9 

66 528 

4S 685 

11 088 

12171 

10 

64 000 

48 387 

10 666 

12 100 

11 

61 420 

48 076 

10 236 

12 019 

12 

58 823 

47 709 

9 804 

11 927 

13 

56 289 

47 333 

9 381 

11 837 

14 

53 691 

46 948 

8 948 

11 737 

15 

51 200 

46 511 

8 533 

11 628 

16 

48 780 

46 082 

8 130 

11 520 

17 

46 444 

45 620 

7 741 

11 405 

18 

44 198 

45 248 

7 349 

11 >12 

19 

42 050 

44 642 

7 008 

11 160 

20 

40 000 

44 130 

6 666 

11032 

21 

38 049 

43 591 

6 341 

10 898 

22 

36 199 * 

43 066 

6 033 

10 766 

23 

34 445 

42 517 

5 741 

10 629 

24 

32 789 

41 946 

5 465 

10 486 

25 

31 219 

41 390 

5 203 

10 347 

26 

29 739 

40 816 

4 957 

10 204 

27 

28 343 

40 225 

4 724 

10 056 

28 

27 027 

39 651 

4 505 

9 913 

29 

25 785 

39 062 

4 298 

9 765 

30 

24 617 

38 461 

4 103 

9615 

31 

23 512 

37 878 

3 919 

9 469 

32 

22 479 

37 285 

3 747 

9 321 

33 

21 491 

36 683 

3 582 

9 171 

34 

20 565 

36 101 

3 428 

9 025 

35 

IS 692 

35 511 

3 282 

8 878 

36 

18 869 

34 916 

3 145 

8 729 

37 

18 090 

34 340 

8 015 

8 585 

38 

17 353 

33 760 

2 892 

8 440 

39 

16 658 

33178 

2 777 

8 294 

40 

16 000 

32 616 

2 667 

8 154 


























346 


Iron Beams. 


Table Showing the Proper Size of Rolled I Beams to be Used 
for Different Loads and Spans. 

Load 



Distance between Supports, in 

Fegt. 



Load 

in 














uniformly 

Ceutre. 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

| 20 

diet. 

1,000 

5/ 

51 

51 

51 

0/ 

6/ 

6/ 

7 

7 

7 

7 

7 

7 

2,000 

1,500 

51 

51 

61 

61 

6/ 

7 

7 

7 

7 

7 

8/ 

8/ 

8/ 

3,000 

2,000 

61 

61 

61 

7 

7 

7 

7 

7 

8/ 

SI 

8/ 

8 A 

8 A 

4,000 

2,500 

61 

61 

M 

4 

7 

7 

7 

8/ 

8/ 

SI 

Sh 

8 A 

8 A 

10*/ 

5,000 

3,000 

61 

7 

7 

7 

7 

8/ 

8/ 

Sh 

Sh 

Sh 

10*/ 

10*/ 

10*/ 

6,000 

3,500 

7 

7 

7 

7 

8/ 

8/ 

Sh 

Sh 

Sh 

10* Z 

10*/ 

10*/ 

10*/ 

7,000 

4,000 

7 

7 

7 

8/ 

SI 

Sh 

Sh 

10*/ 

10*/ 

10*/ 

10*/ 

10*/ 12*/ 

8,000 

4,500 

7 

7 

8/ 

8/ 

Sh 

Sh 

10*/ 

10*/ 

10*/ 

10*/ 

10*/ 

12*/ 

12*/ 

9,000 

5,000 

7 

SI 

8/ 

Sh 

Sh 

10*/ 

10*/ 

10*/ 

10*/ 

10*/ 

12*/ 

• | 

12*/ 

12*/ 

10,000 

6,000 

8/ 

SI 

Sh. 

Sh 

10*/ 

10*/ 

10*/ 

10*/ 

12*/ 

12*/ 

12*/ 

12*/ 

15/ 

12,000 

7,000 

SI 

8 A 

Sh 

10*/ 

10*/ 

10*/ 

10*/ 

12*/ 

12*/ 

12*/ 

15/ 

15/ 

15/ 

14,000 

8,000 

8 A 

Sh 

10*/ 

10*/ 

10*/ 

10*/ 

12*/ 

12*/ 

12*/ 

15/ 

15/ 

15/ 

15/ 

16,000 

9,000 

8 A 

10 */ 

10*/ 

10*/ 

10*/ 

12*/ 

12*/ 

12*/ 

15/ 

15/ 

15/ 

15/ 

15 A 

18,000 

10,000 

10 *Z 

10*/ 

10*/ 

10*/ 

12*/ 

12*/ 

12*/ 

15/ 

15/ 

15/ 

15/ 

15 A 

15 A 

20,000 

11,000 

10 */ 

10*/ 

10*/ 

12*/ 

12*/ 

12*/ 

15/ 

15/ 

15/ 

15/ 

15 h 

15 A 

15 A 

22,000 

12,000 

10*/ 

10*/ 

10*/ 

12*/ 

12*/ 

15/ 

15/ 

15/ 

15/ 

15 A 

15 A 

15 A 

15 A 

24,000 

13,000 

10i / 

10*/ 

12*/ 

12*/ 

15/ 

15/ 

15/ 

15/ 

15 h 

15 h 

15 A 

15 A 


26,000 

14,000 

10*1 

12*/ 

12*/ 

12*/ 

15/ 

15/ 

15/ 

15 h 

15 h 

15 h 

15 A 



28,000 

15,000 

10* l 

12*/ 

12*/ 

15/ 

15/ 

15/ 

15/ 

15 h 

15 h 

15 h 




30,000 

16,000 

12*/ 

12*/ 

12*/ 

15/ 

15/ 

15/ 

15 h 

15 h 

15 h 

15 h 




32,000 

17,000 

12*/ 

12*/ 

15/ 

15/ 

15/ 

15/ 

15 h 

15 h 

15 h 





34,000 

18,000 

12*/ 

12*/ 

15/ 

15/ 

15/ 

15 h 

15 h 

15 h 






36,000 

19,000 

12*/ 

15/ 

15/ 

15/ 

15/ 

15 h 

15 h 

15 A 






38,000 

20,000 

12*/ 

15/ 

15 / 

15/ 

15 h 

15 h 

15 h 







40,000 





h means heavy, 

/ means light. 





























































Nails and Spikes. 


347 


NAILS AND SPIKES. 

Size, Length, and Number to the Pound. 
(Cumberland Nail and Iron Oo.) 


Ordinary. 


Size. 

Length. 

No. to Lb. 


tt 


2d 

7 

Q 

716 

3 fine. 

Its 

588 

3 

Its 

448 

4 

If 

336 

5 

1# 

216 

6 

2 

166 

7 

2 i 

118 

8 

2£ 

94 

10 

2 f 

72 

12 

3f 

50 

20 

3| 

32 

30 

4f 

20 

40 

4f 

17 

50 

5 

14 

60 

5| 

10 

Light. 


tt 


J4 

if 

373 

*5 

-if 

272 

6 

2 

196 

Brads. 


tt 


6d 

o 

163 

8 

h 

96 

10 

2f 

74 

12 


50 


Clinch. 


Finishing. 

Length. 

No. to Lb. 

Size. 

Length. 

No. to Lb. 

ft 



ft 


2 

152 

4d 

If 

384 

2f 

133 

5 

1| 

256 

2£ 

92 

6 

2 

204 

2f 

72 

8 

2£ 

102 

3 

60 

10 

3 

80 

3f 

43 

12 

3f 

65 


20 

3?- 

46 

ft 


Core. 

2 

96 


tt 


2f 

66 

6d 

2 

143 

2* 

56 

8 

2f 

68 

2f 

50 

10 

2y 

60 

3 

40 

12 

3i 

42 

Spikes. 

20 

30 

3f 

4f 

25 

18 

ft 


40 

4f 

14 

3f 

19 




4 

15 

WII 

2f 

69 

4f 

13 

W H L 

2f 

72 

5 

10 


Slate. 


5J 

9 




6 

7 


ft 


Boat. 

3a 

4 

-Its 

lfs 

288 

244 

ft 


5 

If 

187 

H 

206 

6 

2 

146 


Square and Hexagon Nuts. 

Number of each Size in 100 Lbs. 


(Hoopes & Townsend , Philadelphia.) 


Size 

of 

Bolt. 

Width. 

Thick¬ 

ness. 

Sqi 

No. in 100 
Lbs. 

lare. 

Weight 
each in Lbs. 

Hex: 

No. in 100 
Lbs. 

igon. 

Weight 
each in Lbs. 


x 

x 

8140 

.012 

9300 

.011 

i 

1 J$ 

tt 

A 

3000 

.033 

6200 

.016 

3 

U 

if 

i 

2320 

.043 

3120 

.032 


tt 

TS 

1940 

.052 

2200 

.045 

h 

i 

i 

1180 

.085 

1350 

.074 

l 

M 

t 9 s 

920 

.109 

1000 

.100 

f 

1A 

i 

738 

.135 

830 

.120 


if 

f 

420 

.238 

488 

.205 

| 

Its 

7 

■5T 

280 

.357 

309 

.32 

l 

If 

i 

180 

.556 

216 

.46 

n 

m 

if 

130 

.769 

148 

.68 


2 

H 

96 

1.04 

111 

.90 

i| 


if 

70 

1.43 

85 

1.18 

H 

93 

H 

60 

1.67 

70 

1.43 


These nuts are chamfered and trimmed. 





































































848 


Nails and Spikes. 






Tacks. 






Size. 

Length. 

No. to 
Lb. 

Size. 

Length. 

No. to 
Lb. 

Size. 

Length. 

No. to 
Lb. 


1 oz. 

£ 

16000 

4 oz. 


4000 

14 oz. 

to 

1143 


H 

A 

100(56 

6 

* 

2666 

16 

l 

1000 

2 

* 

8000 

8 

t 

2000 

18 

to 

888 


2* 

is 

6400 

10 

to 

1600 

20 

1 

800 


3 


5333 

12 

3 

1333 

22 

1* 

727 



Railroad Spikes. 

Length and Thickness in a Keg of 150 Pounds. 




Length. 

Thickness. 

Number. 

Length. 

Thickness. 

Number. 

4* 

T8 

527 

5* 

* 

356 

4* 

i 

400 

6* 

l $5 

290 

5 

1 

710 

5* 

5 

■?? 

219 

5 

TB 

489 

6 

* 

311 

5 

390 

6 


263 

5 

A 

296 

6 

5 

5 

197 

5 

6 

$ 

258 






Splices and Bolts for One Mile of Track. 

Rails 30 feet long take 704 splices, 1408 bolts. • 


<c 


4< 

28 

44 

(4 

754 

44 

1508 

44 

u 

27 

44 

14 

782 

C« 

1564 

44 

w 

25 

<( 

44 

844 

(4 

1688 

44 

M 

24 

M 

44 

8S0 

44 

1760 

44 


Railroad Iron. 

To find the number of tons of rails for one mile of track, divide the weight 
per yard by 7 and multiply by 11. Thus: for 56 lb. rail, 56 -j- 7 = 8, and 
•8 X 11 = 88 tons per mile. 


Hoopes «fc Townsend’s Regular Sizes. 




Square. 


Hexagon. 


idth. 

Thick¬ 

ness. 

No. in 
100 lbs. 

Wt. each 
in lbs. 

Width. 

Thick¬ 

ness. 

No. in 
100 lbs. 

Wt.each 
iu lbs. 

£ 

1 

4 

6680 

.015 

i 

x 

4 

8600 

.012 


TB 

3540 

.028 

* 

A 

4260 

.023 

3 

1 

2050 

.049 

3 

i 

2500 

.040 

7 

TS 

1 

1380 

.072 

3 

TIT 

2180 

.046 

l 

* 

840 

.119 

1 

I 9 (S 

900 

.111 

n 


650 

.154 

1 

1% 

880 

.114 

H 

1 

410 

.244 

n 

3 

535 

.187 

ii 

1 

270 

.370 

n 

i 

295 

.339 

H 

7 

T 

215 

.465 

H 

1 

224 

.446 

11 

l 

140 

.714 

13 

H 

150 

.667 

2 

H 

95 

1.05 

2 

n 

100 

1.00 

91 

“4 

H 

72 

1.39 

2 

if 

96 

1.04 

2* 

H 

45 

2.22 

2} 

i£ 

72 

1.39 

3 

H 

32 

3.12 

23 

n 

43 

2.33 

































































Roofing Slate. 


349 


Bolts with Square Henris and Nuts. 

Weight of 100 of the Enumerated Sizes. 


(Iloopes & Townsend , Philadelphia.) 


Lengths. 

x in. 

£ in. 

£ in. 

£ in. 

i in. 

t in. 

1 in. 

1£ in 

Inch. 









H 

4.16 

10.62 

23.87 

39.31 





If 

4.22 

11.72 

25.06 

41.38 





2 

4.75 

12.38 

26.44 

45.69 

73.62 




2* 

5.34 

12.90 

28.62 

49.50 

76. 




2* 

5.97 

14.69 

29.50 

51.25 

79.75 




2# 

6.50 

16.47 

31.16 

53. 

83. 




3 


17.87 

32.44 

56. 

85.38 

127.25 



8* 


18.94 

39.75 

63.12 

93.44 

140.56 



4 


20.59 

42.50 

74.87 

108.12 

148.37 

228. 

296. 

4* 


21.69 

44.87 

79.62 

113.12 

158.76 

239. 

310. 

5 


23.62 

48.81 

83. 

122. 

167.25 

250. 

324. 

5* 


25.81 

51.38 

87.88 

128.62 

174.88 

261. 

338. 

6 


26.87 

53.31 

92.38 

131.75 

204.25 

272. 

352. 

6* 

•••••• 

•••••• 

56.87 

96.88 

139.56 

214.69 

283. 

366. 

1 



59.12 

99.87 

145.50 

228.44 

294. 

370. 

7 ! 

* a 



61.87 

105.75 

150.88 

235.31 

305. 

384. 

8 



64.44 

109.50 

157.12 

239.88 

316. 

398. 

9 



70.50 

118.12 

169.62 

258.12 

338. 

426. 

10 



77. 

128.13 

184. 

276.18 

360. 

454. 

11 



82.88 

136.19 

195.13 

295.69 

382. 

482. 

12 



86.37 

144.87 

209.75 

311.94 

404. 

510. 

m 



92. 

155.50 

219.37 

335.81 

426. 

538. 

14 



97.75 

163.58 

237.50 

351.88 

448. 

566. 

15 



103.25 

170.75 

249.06 

391.75 

470. 

594. 


FLAGGING. 

Weight per Cubic Foot, 168 Pounds. 


Weight per Square Foot. 


Thickness. 

1 

2 

3 

4 

5 

6 

7 

8 inch. 

Weight. 

14 

28 

42 

56 

70 

84 

98 

112 lbs. 


ROOFING SLATE.' 


General Rule for tlie Computation of Slate. 

From the length of the slate fake 3 inches, or as many as the third covers 
the first; divide the remainder by 2, and multiply the quotient by the width 
of the slate, and the product will be the number of square inches in a single 
slate. Divide the number of square inches thus procured by 144, the number 
of square inches in a square foot, and the quotient will be the number of feet 
and inches required. A square of slate is what will cover 100 square feet when 
laid upon the roof. 

Weight per Cubic Foot, 174 Pounds. 


Weight per Square Foot. 


Thickness .... 

i 

& 

i 

§ 

i 

i 

i 

1 inch. 

Weight. 

1.81 

2.71 

3.62 

5.43 

7.25 

9.06 

10.87 

14.5 lbs 


Thickness . 
Weight . . 





















































































350 


Eivets. 


Table of Sixes and Number of Slate in One Square. 


Size 

in Inches. 

No. of Slate 
in Square. 

Size 

in Inches. 

No. of Slate 
in Square. 

Size 

in Inches. 

No. of Slate 
in Square. 

6 X 12 

53 

8X16 

277 

12 X 20 

141 

7 

12 

457 

9 

16 

246 

14 

20 

121 

8 

12 

400 

10 

16 

221 

11 

22 

137 

9 

12 

355 

12 

16 

184 

12 

22 

126 

10 

12 

320 

9 

18 

213 

14 

22 

108 

12 

12 

266 

10 

18 

192 

12 

24 

114 

7 

14 

374 

11 

18 

174 

14 

24 

98 

8 

14 

327 

12 

18 

160 

16 

24 

86 

9 

14 

291 

14 

18 

137 

14 

26 

89 

10 

14 

261 

10 

20 

169 

16 

26 

78 

12 

14 

218 

11 

20 

154 





Iron Rivets. 

Weight per 100, 


Length 

under 

Diameters. 

Head. 

1 

4 

1 

1 

9 

i 

i 

7 

T 

1 

1 

1.895 

4.848 

.966 

16.79 

26.49 

39.3 

55.2 


2.067 

5.235 

10.34 

17.86 

27.99 

41.4 

57.9 


2.238 

5.616 

11.04 

18.96 

29.61 

43.5 

60.7 

i 

2.410 

6.003 

11.73 

20.03 

31.13 

45.6 

63.4 


2.582 

6.402 

12.43 

21.04 

32.74 

47.8 

66.2 


2.754 

6.789 

13.12 

22.11 

34.25 

49.9 

68.9 

1 

2.926 

7.179 

13.81 

23.21 

35.86 

52.0 

71.7 

7 

Q 

3.098 

5.566 

14.50 

24.28 

37.37 

54.1 

74.4 

2 

3.269 

7.956 

15.19 

25.48 

38.99 

56.3 

77.2 

x 

3.411 

8.343 

15.88 

26.56 

40.40 

58.4 

79.9 

x 

3.613 

8.733 

16.57 

27.65 

42.11 

60.5 

82.7 

t 

3.785 

9.120 

17.26 

28.73 

43.67 

62.6 

85.4 


3.957 

9.511 

17.95 

29.82 

45.24 

64.8 

88.2 

* 

4.129 

9.898 

18.64 

30.90 

46.80 

66.9 

90.9 

# 

4.301 

10.29 

19.33 

31.99 

48.36 

69.0 

93.7 

7 

4.473 

10.67 

20.02 

33.08 

49.92 

71.1 

96.4 

3 

4.644 

11.06 

20.71 

34.18 

51.49 

73.3 

99.2 

x 

4.816 

11.44 

21.40 

35.27 

53.05 

75.4 

101.9 

1 

4.988 

11.84 

22.09 

36.35 

54.61 

.77.5 

104.7 

t 

5.160 

12.23 

22.78 

37.44 

56.17 

79.6 

107.4 

' * 

5.332 

12.62 

23.48 

38.52 

57.74 

81.8 

110.2 

* 

5.504 

13.01 

24.17 

39.60 

59.30 

83.9 

112.9 

* 

5.676 

1339 

24.86 

40.69 

60.86 

86.0 

116.7 

7 

8 

5.848 

13.78 

25.55 

41.78 

62.42 

88.1 

119.4 

4 

6.019 

14.17 

26.24 

42.87 

63.99 

90.3 

121.2 

1 

6.191 

14.56 

26.93 

43.94 

65.65 

92.4 

123.9 

I 

4 

6.363 

11.95 

27.62 

45.01 

67.11 

94.5 

126.6 

100 

Heads. 

.519 

1.74 

4.14 

8.10 

13.99 

22.27 

33.15 


Length of rivet required to make ond head = 1£ diameters of round bar. 













































Skylight and Floor Glass. 


351 


Price per Pound or Tou.i 


Per lb. 

Price 
per Ton. 

Per lb. 

Price 
per Ton. 

Per lb. 

Price 
per Ton. 

Per lb. 

Price 
per Ton 

■fa cts. 

$2.24 

3 t \j cts. 

$69.44 

6 ^j cts. 

$136.64 

9^ cts. 

$203.84 

fa 

4.48 

TO 

71.68 

1 ? 5 

138.88 

i\5 

206.08 

fa 

6.72 

fa 

73.92 

16 

141.12 

1 3 {J 

208.32 

TO 

8.96 

TO 

76.16 

16 

143.36 

15 

210.56 

fa 

11.20 

TO 

78.40 

16 

145.60 

16 

212.80 

TO 

13.44 

TO 

80.64 

6 

TO 

147.84 

16 

215.04 

TO 

15.68 

TO 

82.88 

7 

16 

150.08 

16 

217.28 

1(5 

17.92 

fa 

85.12 

16 

152.32 

16 

219.52 

TO 

20.16 

t 9 ti 

87.36 

16 

154.56 

16 

221.76 

1 

22.40 

4 

89.60 

7 

156.80 

10 

224.00 

fa 

24.64 

TO 

91.84 

16 

158.04 

iV 

226.24 

fa 

26.88 

TO 

94.08 

16 

161.28 

fa 

228.48 

1(5 

29.12 

fa 

96.32 

TO 

163.52 

TO 

230.72 

1(5 

31.36 

1*0 

98.56 

4 

15 

165.76 

16 

232.96 

t 5 o 

33.60 

TO 

100.80 

5 

1 6 

168.00 

5 

15 

235.20 

1(5 

35.84 

6 

16 

103.04 

6 

16 

170.24 

6 

16 

237.44 

TO 

38.08 

16 

105.28 

7 

16 

172.48 

16 

239.68 

TO 

40.32 

16 

107.52 

16 

174.72 


241.92 

' TO 

42.56 

TO 

109.76 

16 

176.96 

. 16 

244.16 

2 

44.80 

5 

112.00 

8 

179.20 

li 

246.40 

'TO 

47.04 

l l 6 

114.24 

1 

T5 

181.44 

16 

248.64 

"lO 

49.28 

1 2 6 

116.48 

16 

183.68 

1c6 

250.88 

fa 

51.52 

1 3 6 

118.62 

io 

185.92 

1 3 5 

253.12 

16 

53.76 

16 

120.96 

1*6 

188.16 

TO 

255.36 

1 0 o 

56.00 

TO 

123.20 

TO 

190.40 

TO 

257.60 

TO 

58.24 

TO 

125.44 

16 

192.64 

fa * 

259.84 

TO 

60.48 

TO 

127.68 

7 

16 

194.88 

fa 

262.08 

10 

62.72 

TO 

129.92 

8 

16 

197.12 

fa 

264.32 

to 

64.96 

T 9 (J 

132.16 

9 

16 

199.36 

16 

266.56 

3 

67.20 

6 

134.40 

9 

201.60 

12 

268.80 


SKYLIGHT AND FLOOR GLASS, 

Weight per Cubic Foot, 156 Pounds. 

{Lennox Plate Glass Go., Ward & Co., Agents, Philadelphia.) 


Weight per Square Foot. 


Thickness. . . 

1 

8 

fa 

1 

4 

i ' 

1 

a 

* 

1 

1 inch. 

Weight .... 

1.62 

2.43 

3.25 

4.88 

6.50 

8.13 

9.75 

13 lbs. 



















































352 


Differential Balance. 


DIFFERENTIAL BALANCE. 

Fi«. 94 represents a convenient, balance-scale for weighing heavy weights. 
It is much used in iron-foundries, where the balance with the weight is 
hoisted in a crane for weighing. 



The object of the links and the short lever is to bring the weight close to 
the fulcrum, or, more correctly, to obtain a short lever-arm /. 

There is not room enough on the main-balance to bring the direction of the 
action of the weight IF sufficiently close to the fulcrum for weighing heavy 
weights. The levers a and 6 ought to be equal to a’ and b' respectively. 

a + b = a' + 6', and a — b — a' — b\ 

The difference between a and b is generally not made so great as shown in 
the illustration. For a lever L — 8 feet, the lever / is only a fraction of an 
inch, and can be made as small as desired by making (a — b) small. The scale 
should be well balanced by the ball B , without the weights IF and w. 




















Spheres. 


353 


Spheres, Balls—Surfaces, Capacity ami "Weight of. 


Diameter. 

Surface. 

Capacity'. 

Cast iron. 

Lead. 

Water. 

Inches. 

Sq. inches. 

Cub. Inches. 

Pounds. 

Pounds. 

Pounds. 

1 in. 

3.1416 

0.5236 

0.1365 

0.2147 

0.0188 

1.125 

3.9760 

0.7455 

0.1943 

0.3062 

0.0264 

1.25 

4.9087 

1.0226 

0.2673 

0.4200 

0.0368 

1.375 

5.9395 

1.3611 

0.3550 

0.5579 

0.0490 

1.5 

7.0686 

1.7671 

0.4607 

0.7248 

0.0636 

1.625 

8.2957 

2.2467 

0.5861 

0.9227 

0.0809 

1.75 

9.6211 

2.8061 

0.7325 

1.1528 

0.1050 

1.875 

11.044 

3.4514 

0.8000 

1.4156 

0.1242 

2 in. 

12.566 

4.1888 

1.0920 

1.7180 

0.1508 

2.125 

14.186 

5.0243 

1.3124 

2.0631 

0.1809 

2.25 

15.904 

5.9640 

1.5592 

2.4482 

0.2147 

2.375 

17.720 

7.0143 

1.8334 

2.8811 

0.2525 

2.5 

19.635 

8.1812 

2.1328 

3.3554 

0.2945 

2.625 

21.647 

9.4708 

2.4725 

3.8892 

0.3410 

2.75 

23.758 

10.889 

2.8400 

4.4623 

0.3920 

2.875 

25.967 

12.442 

3.2512 

5.1056 

0.4479 

3 in. 

28.274 

14.137 

3.6855 

5.7982 

0.5089 

3.125 

30.6S0 

15.979 

4.1721 

6.5568 

0.5752 

3.25 

33.183 

17.974 

4.6835 

7.3623 

0.6471 

3.375 

35.785 

20.129 

5.2612 

8.2521 

0.7246 

3.5 

38.484 

22.449 

5.8525 

9.2073 

0.8081 

3.625 

41.282 

24.941 

6.5089 

10.231 

0.8979 

3.75 

44.179 

27.612 

7.2135 

11.323 

0.9941 

3.875 

47.173 

30.466 

7.9556 

12.500 

1.0968 

4 in. 

50.265 

33.510 

8.7361 

13.744 

1.2064 

4.25 

56.745 

40.194 

10.510 

16.482 

1.4470 

4.5 

63.617 

47.713 

12.439 

19.569 

1.7177 

4.75 

70.882 

56.115 

14.666 

23.035 

2.0202 

5 in. 

78.540 

65.450 

17.063 

26.843 

2.3562 

5.25 

86.590 

75.766 

19.810 

31.089 

2.7276 

5.5 

95.033 

87.114 

22.720 

35.729 

3.1361 

5.75 

103.87 

99.541 

26.000 

40.856 

3.5835 

6 in. 

113.10 

113.10 

29.484 

46.385 

4.0716 

6.5 

132.73 

143.79 

37.453 

58.976 

5.1765 

7. 

153.94 

179.59 

46.820 

73.659 

6.4653 

7.5 

176.71 

220.89 

57.587 

90/598 

7.9520 

8 in. 

201.06 

268.08 

69.889 

109.95 

9.6509 

8.5 

226.98 

321.55 

83.839 

131.38 

11.576 

9 in. 

254.47 

381.70 

99.510 

156.55 

13.741 

9.5 

283.53 

448.92 

117.03 

184.12 

16.161 

10 

314.16 

523.60 

136.50 

214.75 

18.850 

11 

380.13 

696.91 

181.76 

285.83 

26.289 

12 

452.39 

904.78 

235.87 

371.09 

32.572 

13 

530.92 

1150.3 

299.62 

471.80 

41.411 

14 

615.72 

1436.7 

374.56 

589.27 

51.721 

15 

706.84 

1767.1 

460.69 

724.78 

63.616 

16 

804.24 

2144.6 

559.11 

879.61 

77.206 

17 

853.96 

2572.4 

670.71 

1055.0 

92.607 

18 

1017.8 

3053.6 

796.08 

1252.4 

109.93 

19 

1134.1 

3591.3 

936.27 

1472.9 

129.29 

20 

1256.6 

4188.8 

1092.0 

1718.0 

150.80 


23 

















35*1 Weight of Roixfo Iron, per Foot. 



Sida in 

Weight in 

Side in 

Weight in 

Diameter 

Weight in 

Diameter 

Weight is 

inches 

pounds. 

inches. 

pounds 

in incites. 

pounds. 

is inches. 

pounds. 

T V 

0-013 

31 

44-418 

1 

T & 

0-010 

3f 

34-886 

* 

0*53 

33 

47*534 

h 

0*041 

33 

37-332 

A 

0-118 

31 

50-756 

3 

76 

0*119 

31 

39-864 

3 

0-211 

4 

54-084 

3 

0-165 

4 

42-464 

t 

0-475 

41 

57-517 

s 

0-373 

41 

45*174 

h 

0-845 

4* 

61-055 

1 

0-663 

43 

47*952 

I 

1-320 

41 

64-700 

1 

1-043 

4t 

50*815 

3 

1-901 

41 

68*448 

3 

1-493 

41 

53-760 

z 

2-588 

41 

72-305 

3 

2-032 

41 

56*788 

l 

3*380 

43 

76-264 

l 

2-654 

43 

59*900 

n 

4-278 

41 

80-333 

n 

3-360 

41 

63*094 

n 

5-280 

5 

84-480 

n 

4-172 

5 

66 752 

ii 

6-390 

51 

88-784 

it 

5*019 

51 

69*731 

ir 

7-604 

53 

93-168 

ii 

5*972 

53 

73-172 

ii 

8*926 

51 

97-657 

it 

7*010 

5t 

76-700 

13 

10-325 

51 

102-24 

13 

8*128 

51 

80-304 


11-883 

51 

106-95 

it 

9*333 

51 

84-001 

2 

13-520 

53 

111-75 

2 

10-616 

53 

87-776 

2* 

15-263 

51 

116-67 

21 

11-9S8 

51 

91-634 

2.1 

17112 

6 

121-66 

23 

13-440 

6 

95-552 

21 

19-066 

63 

132-04 

2 | 

14-975 

63 

103-70 

2\ 

21-120 

61 

142-82 

21 

16-688 

61 

112-16 

21 

23-292 

63 

154-01 

2 t 

18-293 

63 

120-96 

23 

25-56 

7 

165-63 

23 

20-076 

7 

130-05 

21 

27-939 

71 

190-14 

21 

21*944 

71 

149-33 

3 

30-416 

8 

216-34 

3 

23-888 

8 

169-85 

31 

33-010 

81 

244-22 

31 

25-926 

81 

191-81 

33 

35-704 

9 

273-79 

33 

28-040 

9 

215-04 

31 

38-503 

10 

337-92 

3t 

30-240 

10 

266-29 

31 

41-408 

12 

486-66 

31 . 

32-512 

12 

382-21 


llitle for Finding the Weight of Pipes. 

The diameter of the pipe in inches, measured from inside to outside, mul¬ 
tiplied by the coefficient for the metal, will be the weight in pounds per linear 
foot. 

Coefficients. 


Lead, . . 

. . 0.1005 

Brass, rolled, 

. 0.0985 

Copper, . 

. . 0.0989 

Iron, rolled, 

. 0.0876 

Brass, cast, . 

. . O.OS82 

Cast iron, 

. 0.0811 

Cast steel, 

. . 0.0891 

Tin, rolled, 

. 0.0821 

Clay, burnt, 

. . 0.0214 

Zinc, rolled, . 

. 0.0808 





































"WEIGHT PER FOOT, IN POUNDS, OF CAST-IRON CYLINDERS AND PIPES. 


355 


Diam. 

0 

Vs 

X 

1 % 

X A 

% 

% 

Vs 

Diam 

0 

00000 

•03804 

15418 

34675 

61669 

96352 

1-3876 

1-8975 

0 

1 

2-5132 

3-1227 

3-9047 

4-6620 

5-5512 

6-5476 

7-5414 

8-7012 

1 

2 

9-S989 

11-145 

12-491 

13-947 

15-419 

16-999 

18-658 

20-392 

2 

3 

22-205 

24-093 

26-059 

28-104 

30-225 

32-420 

34-695 

37-038 

3 

4 

39-544 

41-984 

44-566 

47-227 

49-963 

52-778 

55-629 

58-637 

4 

5 

61-584 

64-807 

68-005 

71-282 

74-537 

78.068 

81-577 

84-848 

5 

6 

88-825 

92-564 

96-3S0 

100-27 

104-24 

108-29 

112-42 

116-62 

6 

7 

120-90 

125-26 

129-69 

134-20 

138-79 

143-45 

148-19 

153-02 

7 

8 

157-91 

162-88 

168-15 

173-06 

178-29 

183-55 

188-91 

194-34 

8 

9 

199-86 

205-44 

211-11 

216-86 

222-68 

228-57 

234-56 

240-50 

9 

10 

246-73 

252-94 

259-23 

265-59 

272-03 

278-54 

285-13 

291-81 

10 

11 

298-55 

305-38 

312-28 

319-24 

326-28 

333-40 

340-64 

347-92 

11 

12 

355*29 

362-72 

370-23 

377-S3 

390-50 

393-26 

401-08 

408-69 

12 

13 

416-98 

425-02 

433-15 

441-39 

449-64 

458-04 

466-46 

475-00 

13 

14 

483-73 

492-24 

501-02 

509-84 

518-77 

527-72 

536-80 

545-94 

14 

15 

528-15 

564-44 

573-81 

583-76 

592-78 

602-36 

612-04 

621-71 

15 

16 

631-64 

641-54 

651-53 

661-58 

671-73 

681-94 

692-24 

702-61 

16 

17 

712-79 

723-59 

734-19 

744-86 

755-80 

766-44 

777-38 

788-35 

17 

18 

799-30 

810-56 

821-79 

838-17 

844-45 

855-86 

867-42 

879-04 

18 

19 

S90-70 

902-48 

914-29 

926-23 

938-20 

950-27 

962-42 

974-64 

19 

20 

986-95 

999-30 

1011-6 

1024-3 

1036-9 

1049-5 

1062-3 

1075-0 

20 

21 

1088-1 

1104-2 

1114-6 

1127-3 

1140-5 

1153-8 

1107-2 

1180-7 

21 

22 

1194-2 

1207-8 

1221-5 

1235-2 

1249-1 

1263-0 

1277-0 

1291-1 

22 

23 

1305-2 

1319-4 

1333-7 

1348-1 

1362-6 

1376-9 

1391-7 

1406-4 

23 

24 

1421-5 

1436-0 

1451-0 

1466-1 

1481-0 

1496-1 

1511-4 

1526-7 

24 

25 

1492-1 

1557-5 

1572-1 

1588-7 

1604-4 

1620-2 

1635-8 

1651-9 

25 

26 

1667-9 

1683-9 

1700-1 

1716-4 

1732-7 

1749-1 

1765-5 

1782-1 

26 

27 

1798-7 

1815-5 

1832-2 

1849-0 

1865-9 

1882-9 

1900-0 

1917-2 

27 

28 

1934.4 

1951-7 

1969-1 

1986-5 

2004-1 

2021-7 

2039-4 

2057-2 

28 

29 

2075-1 

2093-0 

2111-0 

2129-1 

2147-2 

2165-4 

2183-8 

2202-2 

29 

30 

2220-6 

2239-2 

2257-8 

2276-5 

2295-2 

2314-1 

2333-1 

2352-0 

30 

31 

2371-1 

2390-3 

2409-6 

2428-9 

2448-3 

2467-9 

2461-3 

2506-9 

31 

32 

2526-6 

2545-7 

2566-2 

2586-2 

2606-1 

2626-3 

2646-4 

2666-7 

32 

33 

26S7-0 

2707-4 

2727-8 

2748-4 

2769-0 

2789-7 

2810-4 

2831-3 

33 

34 

2852-3 

2873-3 

2894-4 

2927-3 

2936-8 

2958-1 

2979-5 

3001-0 

34 

35 

3022-5 

3044-2 

3065-9 

3087-7 

3109-5 

3131-5 

3143-7 

3175-5 

35 

36 

3197-5 

3219-4 

3242-0 

3264-3 

3286-9 

3309-5 

3332-2 

3354-3 

36 

37 

3377-8 

3400-4 

3423-3 

3446-6 

3469-5 

3492-7 

35160 

3539-2 

37 

38 

3562-9 

3586-1 

3609-6 

3633-5 

3657-0 

3680-9 

3704-8 

3728-6 

38 

39 

3752-2 

3776-8 

3801 0 

3835-1 

3849-7 

3873-8 

3S98-3 

3922-8 

39 

40 

3947-7 

3972-5 

3987-0 

4022-1 

4046-9 

4071-6 

4094-1 

4122-3 

40 

41 

4147-5 

4173-0 

4198-4 

4223-8 

4249-2 

4275-0 

4300-7 

4326-5 

41 

42 

4352-3 

4378-4 

4404-1 

4430-2 

4456-6 

4482-6 

450&-0 

4540-5 

42 

43 

4562'2 

4588-6 

4615-2 

4641-9 

4668-6 

4695"6 

4722-7 

4749-7 

43 

44 

4778-7 

4803-7 

4831-1 

4858-4 

4S85-7 

4913-3 

494M 

4968-7 

44 

45 

4996-3 

5024-0 

5051 4) 

5079-9 

5107-8 

5136-1 

5164-1 

5192-4 

45 

46 

5220-9 

5249-2 

5277-8 

5306-3 

5335-0 

5363-6 

5393-5 

5421-4 

46 

47 

5450-2 

5473-1 

5508-4 

5537-6 

5566-8 

5596-1 

5825 6 

5655-1 

47 

48 

5684-6 

5714*1 ( 

5744-1 

5773-9 

5803-7 

5833-5 

5863-7 

5893-8 

48 


A solid cast-iron cylinder 42 in. diameter weighs 4482 6 pounds per foot. 
Subtract inside cylinder 40*4 in. diameter weight 3972-5 
Weight of pipe 1 A in- thick will be 5101 



























Weight of Flat Rolled Iron per Foot. 


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Weight of Flat Rolled Iron per Foot, 

The thickness is in the first column, and the breadth in the top line. 


Weight of Flat Rolled Iron per foot. 


357 


«e 

47-76 

44-82 

42-69 

39-55 

38-01 

36-49 

32-86 

30-42 

O 

04 

lb 

04 

25-35 

22-81 

04 

o 

04 

M 

b- 

lb 

rH 

o 

04 

tb 

rH 

iH 

CO 

cb 

rH 

10-23 

7-594 

5-069 

3-801 

to 

co 

tp 

04 


tO 

o 

o 

M 

43-32 

41-09 

38-86 

CO 

Ml 

CO 

rH 

o 

Hi 

CO 

CO 

tp 

rH 

CO 

29-15 

04 

b- 

O 

04 

24-29. 

21-86 

co 

M 

04 

rH 

O 

O 

bb 

rH 

iH 

tp 

M 

r-H 

■«ft 

r-H 

04 

rH 

9-716 

b- 

CO 

04 

lb 

CO 

tO 

CO 

M 

3-643 

04 

04 

M 

04 

Hn 

to 

H 

41-82 

39-49 

37-17 

34-85 

CO 

‘O 

04 

co 

o 

04 

co 

CO 

CO 

b— 

04 

O 

tp 

>b 

04 

23-23 

20-91 

18-59 

CO 

04 

cb 

rH 

13-94 

11-61 

9-294 

6-970 

b^ 

M 

co 

M 

3-485 

CO 

04 

CO 

04 

MH 

o 

42-13 

04 

04 

04 

CO 

O 

b— 

b^ 

CO 

35-48 

33-26 

31-05 

28-83 

CO 

cb 

04 

24*39 

CO 

rH 

04 

04 

96-61 

M 

N 

lb 

rH 

15-52 

13-31 

11-09 

8-871 

6-653 

4-436 

3-327 

CO 

rH 

04 

04 

to 

40-13 

38-02 

O 

04 

tb 

CO 

33-79 

31-68 

29-57 

27-46 

25-35 

23-23 

04 

rH 

rH 

04 

19-01 

16-90 

co 

M 

rH 

12-67 

10-56 

8-449 

6-386 

M 

04 

04 

M 

3-16S 

04 

rH 

es» 

Mff 

M 

38-12 

36-12 

34-11 

32-10 

30-09 

28-09 

CO 

O 

O 

04 

CO 

O 

M 

04 

IH 

o 

04 

04 

20-07 

18-06 

16*05 

M 

O 

M 

rH 

M 

O 

04 

rH 

10-03 

8-026 

6-019 

4-013 

3-009 

2-0061 

M^ 

M 

04 

rH 

o 

ro 

34-22 

32-31 

rH 

M 

o 

co 

28-51 

26-61 

rH 

b^ 

M 

04 

22-81! 

*H 

04 

o 

04 

19-01 

17-11 

15-21 

rH 

CO 

cb 

rH 

11-41 

9-507 

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O 

co 

t>. 

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tb 

04 

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co 

cb 

2-851 

rH 

O 

C4 

rn 

M^ 

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34-11 

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CO 

04 

CO 

30-52 

2S-72 

26-92 

25-13 

23*33 

21*54 

M 

b^ 

04 

rH 

tO 

04 

lb 

rH 

16-15 

14-36 

CC 

■O 

04 

rH 

10-77 

8-975 

7-181 

5-386 

3-591 

2-693 

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04 

b- 

rH 

■tf 

32-10 

30-41 

28-72 

M 

o 

tb 

04 

25-35 

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o 

CO 

04 

21-97 

20-28 

18-59 

16-90 

15-21 

04 

tp 

cb 

rH 

11-83 

10-14 

8-445 

6-759 

04 

co 

o 

»b 

3-380 

2-535 

O 

04 

rH 

t-bo 

CO 

30-84 

29-46 

27-57! 

26-19 

24-55 

22-92 

21-28 

19-64 

18-00 

16-37 

14-74 

13-09 

11-46 

9.822 

8-183 

16-547 

4-910 

M 

b- 

04 

cb 

2-455 

b- 

CO 

co 

rH 

w 

CO 

29-59 

28-51 

26-43 

CO 

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04 

23-76 

22-18 

20-59: 

19-01 

17-42 

15-84 

04 

M 

rH 

12-67 

11-09 

9-505 

rH 

04 

'04 

lb 

6-336 

4-752 

3-168 

2-376 

M 

CO 

tp 

rH 

idfX) 

CO 

CO 

CO 

04 

27-56 

25-78 

24-50 

22-97 

21-44 

06-61 

18-37 

M 

CO 

cb 

rH 

15-31 

04 

b- 

cb 

12-20 

10-72 

9-188 

b^ 

‘O 

CO 

lb 

6-125 

M 

04 

>o 

M 

3-062 

2-297 

rH 

CO 

tp 

fH 

Hm 

CO 

128-09 

26-61 

25-13 

23-65 

22-1S 

20-70 

19-22 

M 

tH 

lb 

rH 

16-26 

CO 

b- 

M 

rH 

13-31 

11-83 

10-35 

rH 

b- 

CO 

cb 

7-393 

5-914 

4-436 

2-957 

CO 

rH 

04 

04 

04 

M 

rH 

cofx 

CO 

CO 

rH 

ib 

04 

25-66 

24-23 

22-81 

21-3S 

19-96 

18-53 

17-10 

CO 

CO 

tb 

rH 

14-25 

12-83 

11-40 

9-980 

M 

tO 

to 

cb 

7-129 

5-703 

tH 

lH 

04 

M 

2-851 

CO 

co 

rH 

04 

co 

04 

M 

rH 

M-f 

CO 

26-08 

rH 

b- 

M 

04 

M 

CO 

CO 

04 

21-97 

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19-22 

17-85! 

16-47 

15-10 

cc 

b- 

cb 

rH 

12-36 

10-98 

9-610 

8-237 

6-865 

04 

04 

M 

tb 

4-119 

2-746 

2-059 

CO 

iH 

CO 

rH 

MX 

CO 

O 

to 

04 

co 

b- 

CO 

04 

22-44 

21-12 

19-80 

18-48 

17-16 

15-84 

14-52 

13-20 

11-88 

10-55 

9-740 

O 

04 

04 

lb 

109-9 

5-280 

3-960 

O 

M 

CC 

04 

1-980 

O 

04 

CO 

rH 

CO 

1- 

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04 

rH 

CO 

04 

04 

tp 

04 

20-28 

19-01 

M 

b- 

tb 

b- 

M 

cb 

04 

M 

04 

cb 

rH 

12-67 

11-41 

10-14 

8-871 

M 

o 

co 

lb 

b- 

CO 

CO 

cb 

5-069 

3-802 

to 

co 

to 

04 

rH 

O 

04 

rH 

i 1*267 

tHX 

04 

23-07 

21-86 

20-65 

19-43 

18-22 

O 

o 

tb 

rH 

04 

b- 

tb 

rH 

14-57 

13-36 

12-14 

10-93 

9-716 

8-502 

b- 

CO 

04 

lb 

6-072 

4-85S 

M 

M 

CO 

cb 

04 

04 

M 

04 

1-822 

O 

rH 

04 

rH 

crhf 

04 

04 

04 

20-91 

19-75 

CO 

to 

CO 

2f-ZI 

16-26 

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i-J 

to 

rH 

M 

04 

co 

rH 

12-78 

11-62 

10-45 

M 

04 

04 

04 

8-132 

0Z6-9 

5-80S 

b- 

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CO 

M 

3-485 

2-320 

rH 

M 

b^ 

rH 

04 

CO 

rH 

rH 

u«> 

O 

rH 

04 

96-61 

18-85 

M 

1^ 

lb 

rH 

16 63 

15-521 

04 

M 

M 

rH 

13-311 

12-20 

11-09 

r-H 

co 

C4 

• 

04 

S-872 

co 

CO 

t^ 

lb 

M 

tO 

CO 

cb 

to 

M 

tp 

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4-436 

04 

CO 

co 

co 

rH 

04 

04 

1-663 

04 

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rH 

t-h 

Mm 

04 

20-06 

10-61 

tO 

04 

lb 

r— i 

16-89 

15-84 

CO 

1^ 

Ml 

rH 

13-73 

12-67 

11-62 

10-56 

M 

O 

tp 

04 

CO 

M 

M 

cb 

7-392 

co 

co 

co 

cb 

o 

CO 

04 

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CO 

co 

cb 

04 

rH 

rH 

04 

M 

CO 

tp 

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CO 

to 

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04 

04 

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w 

rH 

rH 

HM 

rH 

f^X 

rH 

M* 

rH 

MX 

rn 

rH 

tix 



M^ 

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358 


Weight of Flat Rolled Iron per Foot, 


c 

c 

fc 


e 

M 


c 

PS 

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m 

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X 

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CO 

CO 

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to 

CO 

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cz 

o 

IH 

t— i 

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05 

05 

CO 

00 

X- 

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to 

to 

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CO 

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CO 

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Weight or materials. 


359 


Weight Per Square Foot in Pounds. 


Tbicknt/i 
in iooiiti. 

I 

Cast Iron. 

Wrought or 
Sheet Iron. 

1 

Sheet Copper, 

Sheet I.cad. 

Sheet Zinc 

1*6 

2*346 

2*517 

2*890 

3-694 

2-320 

4 

4-693 

5*035 

5*781 

7-382 

4-642 

A 

7*039 

7*552 

8*672 

11-074 

6-961 

i 

9-386 

10-070 

11-562 

14-765 

9-275 

IT 

Is 

11*733 

12-588 

14*453 

18-456 

11.61 

3 

14*079 

15-106 

17-344 

22*148 

13-93 

T 

T5 

16-426 

17-623 

20*234 

25*839 

16*23 

* 

18*773 

20 141 

23-125 

29*530 

18*55 

Iff 

21*119 

22-659 

26-016 

33*222 

20-87 

t 

23*466 

25-176 

28-906 

36-913 

23-19 


25*812 

27-694 

31-797 

40*604 

25-53 

i 

28-159 

30-211 

34-688 

44-296 

27-85 

ij 

30-505 

32-729 

37-578 

47*987 

30-17 

1 

32-852 

35*247 

40-469 

51-678 

32-47 

ff 

35-199 

37*764 

43-359 

55-370 

34*81 

l 

37*545 

40-282 

46-250 

59-061 

37*13 

n 

42-238 

45-317 

52*031 

66-444 

41*78 

u 

46*931 

50-352 

57-813 

73-826 

46-42 

it 

51-625 

55-387 

63-594 

63*594 

51-04 

n 

56-317 

60-422 

69-375 

88*592 

55-48 

it 

61-011 

65-458 

75-156 

95-975 

60*35 

if 

65-704 

70-493 

80-938 

103-358 

65.00 

it 

70-397 

75*528 

86-719 

110-740 

69-61 

2 ! 

75-090 

80-563 

92:500 

118-128 

74*25 


Weight of Copper Rods or Bolts per Foot, 


Diapieter. 

1 

Weight. 

Diameter. 

Weight. 

Diameter. 

Weight. 

Diameter 

Weight. 

Inches. 

Pounds. 

Inches 

Pounds 

Inches. 

Pounds. 

Inches. 

Pounds. 

1 

0-1892 

1 

3-0270 


10-642 

k 

34-487 

T5 

0-2956 

V« 

3-4170 

2 

12-108 

31 

37*081 

t 

0-4256 

H 

3-8912 

2 t 

13-668 

3t 

39*737 

1 

rs 

0-5794 

i?. 

4-2688 

21 

15-325 

3f 

42.568 

i 

0-7567 

n 

4-7298 

23 

17-075 

3f 

45-455 

9 

rs - 

0-9578 

h \ 

5-2140 

2 * 

18*916 

4 

48-433 

t 

1-1824 

n 

5-7228 

2 f 

20-856 

41 

53*550 

U 

1-4307 

h \ 

6-2547 

21 

22-891 

41 

61-321 

i 

1*7027 

n 

6-8109 

2 f 

25-019 

4f 

68-312 

n 

1-9982 

i 9 

lie 

7-3898 

3 

27-243 

5 

76-130 

t 

2-3176 

It 

7-9931 

3t 

29-559 

51 

91*550 

1 « 1 

2-6605 

u 1 

9-2702 

31 

31-972 

6 

109* 










































American Wire Gauge. 


300 


Gauge 

mini. 

Size 

inches. 

Rolled Plates. 

Weight per square foot. 

Drawn Wire. 

Weight per 1000 feet. 

No. 

In. 

Iron. 

Steel. 

Cornier 

Brass. 

Iron. 

Steel. 

Copper 

Biass. 



Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

0000 

.4600 

18.75 

18.97 

21.36 

20.84 

566.3 

571.7 

646.8 

634.1 

000 

.4006 

16.70 

16.90 

19.01 

18.56 

449.1 

453.3 

512.9 

502.9 

00 

.3648 

14.87 

15.05 

16.93 

16.52 

356.1 

359.5 

406.8 

398.8 

0 

.3249 

13.24 

13.40 

15.08 

14.72 

282.4 

285.1 

322.5 

316.3 

1 

.2893 

11.79 

11.93 

13.43 

13.11 

224.5 

226.1 

255.8 

250.8 

2 

.2576 

10.50 

10.63 

11.96 

11.67 

177.6 

179.3 

202.9 

198.9 

3 

.2294 

9.354 

9.464 

10.65 

10.39 

140.8 

142.2 

160.8 

167.8 

4 

.2043 

8.330 

8.428 

9.486 

9.255 

111 .7 

112.7 

127.5 

125.1 

5 

.1819 

7.418 

7.505 

8.44S 

8.242 

88.59 

89.43 

101.2 

' 99.20 

6 

.1620 

6.606 

6.683 

7.523 

7.340 

70.26 

70.92 

80.25 

78.67 

i 7 

.1443 

5.882 

5.952 

6.699 

6536 

55.71 

56.24 

63.64 

62.38 

8 

.1285 

5.238 

5.300 

5.966 

5.821 

44.18 

44.60 

50.46 

49.48 

9 

.1144 

4.665 

4.720 

5.313 

5.184 

35.04 

35247 

40.02 

39.24 

10 

.1019 

4.154 

4.203 

4.731 

4.616 

2s.26 

28.05 

31.73 

31.11 

11 

.0907 

3.700 

3.743 

4.213 

4.110 

22.03 

22.24 

25.16 

24.63 

12 

.080S 

3.294 

3.S33 

3.752 

3.661 

17.47 

17.64 

19.95 

19.57 

13 

.0720 

2.934 

2.968 

3.341 

3.260 

13.85 

13.99 

15.82 

15.52 

14 

.0641 

2.613 

2.643 

2.978 

2.903 

10.99 

11.03 

12.55 

12.31 

15 

.0571 

2.327 

2.354 

2.650 

2.585 

8.717 

8.899 

9.953 

9.761 

16 

.0508 

2.072 

2.096 

2.359 

2.302 

6.913 

6.978 

7.896 

7.741 

17 

.0152 

1.845 

1.867 

2 .JUI1 

2.050 

5.481 

5.532 

6.26 L 

6.137 

18 

.0403 

1.643 

1 662 

1.372 

1.826 

4.347 

4.387 

4.965 

4.867 

19 

.0359 

1.463 

1.480 

1.666 

1.626 

3.447 

3.479 

3.937 

3.861 

20 

.0320 

1.303 

1.318 

1.484 

1.448 

2.735 

2.761 

3.125 

3.064 

21 

.0285 

1.160 

1.174 

1.321 

1.289 

2.168 

2.188 

2.476 

2.428 

22 

.0253 

1.033 

1.045 

1.176 

1.148 

1.720 

1.736 

1.964 

1.926 

23 

.0226 

.9203 

.9310 

1.048 

1.023 

1.363 

1.376 

1.557 

1.527 

24: 

.0201 

.8195 

.8291 

.9334 

.9105 

1.081 

1.091 

1.235 

1.211 


.0179 

.7298 

.7383 

.8311 

.8109 

.8575 

.8656 

.9795 

.9603 

20 

.0159 

.6499 

.6575 

.7401 

.7221 

.6801 

.6864 

.7768 

.7616 

27 

.0142 

.57S7 

.5855 

.6591 

.6430 

.5393 

.5444 

.6160 

.6039 

28 

.0126 

.5151 

.5214 

.5S69 

.5726 

.4277 

.4317 

.4885 

.4789 

29 

.0113 

.4580 

.4643 

.5227 

.5099 

.3391 

.3422 

.3873 

.3297 

30 

.0100 

.4087 

.4135 

.4654 

.4541 

.3699 

.2714 

.3072 

.3012 

31 

.0089 

.3640 . 

.3683 

.4145 

.40 J 4 

.2134 

.2153 

.2437 

.2389 

34 

.0080 

.3241 

.3279 

.3691 

.3601 

.1691 

.1707 

.1932 

.1894 

33 

.0071 

.2887 

.2920 

.3287 

.3207 

.1341 

.1354 

.1532 

.1502 

34 

.0063 

.2570 

.2600 

.2927 

.2856 

.1063 

.1073 

.1216 

.1192 

35 

.0056 

.2289 

.2316 

.2606 

.2543 

.0845 

.0853 

.0965 

.0947 

36 

.0050 

.2039 

.2062 

.2322 

.2265 

.0669 

.0075 

.0764 

.0750 

37 

.0045 

.1816 

.1837 

.2067 

.2017 

.0531 

.0536 

.0606 

.0594 

38 

.0040 

.1617 

.1636 

.1841 

.1796 

.0118 

.0424 

.O4S0 

.047 L 

39 

.0035 

.1440 

.1456 

.1610 

.1600 

.0334 

.0337 

.0381 

.0374 

40 

.0031 

.1282 

.1297 

.1460 

.1424 

.0208 

.0267 

.0302 

.0297 

Spec 

grav. 

7.828 

7.92 

8.917 

8.70 

7.85 

7.93 

8.96 

8.78 


The American Wire Gauge is introduced and manufactured by J. R. Brown & 
Sharpe, of Providence, It. I., and is to be had in the principal hardware stores in the 
I country. It is adopted by most manufacturers of plates and wire, and is now con- 
isidered the American Standard Gauge. 







































Birmingham Gauge. 


361 


Birmingham Gauge for Wire, Sheet Iron and Steel. 

Weight per Square Foot in Pounds. 


Thickness by 

Thickness in 

Sheet and 

Sheet Cast 

Sheet 

Sheet Lead. 

Thickness in 

the 

gauge. 

inches. 

Boiler Iron. 

Steel. 

Copper. 

inches. 

No. 0000 

0.454 

18.267 

18.259 

20.566 

26.75 

7 

: 16 


000 

0.425 

17.053 

17.280 

19.252 

25.06 

27 

: 64 


00 

0.3S0 

15.247 

15.451 

17.214 

22.42 

3 

: 8 


0 

0.340 

13.7 

J4.0 

15.6 

20.06 

11 

: 32 


1 

0.300 

12.1 

12.4 

13.8 

17.72 

5 

: 16 


2 

0.284 

11.4 

11.7 

13.0 

16.75 

9 

: 32 


3 

0.259 

10.4 

10.6 

11.9 

15.26 

1 

: 4 


4 

0.238 

9.60 

9.80 

11.0 

14.02 

7 

: 32 


5 

0.220 

•8.85 

9.02 

10.1 

12.98 

7 

: 32 


6 

0.203 

8.17 

8.33 

9.32 

11.98 

7 

: 32 


7 

0.180 

7.24 

7.38 

8.25 

10.63 

3 

: 16 


8 

0.165 

6.65 

6.78 

7.59 

9.73 

3 

: 16 

.2 

9 

0.148 

5.96 

6.08 

6.80 

8.72 

5 

: 32 

d 

C3 

10 

0.134 

5.40 

5.51 

6.16 

7.90 

5 

: 32 

a 

11 

0.120 

4.83 

4.93 

5.51 

7.08 

1 

: 8 

A 

i 

12 

0.109 

4.40 

4.50 

5.02 

6.42 

1 

: 8 

O 

s-» 

13 

0.095 

3.83 

3.91 

4.37 

5.60 

3 

: 32 

o 

0) 

14 

0.083 

3.34 

3.41 

3.81 

4.90 

3 

: 32 

3 

15 

0.072 

2.90 

2.96 

3.31 

4.25 

1 

: 16 

0> 

16 

0.065 

2.62 

2.67 

3.00 

3.83 

1 

: 16 

w 

17 

0.058 

2.34 

2.39 

2.67 

3.42 

1 

: 16 


18 

0.049 

1.97 

2.01 

2.25 

2.90 

1 

: 16 

tc 

d 

19 

0.042 

1.69 

1.72 

1.93 

2.48 

3 

: 64 

c3 

o 

20 

0.035 

1.41 

1.42 

1.61 

2.04 

3 

: 64 


21 

0.032 

1.29 

1.31 

1.47 

1.89 

3 

: 64 


22 

0.028 

1.13 

1.15 

1.29 

1.65 

1 

: 32 


23 

0.025 

1.00 

1.02 

1.14 

1.47 

1 

: 32 

3 

24 

0.022 

0.885 

0.903 

1.01 

1.30 

1 

: 32 

£ 

<x> 

25 

0.020 

0.805 

0.820 

0.918 

1.18 

1 

: 32 

5Z5 

26 

0.018 

0.724 

0.738 

0.826 

1.06 

1 

: 64 


27 

0.016 

0.644 

0.657 

0.735 

0.945 

1 

: 64 


28 

0.014 

0.563 

0.574 

0.642 

0.826 




29 

0.013 

0.523 

0.533 

01597 

0.767 




30 

0.012 

0.483 

0.493 

0.551 

0.708 




31 

0.010 

0.402 

0.410 

0.480 

0.600 




32 

0.009 

0.362 

0.370 

0.420 

0.532 




33 

0.008 

0.322 

0.328 

0.370 

0.472 




34 

0.007 

0.282 

0.288 

0.323 

0.413 




35 

0.005 

0.230 

0.235 

0.262 

0.309 




36 

0.004 

0.170 

0.173 

0.194 

0.236 




Birmingham Gauge for Silver and Gold. 


No. 

Thick. 

Inch. 

No. 

Thick. 

Inch. 

No. 

Thick. 

Inch. 

No. 

Thick.! 

Inch. 

No. 

Thick. 

Inch. 

No. 

Thick 

Inch. 

1 

.004 

7 

.015 

13 

.036 

19 

.064 

25 

.095 

31 

.133 

2 

.005 

8 

.016 

14 

.041 

20 

.067 

26 

.103 

32 

.143 

3 

.008 

9 

.019 

15 

.047 

21 

.072 

27 

.113 

33 

.145 

4 

.010 

10 

.024 

16 

.051 

22 

.074 

28 

.120 

31 

.148 

5 

.013 

1 1 

.029 

17 

.057 

23 

.077 

29 

.120 

35 

.158 

6 

.013 

12 

.031 

18 

.061 

24 

.082 

30 

.126 

36 

.167 







































362 


Proportion op Bolts and Nuts. 


Proportions of Bolts and Nuts. Number of Threads per In. 



0 




1® 

Diameter. 

Number of 
per In 

Threads 

ch. 

M 

18? 

w 


ffils 

3 inch. 

4| 

5f 

5 

7xV 

2f 

i 

¥ 

20 

10 

2* 

n 

4f 

4f 

6f 

Ol 

5 

Iff 

18/ 

9 

2f 

3| 

4| 

4f 

5f4 

2 

3 

8 

16 

9 

2f 

3| 

3f 

3* 

5* 

If 

IS 

14 

8 

2 

3 

3f 

3f 

4f 

If 

f 

12 

7 

H 

2* 

31 

3 

4 t V 

If 

I 

11 

7 

if 

2| 

3 

2| 

3f 

If 

3 

¥ 

10 

6 

if 

2f 

2f 

2f 

v-<!^ 

CO 

H 

f 

9 

6 

if 

2f 

2| 

2f 

3f 

If 

1 

8 

5 


2 

2| 

2f 

3r\ 

4tV 

If 

7 

4 

»a 

ii 

if 

2i 

2 

2f 

1 

If 

7 

3f 

In 

if 

if 

If 

2f 

f 

If 

6 

3 

ii 

if 

if 

If 

2* 

f 

If 

5 

. 2f 

5 i 

i 5 
i T? 

if 

If 

2f 

f 

2 

4f 

2f 

i 

1 TiT 

H 

4 A 

If 

t 

2f 

4 

2 

t 

1 

if 


iff 

f 

3 

3f 


fs 

f 

l 

1 

Ws 

f 

3f 

3f 


f 

s 

I 

f 

i 


TS 

4 

3 


& 

f 

1 

s 

. * 

Its 

3 

¥ 

4f 

2f 


a 

1 


t 

8 

¥ 

Its 

T^S 

5 

2f 


fV 

ts 

* 

TS 

H 

T¥ 

5f 

2f 


f 

8 

8 

tV 

TZ 

ff 

f 

6 

2f 



The above proportions of bolts and nuts were established by Sir Joseph Whitworth. 


Weight in Pounds of Nut and Bolt-Head. 


Diameter of Bolt in Inches. 


Head and Nut. 

l 

¥ 

s 

¥ 

1 

2 

5 

¥ 

3 

¥ 

f 

1 

If 

If 

If 

2 

2f 

3 

Hexagon, 

.017 

.057 

.128 

.267 

.43 

.73 

1.1 

2.14 

3.77 

5.62 

8.75 

17.2 

28.8 

Square, . . 

.021 

.070 

.164 

.321 

.553 

.832 

1.31 

2.56 

4.42 

7.00 

10.5 

21 . 

36.4 


\ 



















































Screw-Threads. 


363 


Proportions of Screw-Threads, Nuts and Boltlieads. 


Diam. of 

Threads 

Diamet. 

Width 

I Outside 

Inside 


height 

Screw. 

k AAM i 

per inch. 

of core. 

of flat. 

diamet. 

diamet. 

Diagonal. 

of h’d. 









•i/-t 

20 

•185 

•0062 

•9/16 

•1/2 

•11/16 

• 1/4 

•5/16 

18 

•240 

•0070 

*11/16 

•19/32 

•13/16 

•19/64 

•3/8 

16 

•294 

•0078 

•25/32 

•11/16 

•31/32 

•11/32 

•7/16 

14 

•344 

•0089 

•43/48 

•25/32 

1-1/16 

•25/64 

*1/2 

13 

•400 

•0096 

1- 

•7/8 

1-1/4 

• 7/16 

•9/16 

12 

•454 

•0104 

1-7/64 

•31/32 

1-5/16 

•31/64 

•5/8 

11 

•507 

•0113 

1-7/32 

1-1/16 

1-1/2 

•17/32 

•3/4 

10 

•620 

•0125 

1-7/16 

1-1/4 

1-3/4 

• 5/8 

*7/8 

9 

•731 

•0140 

1-21/32 

1-7/16 

2-1/32 

•23/32 

1* 

8 

•837 

•0156 

1-7/8 

1-5/8 

2-5/16 

•13/16 

1-1/8 

7 

•940 

•0180 

2-3/32 

1-13/16 

2-1/2 

•29/64 

1-1/4 

7 

1-065 

•0180 

2-5/16 

2- 

2-27/32 

1* 

1-3/8 

6 

1-160 

•0210 

2-1/2 

2-3/16 

3-1/10 

1*3/32 

1-1/2 

6 

1-284 

•0210 

2-3/4 

2-3/8 

3-3/8 

1-3/16 

1-5/8 

5-1/2 

1-389 

•0227 

2-15/16 

2-9/16 

3-5/8 

1-9/32 

1-3/4 

5- 

1-490 

•0250 

3-3/16 

2-3/4 

3-29/32 

1-8/8 

1-7/8 

5- 

1-615 

•0250 

3-13/32 

2-15/16 

4-3/16 

1-15/32 

2- 

4-1/2 

1-712 

•0280 

3-5/8 

3-1/8 

4-7/16 

1-9/16 

2-1/4 

4-1/2 

1-962 

•0280 

4-1/16 

3-1/2 

4-31/32 

1-3/4 

2-1/2 

4- 

2-175 

•0310 

4-1/2 

3-7/S 

5-1/2 

1-15/16 

2-3/4 

4- 

2-425 

•0310 

4-29/32 

4-1/4 

6- 

2-1/8 

3- 

3-1/2 

2-628 

•0357 

5-3/8 

4-5/8 

6-9/16 

2-5/16 

3-1/4 

3-1/2 

2-878 

•0357 

5-3/4 

5* 

7-1/8 

2-1/2 

3-1/2 

3-1/4 

3-100 

•0384 

6-7/64 

5-3/8 

7-5/8 

2-11/16 

3-3/4 

3- 

3-317 

•0410 

6-5/8 

5-3/4 

8-3/16 

2-7/8 

4- 

3- 

3-566 

•0410 

7-3/64 

6-1/8 

8-11/16 

3-1/16 

4-1/4 

2-7/8 

3-798 

•0435 

7-1/2 

6-1/2 

9-1 f 4 

3-1/4 

41/2 

2-3/4 

4-027 

•0460 

7-31/32 

6-7/8 

9-3/4 

3-7/16 

4-3/4 

2-5/8 

4-255 

•04S0 

8-3/8 

7-1/4 

10-9/32 

3-5/8 

5- 

2-1/2 

4-480 

•0500 

8-13/16 

7-5/8 

10-13/16 

3-13/16 

5-1/4 

2-1/2 

4-730 

•0500 

9-1/4 

8- 

11-3/8 

4- 

5-1/2 

2-3/8 

4-953 

•0526 

9-11/16 

8-3/8 

11-29/32 

4-3/16 

5-3/4 

2-3/8 

5-203 

•0526 

10-1/8 

8-3/4 

12-7/16 

4-3/8 

6- 

2-1/4 

5-423 

•0555 

10-9/16 

9-1/8 

12-9/10 

4-9/16 


Englishmen make the angle of the thread 55°, with round top and bottom; 
whilst in the U. S. we make the angle of the thread 60°, with flat top and bot¬ 
tom, and of the following proportions, which were recommended by a special 
committee appointed by the Franklin Institute of Philadelphia. For full infor¬ 
mation see Journal of the Institute, May, 1864, and Jan., 1865. 

Notation. All dimensions in inches. 


D = outside diameter of screw. 
d — diameter of root of thread, or of 
hole in the nut. 
p = pitch of screw. 
t — number of threads per inch. 
f = flat top and bottom. 
o = outside diameter of hexagon nut 
or bolthead. 

V 16 D -f 10 — 2-909 


t = inside diameter of hexagon, or 
side of square nut or bolthead. 
s — diagonal of square nut or bolthead. 
h = height of rough or unfinished bolt- 
head. 

The height of finished nut or bolt- 
head is made equal to the diameter D 
of the screw.* 

t = — ■ s = 1-414 1 . 

P 


. _ 1299 . 3 2> 1 

(x D —•-- t == — 4- 

t 2^8 


o = 1-155 t. 


/= 2L. 
J 8 


* Whitworth makes the height of the nut about half the hexagon diameter o. 



































Brass Tubes, Strength of Iron and Copper. 


364 


Weight, Size, Price ami Surface of Copper and Brass Tubes, 

10 feet, long. 


Outside 

Bir.W. 

Gauge 

Weight of tube. 

Price 

Whole 

Outside 

Rir "NY - 

Weight of tube. 

Price 

Whole 

Diam¬ 

eter. 

Brass. 

Cop. 

per 

tube. 

Sur¬ 

face. 

Diam¬ 

eter. 

Gauge 

Brass. 

Cop. 

per 

tube. 

Sur¬ 

face. 

Inches. 

No. 

Lbs. 

Lbs. 

$ ets. 

Si]. Ft. 

Inches. 

No. 

Lbs. 

I.bs. 

$ 

eta. 

Sq. Ft. 

0.625 

18 

3.478 

3.681 

2 30 

1.636 

2. 

14 

18.84 

19.95 

8 

00 

5.236 

0.75 

17 

4.950 

5.241 

2 97 

1.963 

2.125' 

14 

19.07 

20.18 

8 

20 

5-603 

.8125 

17 

5.372 

5.679 

3 00 

2.127 

2.25 

14 

21.18 

22.42 

8 

90 

5.890 

0.875 

17 

5.775 

6.114 

3 40 

2.290 

3.375 

14 

22.32 

23.65 

9 

45 

6.217 

.9375 

16 

6.954 

7.362 

3 60 

2.454 

2.5 

14 

23.53 

24.89 

9 

95 

6.544 

1. 

16 

7.418 

5.854 

3 92 

2.618 

2.625 

14 

24.07 

26.12 

10 

45 

6.872 

1.125 

16 

8.354 

8.835 

4 40 

2.945 

2.75 

14 

25.83 

27.35 

11 

00 

7.200 

1.25 

15 

10.21 

10.83 

4 70 

3.272 

3. 

13 

37.00 

39.17 

13 

70 

7.854 

1.375 

15 

11.23 

11.91 

4 95 

3.600 

3.25 

13 

40.00 

42.34 

14 

85 

8.508 

1.5 

15 

12.28 

13.U0 

5 20 

3.927 

3.5 

13 

43.10 

45.61 

16 

00 

9.163 

1.625 

15 

13.30 

14.08 

5 65 

4.254 

4. 

12 

49.39 

52.30 

17 

00 

10.47 

1.75 

14 

16.5 

17.45 

7 00 

4.581 

4.5 

12 

55.55 

58.8 

19 

10 

11.78 

1.812 

14 

17.08 

18.08 

7 20 

4.745 

5. 

12 

61.44 

65.00 

21 

00 

13.08 

1.875 

14 

17.72 

18.75 

7 50 

4.908 

6 . 

11 

81.58 

86.35 

26 

00 

15.71 

1.937 

14 

18.26 

19.32 

7 75 

5.072 

8. 

11 

108.8 

115.0 

34 

50 

20.95 


Seamless-Drawn Brass Tubes for Plumbing, 

In lengths of 10 feet. Screw-coupling on one end of each length. Price per tube. 


Diameters, inches. 

5 

8 

3 

¥ 

1 

1 

n 

n 


$ cts. 

$ cts. 

$ cts. 

8 cts. 

$ cts. 

$ cts. 

Plain tubes, . 

2 50 

3 00 

4 50 

6 00 

7 00 

8 00 

Tinned tubes, . 

3 00 

3 50 

5 00 

7 00 

8 00 

9 00 


Price of Taps, Dies and Stocks. 


Diameters, inches. 

5 

J 

i 

I 

1 

H 



$ cts. 

$ cts. 

$ cts. 

$ cts. 

$ cts. 

S cts. 

Taps, .... 

2 50 

2 75 

3 00 

3 50 

4 50 

6 00 

Solid dies, . 

3 50 

3 50 

3 50 

3 50 

3 50 

4 00 


Stocks, $8, net. 


Price for Each Extra Coupling. 


Diameters, inches. 

5 

J 

3 

¥ 

7 

J 

1 

H 

n 


$ cts. 

$ cts. 

$ ets. 

$ cts. 

S cts. 

$ cts. 

Straight couplings, 

20 

25 

35 

40 

45 

50 

Elbows, ... 

26 

35 

48 

53 

65 

80 

Tees, .... 

30 

40 

65 

60 

85 

1 00 

Cross couplings, 

45 

60 

85 

90 

1 50 

2 20 


The prices are only approximate. The price of copper tubes is 12 to 13 per cent, 
more than of brass. 

Brass and copper tubes are manufactured at the American Tube Works, Boston, 
Mass.; Merchant & Co., 507 Market street, Philadelphia, agents. 


Proportionate Tensile Strength of Rolled Iron and Copper, 

In pounds per square inch, at different temperatures , Fahr. and Centigrade. 


Fahr. 

Cent. 

Iron. 

Copper. 

Fahr. 

Cent. 

Iron. 

Copper. 

32 

0. 

55,000 

32.800 

800 

427 

51,800 

17,200 

100 

37.7 

58,200 

32,300 

900 

483 

45,000 

14,000 

200 

93.3 

62,800 

31,000 

1000 

540 

37,000 

11,000 

300 

149. 

65,750 

29,500 

1200 

650 

25,000 

7,000 

400 

205. 

67,000 

27,400 

1500 

820 

16,500 

3,000 

500 

260. 

66,000 

25,300 

2000 

1090 

7,000 

0.0000 

600 

316. 

62,700 

23,000 

2500 

1370 

2,500 

Fused to 

700 

370. 

57,800 

20,100 

3000 

1650 

Fused. 

liquid. 

























































































Nails, Rivets, Irou, Copper, Zinc. 


365 


Composition Nails, Copper and Iron Rivets. 


bO 

Composition Nails. 

Braziers’ Copper Rivets. 

O; 

to 

Iron Rivets. 

o 

Thick. 

Length. 

In 1 
ft. 

Diameter. 

Length. 

In 10 
fts. 

eg 

o 

Diameter. 

1 Length. 

In 10 
fts. 

No. 

Inches. 

Inches. 

Num 

Inches. 

Inches. 

Num. 

No. 

Inches. 

Inches. 

Num. 

1 

0.04 

3/4 

290 

3 A0 

1/2 

2384 

0 

3/16 

1/2 

3280 

2 

0.05 

7/8 

260 

1/4 

1/2 

1018 

1 

1 /4 

1/2 

1276 

3 

0.06 

1 inch. 

212 

1 1 4 

9/16 

983 

2 

1/4 

9/16 

1130 

4 

0.07 

1.1/8 

201 

5 AG 

9/16 

573 

3 

5/16 

9/16 

654 

5 

0.08 

1.1/4 

199 

5/16 

5/8 

516 

4 

5/16 

5/8 

589 

6 

0.09 

1 inch. 

190 

3/8 

7/8 

357 

5 

3/8 

7 /8 

407 

7 

0.10 

1.1/8 

184 

3/S 

15/16 

334 

6 

3 /8 

15/16 

380 

8 

0.10 

1.1/4 

168 

7/16 

1 inch. 

210 

7 

7/16 

1 inch. 

239 

9 

0.11 

1.1 /2 

110 

1/2 

1.3 A 6 

141 

8 

1/2 

1.3/16 

160 

10 

0.11 

1.5/8 

101 

9/16 

1.5/16 

99.5 

9 

9/16 

1.5/16 

112 

11 

0.12 

1.3/4 

74 

5/8 

1.7/16 

71.9 

10 

5/8 

1.7/16 

81.7 

12 

0.12 

2 inches 

64 

11/16 

1.9/16 

53.8 

11 

ii/16 

1.9/16 

61.3 

13 

0.13 

2.1 / 4 

59 

3/4 

1.3/4 

41.6 

12 

3/4 

1.3/4 

47.3 

14 

0.14 

2.1/2 

51 

13 A6 

1.13/16 

32.8 

13 

13/16 

1.13 /16 

37.3 

15 

0.15 

2.3/4 

43 

7/8 

2.1 A6 

26.3 

14 

7/8 

2.1 A6 

30. 

16 

0.16 

3 inches 

35 

1 inch. 

2.3/8 

16.7 

15 

1 inch. 

2.3 f 8 

19. 


H.ei»gth in Indies of Penny Nails. 


1 in. 
2d. 


1.25 

1.5 

1.75 

2 

2.25 

2.5 

2.75 

3 

3.25 

3.5 

4 

4.25 

5 

5.5 

3d. 

4d. 

5 d. 

6 d. 

Id. 

8 d. 

9 d. 

10 

12 

16 

20 

30 

40 

50 


6 

60 


Sheet Zinc and Iron. 


Sheet Zinc. 

Size 84 in. by 24, 28, 32, 36 and 40 inches. 

Russia Sheet Iron. 

Size 28 X 56 in. = 10.88 sq. feet. 

Zinc 

Width of Sheet. 

Bir. W. 

Russian 

Weight per 

Bir. W. 

gauge. 

24 

33 

40 

gauge. 

gauge. 

Sheet. 

Sq. Ft. 

gauge. 

No. 

Pounds. 

Pounds. 

Pounds. 

No. 

No. 

Pounds. 

Pounds. 

No. 

8 

6.23 

9.68 

12.1 

28 

7 

6.25 

0.574 

29 

9 

7.20 

11.2 

14.0 

27 

8 

7.25 

0.666 

28 

10 

8.00 

124 

15.6 

26 

9 

8. 

0.735 

27 

11 

8.90 

13.8 

17.3 

25 

10 

9. 

0.827 

26 

12 

10.1 

15.7 

19.7 

24* 

11 

10 . 

0.91S 

25 

13 

11.1 

17.3 

21.6 

23 

12 

10.75 

0.987 

24* 

14 

12.4 

19.3 

24.1 

22 

13 

11.75 

1.08 

24 

15 

16.2 

25.2 

31.6 

21 

14 

12.5 

1.15 

23* 

16 

17.4 

27.1 

33.9 

20 

15 

13.5 

1.24 

221 

18 

21.9 

34.0 

42.6 

18 

16 

14.5 

1.33 

2H 


Multiply the weight 
of the Pattern by 


weight of the Castings. 


To find tile Weiglit of Castings, hy the Weight of Pine Patterns 

RULE.— C 12 for Cast Iron, 

19 foJ Lea? l anfl the P rodnct is the 

12.2 for Tin, 

11.4 for Zinc, 

Reductions for Round Cores and Core-prints. 

Rule. Multiply the square of the diameter by the length of the Core in 
inches, and the product by 0.017, is the weight of the pine core, to be deducted 
from the weight of the pattern. 


Shrinking of Castings. 


( 

Cast Iron, . 


Pattern-Makers' Rule 1 

Brass, . . 

tV 

should be for \ 

Lead, • • • 

1 

? 


Tin, . , . 

X 

T? 

V 

Zinc, . . . 

3 

T"t> 


of an inch longer per 
linear foot. 






































































Geaiuno. 



Notation. 

P — pitch,—the distances between the centres of two teeth in the 
pitch circle. 

D = diameter ^ 

C = circumference 
M= number of teeth 
N = number of revolutions 
d — diameter 
c = circumference 
m = number of teeth 
n = number of revolutions 


.of the wheel. 


•of the pinion. 


Pitch 


No. of teeth 




C 

M 

n D 

~W 

SL 

p 

n D 


Circum. 



8 


D : d = C : c = M : m — n : N 


Formula 2. Pitch P=* 


Example 1. A wheel of D = 40 inches in diameter, is to have M =* 
75 teeth. Required tho pitch P — ? 

14X^0 __ j.gg inches nearly. 

75 J 

Example 2. The pitch of teeth in a wheel, is to be P — 1*71 inches, and 
having M = 48 teeth. Required the diameter D = ? of the wheel. 

1.71 v d.Q 

Formula 7. Diam. D of pitch circle =--—— = 26*14 in. 

3-14 





























Gearing. 


367 


Construction of Teeth for Wheels* 

Draw the radius R r, and pitch circle PR. Through r draw the line o of at 
an angle of 75° to the radius R r. 


, i 180 

{ wheel, v = —— 

M 

180 

pinion, V =■ ^ 

D : d = sin. V: sin. v. 

d sin. V 


Diameter of the - 


wheel, D = 


sin. v 
D sin. v 


P inion ’ d = 1KT 

Pitch (chord) of teeth f wheel, P = D sin. v. 
in the pitch circle 1 pinion, P— d sin. V. 

Approximate pitch in the wheel P — 0*028 D. - 


Number of teeth 


wheel, M = 

dM 


pinion, m 


D 


Thickness of tooth, a = 0*46 P* 
Bottom clearance, b = 0*4 P. 
Depth to pitch line, c =* 0*3 P. 

TV , n _ P (m + C'l 

Distance r o. a — 

2 [m — 11 ) 

Distance r o’, e = 0*11 P 


5 

6 

7 

8 


10 

11 

12 

13 

14 


* If a wheel of more than 80 teeth Is to gear a pinion of less than 20 teeth, 
and the wheel and pinion are of the same kind of materials; take the thickness 


of the tooth in the 


f wheel, a = P ^0*42 + - - 15 

J ’ V 700 / 

(pinion ,a = 0*5 P^l — - 16 


A rack is to be considered as a wheel of 200 teeth. 

























308 


Gearing 


Approximate pitch P — 0*028X48—1*34 in. - 


Example with. Plate I • 

Example. A wheel of D = 48 inches diameter is to gear a pinion about 8 
revolutions to 1. Required a complete construction of the gearing? 

7 

8 

9 


Number of teeth 
in the 


wheel, if - ?™ = 112. 


1*34 
112 


tt i/» ,1 / wheel, v = 

Half the an- ( 

gle between •< 

two teeth in I . . ^ 

the P m,on F = 

Diameter of pinion d — 


pinion, m = 
180 


- 14 


112 

180 


=1°3G'. sm=0*028. 


=12051'. sm=0*2?24. 
= 6*043 in. 


14 

48X0*028 


0*2224 


Draw the pitch circle for the wheel and pinion so that they arc tangent to one 
another at r on a straight line between the centres of the circles. 

Pitch in the gearing P = 48X0*028=1*344 in. - 5 

Take this chordial pitch in a pair of compasses, and set it off in the pitch 
circles. 

14 )=0*592iu. 


Thickness of 
tooth 


wheel a = 1*344 ( 0*42 + 


(o*- 


700 

14 


15 


- pinion a = 0*5 1*344^1-^^=0*645 in. 16 

Set off the thickness of tooth in the corresponding pitch circles. 

Bottom clearance b = 0*4X1*344 = 0*5376 in. • 11 

Depth to pitch line c = 0*3 1*344=0*4032 in. - - 12 

, 1*344(112 f 6) n .K QK , . 

Distances r o and j 2(112—11) ~ in> * ^ 

r o' in the wheel | „ __ 

e = 0*11X1*344 v/112 = 0*7126 in. 14 

Setoff these distances on the line o o' from r,—d beyond and e within the 
pitch circle for the wheel; then o is the centre and o m radius for the flank in. 
o' the centre and o' n radius for the face n. Draw circles through o and o' con¬ 
centric with the pitch circle of the wheel. 


Distances r o and 
r o' in the pinion 


{ 


d = 4-48 in. 


13 

14 


2(14—11) 

e = 0*11X1*344 v 3 /14 = 0*356 in. 

Proceed with the pinion in a similar manner as with the wheel. 

On the plate is a scale of inches and decimals, which will bo convenient for 
the above measurements. 


















4 / 7 //'///// 


Ma/c /. 


J.ttyystrom. ( 








































































































Stef .>*■ Teeth. 


369 


Oil tlie Strength of Tcetli in Cast-iron Wheels. 

The strength and durability of cast-iron teeth require that they shall transmit 
a force of 80 lbs. per inch of pitch and per inch breadth of face. 

Let IP = indicated horse-power transmitted. 

“ D = diameter of pitch circle in inches. 

“ N — number of turns of wheel per minute. 

“ F = stress on one tooth in lbs. . 

We have— 

F —126050 

D N 

Having thus found F, we can assume the pitch and find the breadth of face, or 
we can assume the breadth of face and find the pitch. 

Williams & Brown, Tenth and Chestnut streets, rhila., Pa., have for sale, at 50 
cents each,Epicycloidal and Involute Odontographs devised so as to furnish with¬ 
out formulae all the data required to correctly lay out both forms of teeth of 
wheels. The writer’s methods do not agree with Mr. Nystrom’s.—W. D. M. 


To find tlie Diameter of Axles and Shafts. 

Notation. 


d = diameter in inches, in the bearing; and the length of the bearing 1.5 d. 
W = weight in pounds, acting in the bearing. 


Empirical Formulas. 


Water¬ 

wheels. 


iAYi/ 

d = of cast iron. 

18 

V w 

d = -— of wrought iron. 
21 


Common 
Machinery 
in good 
order. 


d = —- of cast iron. 

24 

l/ TJ7' 

d = —- of wrought iron. 
28 


Example 1. A water-wheel weighs 58,680 pounds, and is supported in two bear¬ 
ings. Required the diameter of the wheel axles? The weight acting in each 
bearing will be 58680 : 2 — 29340 pounds, and 


diameter d = 


1 /29340 

21 


= 8.15 inches of wrought iron. 


Example 2. Required the diameter of an axle in a wheel, when the weight 
= 4864 pounds? If the wheel is supported' in two bearings W = 4864 : 2 = 2432 
pounds. 


diameter d = 


V 2432 
28 


= 1.7G inches of wrought iron. 


24 
















370 


Strengi eriaes. 


Example 3. The pressure on th< stea i piston in a working beam engine is 
25,000 pounds. Required the diameter of the beam journals? 


/ 


Rational Method. 

(See The Relative Proportions of the Steam Engine, Marks, J. B. Lippincott Co.) 


Projected area = -= 50 sq. inches for end bearings. 


Projected area = 


50000 


500 


100 sq. inches for centre bearings. 




D 


iypR 




H 


n 


n = number of turns per minute. 

D = inches wrought iron. 

R = radius of crank in feet. 

F = force from the steam piston, lbs. 


E 
n ■■ 


E:d= V R : J r ' 

£ = 4 - 35.7 # 

v n 

= horse-power transmitted. 

= number revolutions per minute. 


When an axle or shaft not only serves as a fulcrum, but effect is transmitted 
by the act of twisting it, the diameter is to be calculated as follows. 

Example 4. The pressure on the piston in a steam engine is F = 45,600 
pounds, applied direct on a crank of 11= 3 feet radius. Required the diameter 
of the shaft and crank pin ? 

■jy 451)00X3 


Diameter of the shaft D 


= 12*9 inches. 


For diameter of crank-pin, see The Relative Poportiont of the Steam-Engine. 


Example 5. A steam engine of 3G8 horse-power is to make 32 revolutions per 
minute. Required the diameter of the main shaft? 


Diameter Z) = 5 


3 /368 
V 32 


=^111 inches. 


Example f>. A cog wheel of R = 6‘5 feet radius is to gear with a pinion of r = 
1-25 feet radius, and to transmit an effect of 231 horse-power with 42 revolutions 
per minute. Required the diameter of the wheel and pinion shafts? The force 
F is acting uniformly at the periphery. 


Diameter of wheel shaft JD — 4’35 


V 2 : 

V 42 


231 


7*66 inches 


D : d — 1/ R : v r 


Diameter of pinion shaft d — 7 


•66 v 7 

V (1-5 


4*41 inches. 
- 1 








































Coefficients f a Ca city and Weight. 


371 


Coefficients for Capacity amt Weight, 



Names of Substances. 

FFF. 

Fit. 

iii. 

Ft\ 

F*. 

u*. 

F *. 

t*. 

Cubic inches, 



1728 

12 

i 

1356 

9-42 

0-78' 

903-7 

0-523 

Cubic feet, - 



1 

-..694 

*.58 

0-785 

-..549 

’.44 

0-523 

-.3 

Gallons, - - 



7-476 

0-052 

-...433 

5-868 

• .498 

*.34 

3-91 

•..226 

Water, fresh, 



62-5 

0-433 

0-036 

49 

0-34 

*. 283 

32-7 

0-019 

Water, salt, - 



64-3 

0-445 

0-037 

' 50-4 

0-35 

0-029 

330 

0-02 

Oil, - - - - 



57-5 

0-4 

0-033 

45-1 

0-313 

0-026 

30 

0-017 

Cast-iron, 



450 

3-12 

0-26 

353 

2-45 

0-204 

235 

0-136 

Wrought-iron, 



487 

3-37 

0-281 

382 

2-65 

0-221 

255 

0-147 

Steel, ... 



490 

3-4 

0-283 

385 

2-67 

0-222 

257 

0-149 

Brass, - - - 



532 

3-68 

0-307 

417 

2-9 

0-241 

278 

0-161 

Tin, ... 



456 

3-16 

0-263 

358 

2-48 

0-207 

239 

0-138 

Lead, ... 



710 

4-92 

0-41 

557 

3S7 

0-322 

371 

0-215 

Zinc, ... 



440 

3-05 

0-254 

345 

2-4 

0-2 

230 

0-133 

Copper, - - 



556 

3-85 

0-321 

436 

3-03 

0-252 

291 

0-168 

Mercury,. - 



850 

5-9 

0-491 

666 

4-63 

0-385 

445 

0-257 

Stone, common, 


156 

1-08 

0-09 

122 

0-85 

0-071 

82 

0-1)47 

Clay, - • - 



135 

0-936 

0-078 

106 

0-735 

0-061 

70 

0-04 

Earth, compact, 


127 

0-S8 

0-0733 

99 

0-692 

0-058 

06 

0-038 

Earth, loose, 



95 

0-66 

0-055 

74 

0-517 

0-043 

50 

0-029 

Oak, dry, - 



58 

0-4 

0-033 

44 

0-316 

0-02C 

30 

0017 

l’ine, - - - 



30 

0-208 

0-017 

24 

0-163 

0-014 

16 

0-009 

Mahogany, - 



66 

0-457 

0-038 

52 

0-36 

0-03 

34 

0-02 

Coal, stone, - 



54 

0-375 

0-031 

42 

0-294 

0-024 

28-2 

0-016 

Charcoal, - - 

- 


27-5 

0-19 

0-016 

21 

0-15 

0-0TJ 

14-4 

0-008 


To Find the Weight and Capacity by this Table 

RULE. The product of the dimensions in feet or in inches, as noted in the 
columns, multiplied by the tabular coefficient, is the capacity of the solid, or 
weight in pounds avoirdupois. 

Example 1. A cistern is 6 feet long, 27 inches wide, and 20 inches deep. 
How many gallons of liquid can be contained in it? 

6X27X20X0-052 = 168 48 gallons. 

Example 2. A cast-iron cylinder is 4-5 feet long, and 7- 5 inches diametei 
Required the weight of it? 

4-5X7-5*X2-45 = 620 pounds. 


Preliminary IVote oil the Pitch of Gearing. 

The Reviser of this Pocket-Book does not agree with Mr. Nystrom as to the 
necessity or advantage in the use of “chordal pitch.” 

Diametral pitch affords a very easy method of computing the teeth of wheels, 
and should be used as much as possible. 

The tables are left to the judgment of the reader. 




































372 


Standard Pitc car-W heeds. 


STANDARD PITCH OF GEAR-WHEELS. 

Tlie difficulty in finding cog-wheel patterns made at different establish¬ 
ments to gear correctly into one another is well known, and much time and 
money is lost for the want of a standard scale of pitch in gearings. The 
pitch of teeth in a cog-wheel should always be understood to mean the 
chord-pilch, and not the arc-pitch, because equal arc-pitch in wheels of 
widely different diameters will not gear well. 

The pitch of gear-wheels should be even measures of the inch and binary 
fractions thereof, and the number of teeth and diameter of pitch-circle should 
be regulated accordingly. 

The following pitch-table is offered or proposed as standard, in which the 
first column varies with sixteenths of an inch from 0 to 1 inch, with eighths 
from 1 to 3 inches, and with quarters from 3 to 7 inches. The pitch of wheels 
from j l (j to 7 inches should not be made of any other measure than of those 
in the table, and the fractions with the most decimals should be avoided as 
much as possible: 

Standard Pitch for Gear-Wheels. 


Pitch from x l a to 1 inch. 

Pitch from 1 to 3 inches. 

Pitch from 3 to 7 inches. 

Biuary Decimals. 

1 in. Decimals. 

3 in. Decimals. 

i = 00625 
£=0*125 

li = 1125 

3£ = 3-25 

l£ = l-25 

3£ = 3-5 

T 3 g = 0-1875 

If = P375 

3| = 3-75 

£ = 0-25 

H= P5 

4 in. 

T R 5 = 03125 

1§ = 1-625 

4£ = 4'25 

| 0'375 

11= P75 

4£ = 45 

T 7 a = 0-4375 

H = 1-875 

4$ = 4-75 

£ = 0-5 

2 in. 

5 in. 

T 9 „ = 0-5625 

2£ = 2.125 

5£ = 5-25 

£ =■ 0 625 

2£ = 2-25 

5£ = 55 

ii = 0-6875 
| = 0-75 

2£ = 2375 

5} = 575 

2£ = 2-5 

6 in. 

U = 0-8125 
l = 0-875 

2 f = 2-625 

6 £ = 6-25 

2} = 2-75 

65 = 6*5 

i# = 0-9375 

2f = 2.875 

6 £ = 675 

1 in. 

3 in. 

7 in. 


The width of the face should be two and a half the pitch. 


The following table contains the proportions of number of teeth, diameter, 
and chord-pitch of wheels from 6 to 250 teeth. The first column is the num¬ 
ber of teeth, the second is the diameter when the chord-pitch is unity, and 
the third column is the chord-pitch when the diameter is the unity. 

Example 1.—What diameter is required for a wheel of 45 teeth and chord- 
pitch 1 $ inches? Opposite 45 teeth we find the diameter. 

14.3356 X 1.75 = 25.0874 inches in pitch-line. 

Example 2.—A wheel of 62.35 inches diameter in the pitch-line has 198 
teeth. What is the chord-pitch? 

Pilch = 62.35 X 0.015866 = 0.989 of an inch. 

That wheel will not work with the standard gear. 

The number of teeth multiplied by the pitch gives the length of the pitch- 
polygon and not the pitch-circle. The difference between the length of the 
two pitch-lines is greater the less the number of teeth in the wheels. For 
250 teeth and one-inch pitch the difference is only part of an inch in 

the whole pitch-line. 

For properly-constructed cut -gearing there should be no clearance between 
the teeth, as shown in Plate I., but the thickness a of the teeth should be 
half the pitch. 















Proportio : op ' par -Wheels. 873 


No. 

Teeth. 

Diam¬ 

eter 

■when 

P=l. 

Pitch 

■when 

D = 1. 

No. 

Teeth. 

Diam¬ 

eter 

when 

P = l. 

Pitch 

when 

2> = 1. 

No. 

Teeth. j 

Diam¬ 

eter 

when 

P=l. 

Pitch 

when 

D — 1. 

. .C 

A D 

Diam¬ 

eter 

wlien 

P—1. 

Pitch 

when 

D = l. 

6 

2.0000 

.50000 

66 

21.016 

.04758 

126 

40.111 

.02493 

186 

59.208 

.01689 

7 

2.3068 

.43358 

67 

21.334 

.04687 

127 

40.429 

.02473 

187 

59.527 

.01680 

8 

2.6131 

.38268 

68 

21.652 

.04618 

128 

40.748 

.02454 

188 

59.845 

.01671 

9 

2.9238 

.34202 

69 

21.970 

.04552 

129 

41.066 

.02435 

189 

60.163 

.01662 

10 

3.2361 

.30902 

70 

22.289 

.04486 

130 

41.384 

.02416 

190 

60.482 

.01653 

11 

3.5490 

.28177 

71 

22.607 

.04423 

131 

41.702 

.02398 

191 

60.800 

.01645 

12 

3.8637 

.25882 

72 

22.925 

.04361 

132 

42.021 

.02380 

192 

61.118 

.01636 

13 

4.1785 

.23932 

73 

23.243 

.04307 

133 

42.330 

.02362 

193 

61.436 

.01628 

14 

4.4940 

.22252 

74 

23.562 

.04242 

134 

42.657 

.02344 

194 

61.755 

.01619 

15 

4.8097 

.20791 

75 

23.880 

.04187 

185 

42.976 

.02327 

195 

62.073 

.01611 

16 

5.1259 

.19509 

76 

24.198 

.01131 

136 

43.294 

.02310 

196 

62.391 

.01603 

17 

5.4423 

.18374 

77 

24.516 

.04091 

137 

43.612 

.02293 

197 

62.710 

.01595 

18 

5.7588 

.17365 

78 

24.835 

.04026 

138 

43.931 

.02276 

198 

63.028 

.01587 

19 

6.0756 

.16460 

79 

25.153 

.08976 

139 

44.250 

.02260 

199 

63.346 

.01579 

20 

6.3925 

.15643 

80 

25.471 

.03926 

140 

44.567 

.02244 

200 

63.665 

.01571 

21 

6.7095 

.14904 

81 

25.789 

.03878 

141 

44.885 

.02228 

201 

63.983 

.01563 

22 

7.0266 

.14231 

82 

26.108 

.03830 

142 

45.204 

.02212 

202 

64.301 

.01555 

23 

7.3338 

.13636 

83 

26.426 

.03784 

143 

45.522 

.02197 

203 

64.620 

.01547 

24 

7.6613 

.13053 

84 

26.744 

.03739 

144 

45.840 

.02182 

204 

64.938 

.01540 

25 

7.9787 

.12533 

85 

27.062 

.03695 

145 

46.158 

.02167 

205 

65.256 

.01532 

26 

8.2962 

.12054 

86 

27.381 

.03652 

146 

46.477 

.02152 

206 

65.575 

.01525 

27 

8.6138 

.11609 

87 

27.699 

.03611 

147 

46.795 

.02137 

207 

65.893 

.01517 

28 

8.9315 

.11196 

88 

28.017 

.03569 

148 

47.113 

.02122 

208 

66.211 

.01510 

29 

9.2493 

.10811 

89 

28.335 

.03529 

149 

47.432 

.02108 

209 

66.529 

.01503 

30 

9.5668 

.10453 

90 

28.654 

.08490 

150 

47.750 

.02094 

210 

66.848 

.01496 

31 

9.8845 

.10117 

91 

28.972 

.03452 

151 

48.068 

.02080 

211 

67.166 

.01488 

32 

10.202 

.09800 

92 

29.290 

.03414 

152 

48.386 

.02067 

212 

67.484 

.01482 

33 

10.520 

.09506 

93 

29.608 

.03377 

153 

48.705 

.02553 

213 

67.802 

.01475 

34 

10.838 

.09226 

94 

29.927 

.03341 

154 

49.023 

.02039 

214 

68.121 

.01470 

35 

11.156 

.08964 

95 

30.245 

.03306 

155 

49.341 

.02029 

215 

68.439 

.01461 

36 

11.474 

.08716 

96 

30.563 

.03272 

156 

49.660 

.02014 

216 

68.757 

.01454 

37 

11.792 

.08480 

97 

30.881 

.03238 

157 

49.978 

.02001 

217 

69.076 

.01448 

38 

12.110 

.08257 

98 

31.200 

.03205 

158 

50.296 

.01988 

218 

69.394 

.01441 

39 

12.427 

.08049 

99 

31.518 

.03173 

159 

50.614 

.01976 

219 

69.712 

.01434 

40 

12.745 

.07846 

100 

31.836 

.03141 

160 

50.933 

.01963 

220 

70.031 

.01428 

41 

13.064 

.07653 

101 

32.154 

.03100 

161 

51.251 

.01951' 

221 

70.349 

.01421 

42 

13.382 

.07476 

102 

32.473 

.03079 

162 

51.569 

.01939 

222 

70.667 

.01415 

43 

13.700 

.07299 

103 

32.791 

.03049 

163 

51.888 

.01927 

223 

70.985 

.01409 

44 

14.018 

.07134 

104 

33.109 

.03021 

164 

52.206 

.01915 

224 

71.304 

.01402 

45 

14.336 

.06976 

105 

33.427 

.02992 

165 

52.524 

.01904 

225 

71.622 

.01396 

46 

14.654 

.06826 

106 

33.745 

.02963 

166 

52.842 

.01892 

226 

71.940 

.01390 

47 

14.972 

.06679 

107 

34.064 

.02986 

167 

53.161 

.01881 

227 

72.259 

.01384 

48 

15.290 

.06540 

108 

34.382 

.02908 

168 

53.479 

.01870 

228 

72.577 

.01378 

49 

15.608 

.06407 

109 

34.700 

.02882 

169 

53.797 

.01859 

229 

72.895 

.01372 

50 

15.926 

.06279 

110 

35.018 

.02856 

170 

54.116 

.01848 

230 

73.213 

.01366 

51 

16.244 

.06156 

111 

35.337 

.02830 

171 

54.434 

.01837 

231 

73.532 

.01360 

52 

16.562 

.06038 

112 

35.655 

.02805 

172 

54.752 

.01826 

232 

73.850 

.01354 

53 

16.880 

.05925 

113 

35.973 

.02780 

173 

55.070 

.01816 

233 

74.168 

.01348 

54 

17.198 

.05815 

114 

36.292 

.02755 

174 

55.889 

.01805 

234 

74.487 

.01342 

55 

17.517 

.05709 

115 

36.610 

.02731 

175 

55.707 

.01795 

235 

74.805 

.01337 

56 

17.835 

.05607 

116 

36.928 

.02708 

176 

56.025 

.01785 

236 

75.123 

.01331 

57 

18.153 

.05509 

117 

37.246 

.02685 

177 

56.344 

.01775 

237 

75.442 

.01325 

5S 

18.471 

.05414 

118 

37.565 

.02662 

178 

56.662 

.01765 

238 

75.700 

.01320 

59 

18.789 

.05322 

119 

37.883 

.02640 

179 

56.980 

.01755 

239 

Vo.ov 8 

.01814 

60 

19.107 

.05234 

120 

38.204 

.02618 

180 

57.299 

.01745 

240 

76.396 

.01309 

61 

19.425 

.05148 

121 

38.520 

.02596 

181 

57.617 

.01736 

241 

76.715 

.01303 

62 

19.744 

.05065 

122 

38.838 

.02575 

182 

57.935 

.01726 

242 

77.083 

.01298 

63 

20.062 

.04982 

123 

39.156 

.02554 

183 

58.253 

.01717 

243 

77.351 

.01293 

64 

20.380 

.04907 

124 

39.475 

.02533 

184 

58.572 

.01707 

244 

77.670 

.01287 

65 

20.698 

.04831 

125 

39.793 

.02513 

185 

58.890 

.01698 

245 

77.988 

.01282 







































374 Pitch-Line Diameter of Gear-Wheels. 


Number 

Teeth. 

a 

( 

i 

Chordal 

3 

a 

’itch of 

i 

z 

Wheel o 

| 

r Pinioi 

£ 

i. 

i 

1 inch. 

G 

0.2500 

0.5000 

0.7500 

1.0000 

1.2500 

1.5000 

1.7500 

2.0000 

7 

0.2783 

0.5767 

0.8550 

1.1534 

1.4317 

1.7301 

2.0085 

2.3068 

8 

0.3266 

0.6532 

0.9799 

1.3065 

1.6332 

1.9597 

2.2864 

2.6131 

9 

0.3654 

0.7309 

1.0964 

1.4619 

1.8264 

2.1928 

2.5583 

2.9238 

10 

0.4045 

0.8090 

1.2136 

1.6180 

2.0225 

2.4270 

2.8315 

3.236 1 

11 

0.4436 

0.8872 

1.3309 

1.7745 

2.2182 

2.6617 

3.1053 

3.5490 

12 

0.4829 

0.9659 

1.4489 

1.9318 

2.4148 

2.8977 

3.3807 

3.8637 

13 

0.5223 

1.0446 

1.5670 

2.0892 

2.6116 

3.1338 

3.6562 

4.1785 ' 

14 

0.5617 

1.12.35 

1.6852 

2.2470 

2.8087 

3.3705 

3.9323 

4.4940 

15 

0.6012 

1.2024 

1.8037 

2.4048 

3.0061 

3.6072 

4.2085 

4.8097 

16 

0.6407 

1.2814 

1.9222 

2.5629 

3.2037 

3.8443 

4.4851 

5.1259 

17 

0.6803 

1.3605 

2.0408 

2.7211 

3.4014 

4.0816 

4.7620 

5.4423 

18 

0.7198 

1.4397 

2.1596 

2.8794 

3.5992 

4.3191 

5.0390 

5.7588 

19 

0.7594 

1.5189 

2.2783 

3.0378 

3.7973 

4.5567 

5.3162 

6.0756 

20 

0.7991 

1.5981 

2.3972 

3.1962 

3.9953 

4.7943 

5.5934 

6.3925 

21 

0.8387 

1.6774 

2.5160 

3.3547 

4.1934 

5.0320 

5.8708 

6.7095 

22 

0.8783 

1.7566 

2.6350 

3.5133 

4.3916 

5.2699 

6.1483 

7.0266 

23 

0.9167 

1.8334 

2.7502 

3.6669 

4.5836 

5.5003 

6.4171 

7.3338 

24 

0.9576 

1.9153 

2.8730 

3.8306 

4.7883 

5.7459 

6.7036 

7.6613 

25 

0.9973 

1.9946 

2.9920 

3.9893 

4.9866 

5.9839 

6.9813 

7.9787 

2G 

1.0370 

2.0740 

3.1111 

4.1481 

5.1851 

6.2221 

7.2592 

8.2962 

27 

1.0767 

2.1534 

3.2302 

4.3069 

5.3836 

6.4603 

7.5371 

8.6138 

28 

1.1164 

2.2328 

3.3493 

4.4657 

5.5822 

6.6985 

7.8150 

8.9315 

29 

1.1561 

2.3123 

3.4685 

4.6246 

5.7808 

6.9369 

8.0931 

9.2493 

30 

1.1958 

2.3917 

3.5875 

4.7834 

5.9792 

7.1751 

8.3710 

9.5668 

31 

1.2355 

2.4711 

3.7067 

4.9422 

6.1778 

7.4133 

8.6488 

9.8845 

82 

1.2753 

2.5506 

3.8258 

5.1012 

6.3764 

7.6516 

8.9270 

10.2023 

83 

1.3150 

2.6300 

3.9450 

5.2600 

6.5750 

7.8900 

9.2051 

10.5201 

34 

1.3547 

2.7095 

4.0642 

5.4190 

6.7737 

8.1285 

9.4833 

10.8380 

85 

1.3944 

2.7889 

4.1834 

5.5778 

6.9723 

8.3667 

9.7612 

11.1557 

36 

1.4342 

2.8684 

4.3027 

5. / 3t>8 

7.1711 

8.6052 

10.0395 

11.4737 

37 

1.4739 

2.9479 

4.4219 

5.8958 

7.3698 

8.8437 

10.3177 

11.7917 

38 

1.5137 

3.0274 

4.5411 

6.0548 

7.5685 

9.0812 

10.5949 

12.1096 

39 

1.5534 

3.1068 

4.6603 

6.2137 

7.7672 

9.3205 

10.8740 

12.4275 

40 

1.5932 

3JS64 

4.7795 

6.3728 

7.9659 

9.5590 

11.1522 

12.7455 

41 

1.6329 

3.2659 

4.8988 

6.5318 

8.1617 

9.8077 

11.4407 

13.0636 

42 

1.6727 

3.3454 

5.0181 

6.6908 

8.3635 

10.0362 

11.7089 

13.3816 

43 

1.7124 

3.4249 

5.1373 

6.8498 

8.5622 

10.2747 

11.9872 

13.6996 

44 

1.7522 

3.5044 

5.2567 

7.0088 

8.7611 

10.5132 

12.2655 

14.0177 

45 

1.7919 

3.5839 

5.3759 

7.1678 

8.9598 

10.7517 

12.5437 

14.3357 

46 

1.8317 

3.6634 

5.4952 

7 3269 

9.1586 

10.9903 

12.8221 

14.6538 

47 

1.8714 

3.7429 

5.6144 

7.4859 

9.3573 

11.2288 

13.1003 

14.9718 

48 

1.9112 

3.8224 

5.7337 

7.6449 

9.5561 

11.4673 

13.3786 

15.2898 

49 

1.9509 

3.9019 

5.8529 

7.8039 

9.7549 

11.7058 

13.6568 

15.6079 

50 

1.9907 

3.9815 

5.9722 

7.9630 

9.9537 

11.9445 

13.9352 

15.9260 

51 

2.0305 

4.0610 

6.0916 

8.1220 

10.1525 

12.1830 

14.2141 

16.2441 

52 

2.0702 

4.1405 

6.2108 

8.2811 

10.3513 

12.4216 

14.4919 

16.5622 

53 

2.1105 

4.2210 

6.3315 

8.4420 

10.5525 

12.6630 

14.7735 

16.8840 

54 

2.1498 

4.2996 

6.4495 

8.5992 

10.7491 

12.8988 

15.0487 

17.1985 

55 

2.1895 

4.3791 

6.5687 

8.7583 

10.9479 

13.1374 

15.3270 

17.5167 

56 

2.2293 

4.4587 

6.6881 

8.9174 

11.1468 

13.3761 

15.6055 

17.8349 

57 

2.2691 

4.5382 

6.8075 

9.0765 

11.3456 

13.6147 

15.8839 

18.1530 

58 

2.3088 

4.6177 

6.9266 

9.2355 

11.5444 

13.8532 

16.1621 

18.4711 

59 

2.3486 

4.6973 

7.0459 

9.3946 

11.7432 

14.0919 

16.4406 

18.7892 

60 

2.3884 

4.7768 

7.1653 

9.5536 

11.9421 

14.3304 

16.7189 

19.1073 

61 

2.4282 

4.8563 

7.2845 

9.7127 

12.1409 

14.5690 

16.9972 

19.4255 

62 

2.4679 

4.9359 

7.4038 

9.8718 

12.3397 

14.8077 

17.2757 

19.7436 

63 

2.5077 

5.0154 

7.5232 

10.0309 

12.5387 

15.0463 

17.5541 

20.0618 

64 

2.5474 

5.0949 

7.6424 

10.1899 

12.7374 

15.2848 

17.8323 

20.3799 

65 

2.5872 

5.1745 

7.7617 

10.3490 

12.9362 

15.5235 

18.1108 

20.6980 ! 






































375 


Pitch-Like Djamftk.i of Gear-Wheels. 


Number 

Teeth. 

1 

B 

i 

Chordal 

| 

Pitch of 

i 

5 

Wheel o 

| 

r Pinion 

3 

i 

| 

1 inch. 

66 

2.6270 

5.2540 

7.8810 

10.5081 

13.1351 

15.7621 

18.3891 

21.0161 

67 

2.6668 

5.3336 

8.0004 

10.6672 

13.3340 

16.0008 

18.6675 

21.3343 

68 

2.7065 

5.4131 

8.1196 

10.8262 

13.5327 

16.2393 

18.9459 

21.6524 

6!) 

2.7463 

5.4926 

8.2490 

10.9853 

13.7316 

16.4779 

19.2242 

21.9705 

70 

2.7861 

5.5722 

8.3583 

11.1444 

13.9305 

16.7166 

19.5026 

22.2887 

71 

2.8258 

5.6518 

8.4776 

11.3035 

14.1293 

16.9553 

19.7812 

22.6070 

72 

2.8656 

5.7313 

8.5970 

11.4627 

14.3283 

17.1940 

20.0597 

22.9253 

73 

2.9054 

5.8109 

8.7163 

11.6218 

14.5272 

17.4327 

20.3381 

23.2435 

74 

2.9452 

5.8904 

8.8356 

11.7809 

14.7261 

17.6713 

20.6165 

23.5617 

75 

2.9850 

5.9700 

8.9550 

11.9400 

14.9250 

17.9100 

20.8950 

23.8S00 

76 

3.0248 

6.0495 

9.0743 

12.0991 

15.1238 

18.1486 

21.1734 

24.1982 

77 

3.0646 

6.1291 

9.1937 

12.2583 

15.3230 

18.3874 

21.4519 

24.5165 

78 

3.1043 

6.2087 

9.3131 

12.4174 

15.5217 

18.6261 

21.7305 

24.8348 

79 

3.1441 

6.2882 

9.4324 

12.5765 

15.7206 

18.8647 

22.0089 

25.1530 

80 

3.1839 

6.3678 

9.5518 

12.7357 

15.9196 

19.1035 

22.2874 

25.4713 

81 

3.2237 

6.4474 

9.6711 

12.8948 

16.1185 

19.3422 

22.5658 

25.7895 

82 

3.2635 

6.5269 

9.7904 

13.0539 

16.3173 

19.5808 

22.8443 

26.1078 

83 

3.3032 

6.6065 

9.9098 

13.2130 

16.5162 

19.8195 

23.1228 

26.4260 

84 

3.3430 

6.6861 

10.0292 

13.3722 

16.7152 

20.0583 

23.4013 

26.7443 

85 

3.3828 

6.7656 

10.1485 

13.5313 

16.9141 

20.2969 

23.6797 

27.0625 

86 

3.4226 

6.8452 

10.2678 

13.6904 

17.1130 

20.5356 

23.9581 

27.3807 

87 

3.4624 

6.9247 

10.3871 

13.8495 

17.3118 

20.7642 

24.2266 

27.6990 

88 

3.5021 

7.0043 

10.5065 

14.0086 

17.5107 

21.0129 

24.5151 

28.0172 

89 

3.5419 

7.0838 

10.6258 

14.1677 

17.7096 

21.2515 

24.7934 

28.3354 

90 

3.5817 

7.1634 

10.7452 

14.3269 

17.9086 

21.4903 

25.0720 

28.6537 

91 

3.6215 

7.2430 

10.8645 

14.4860 

18.1075 

21.7290 

25.3505 

28.9719 

92 

3.6612 

7.3225 

10.9838 

14.6451 

18.3063 

21.9676 

25.6289 

29.2902 

93 

3.7011 

7.4021 

11.1032 

14.8042 

18.5052 

22.2063 

25.9074 

29.6084 

94 

3.7408 

7.4817 

11.2223 

14.9633 

18.7038 

22.4440 

26.1847 

29.9267 

95 

3.7806 

7.5612 

11.3419 

15.1225 

18.9031 

22.6837 

26.4643 

30.2449 

96 

3.8204 

7.6408 

11.4612 

15.2816 

19.1020 

22.9224 

26.7428 

30.5632 

97 

3.8602 

7.7203 

11.5805 

15.4407 

19.3009 

23.1610 

27.0213 

30.8814 

98 

3.8999 

7.7999 

11.6999 

15.5999 

19.4998 

23.3998 

27.2998 

31.1997 

99 

3.9397 

7.8794 

11.8192 

15.7589 

19.6986 

23.6383 

27.5780 

31.5179 

100 

3.9795 

7.9590 

11.9386 

15.9181 

19.8976 

23.8771 

27.8566 

31.8362 

101 

4.0193 

8.0386 

12.0579 

16.0772 

20.0965 

24.1168 

28.1351 

32.1544 

102 

4.0591 

8.1181 

12.1772 

16.2363 

20.2954 

24.3544 

28.4135 

32.4726 

103 

4.0964 

8.1927 

12.2891 

16.3954 

20.4917 

24.5881 

28.6845 

32.7909 

104 

4.1361 

8.2723 

12.4085 

16.5546 

20.6907 

24.8269 

28.9630 

33.1091 

105 

4.1784 

8.3568 

12.5353 

16.7137 

20.8821 

25.0695 

29.2479 

33.4273 

106 

4.2182 

8.4364 

12.6546 

16.8728 

21.0910 

25.3092 

29.5273 

33.7455 

107 

4.2579 

8.5159 

12.7739 

17.0318 

21.2897 

25.5477 

29.8057 

34.0637 

108 

4.2977 

8.5954 

12.8932 

17.1909 

21.4886 

25.7863 

30.0841 

34.3819 

109 

4.3375 

8.6750 

13.0126 

17.3501 

21.6876 

26.0251 

30.3626 

34.7001 

110 

4.3773 

8.7546 

13.1319 

17.5092 

21.8865 

26.2638 

30.6411 

35.0183 

111 

4.4171 

8.8341 

13.2512 

17.6683 

22.0854 

26.5024 

30.9195 

35.3366 

112 

4.4569 

8.9137 

13.3706 

17.8275 

22.2844 

26.7412 

31.1981 

35.6550 

113 

4.4967 

8.9933 

13.4900 

17.9867 

22.4834 

26.9800 

31.4766 

35.9733 

114 

4.5364 

9.0729 

13.6094 

18.1458 

22.6823 

27.2187 

31.7552 

36.2916 

115 

4.5762 

9.1525 

13.7288 

18.3050 

22.8812 

27.4575 

32.0338 

36.6100 

116 

4.6160 

9.2321 

13.8482 

18.4642 

23.0802 

27.6963 

32.3123 

36.9283 

117 

4.6558 

9.3116 

13.9675 

18.6233 

23.2791 

27.9349 

32.6908 

37.2466 

118 

4.6956 

9.3912 

14.0869 

18.7825 

23.4781 

28.1737 

32.8694 

37.5650 

119 

4.7354 

9.4708 

14.2063 

18.9417 

23.6771 

28.4125 

33.1479 

37.8833 

120 

4.7752 

9.5504 

14.3256 

19.1008 

23.8760 

28.6512 

33.4264 

38.2016 

121 

4.8149 

9.6299 

14.4449 

19.2599 

24.0748 

28.8898 

33.7048 

38.5198 

122 

4.8548 

9.7095 

14.5643 

19.4191 

24.2739 

29.1286 

33.9833 

38.8381 

123 

4.8945 

9.7891 

14.6837 

19.5782 

24.4728 

29.3673 

34.2619 

39.1564 

124 

4.9393 

9.8687 

14.8081 

19.7374 

24.6767 

29.6061 

34.5454 

39.4747 

125 

4.9741 

9.9482 

14.9224 

19.8965 

24.8706 

29.8447 

34.8189 

39.7930 































376 


Pitch-Line Dw. rr.. 


Gear-Wheels. 


Chordal I itch of-Wheel or Pinion. 


uuiuer 

Teeth. 

1 

¥ 

i 

9 

* 

I 

2 

7 

3 

1 inch. 

126 

5.0139 

10.0278 

15.0417 

20.0.556 

25.0695 

30.0S34 

35.0973 

40.1112 

127 

5.0537 

10.1074 

15.1611 

20.2148 

25.2685 

30.3222 

35.3759 

40.4265 

128 

5.0934 

10.1866 

15.2803 

20.3736 

25.4673 

30.5608 

35.6542 

40.7478 

126 

5.1332 

10.2665 

15.3697 

20.5330 

25.6662 

30.7995 

35.6327 

41.0660 

130 

5.1730 

10.3461 

15.5161 

20.6922 

25.8652 

31.0383 

36.2113 

41.3843 

131 

5.2128 

10.4256 

15.6384 

20.8513 

26.0641 

31.2769 

36.4897 

41.7026 

132 

5.2526 

10.5052 

15.7578 

21.0104 

26.2630 

31.5156 

36.7682 

42.0208 

133 

5.2924 

10.5848 

15.8792 

21.1696 

26.4620 

31.7544 

37.5468 

42.3361 

134 

5.3321 

10.6643 

15.666-1 

21.3287 

26.6608 

31.6630 

37.3251 

42.6574 

135 

5.3719 

10.7439 

16.1158 

21.4876 

26.8598 

32.2318 

37.6037 

42.6757 

136 

5.4117 

10.8284 

16.2351 

21.6466 

27.0586 

32.4903 

37.8820 

43.2939 

137 

5.4515 

10.9030 

16.3545 

21.8061 

27.2576 

32.7091 

38.1606 

43.6122 

138 

5.4613 

10.6826 

16.4739 

21.9653 

27.4566 

32.9479 

38.4392 

43.6305 

139 


11.0624 

16.5936 

22.1249 

27.6561 

33.1873 

38.7185 

44.2498 

140 

5.5709 

11.1418 

16.7127 

22.2836 

27.3545 

33.4254 

38.9963 

44.5671 

141 

5.6106 

11.2213 

16.8319 

22.4427 

28.0533 

33.6040 

39.2746 

44 8854 

142 

5.6504 

11.3006 

16.6513 

22.6019 

28.2523 

33.9528 

36.5532 

45.2037 

143 

5.6900 

11.3800 

17.0700 

22.7601 

28.4501 

34.1401 

36.8301 

45.5202 

144 

5.7300 

11.4600 

17.1900 

22.9201 

28.6501 

34.3801 

40.1101 

45.8402 

145 

6.7698 

11.5366 

17.3094 

23.0793 

28.8461 

34.6186 

453887 

46.1585 

146 

5.8096 

11.6162 

17.4288 

23.2384 

29.0480 

34.8576 

40.6672 

46.4768 

147 

5.8464 

11.6688 

17.5482 

23.3976 

29.2470 

35.0964 

40.6458 

46.7951 

148 

5.8891 

11.7783 

17.6674 

23.5567 

29.4458 

35.3350 

41.2241 

47.1134 

146 

5.6286 

11.8579 

17.7868 

23.7156 

29.6448 

35.5738 

41.5027 

47.4317 

150 

5.6687 

11.9375 

17.8962 

23.8750 

29.8437 

35.8125 

41.7812 

47.7500 

151 

6.0080 

12.0171 

18.0251 

24.0342 

30.1222 

36.0513 

42.1393 

48.0683 

152 

6.0483 

12.0966 

18.14-16 

24.1633 

30.2416 

36.2896 

42.3382 

48.3866 

i 153 

6.0881 

12.1762 

18.2645 

24.3525 

30.4406 

36.5287 

42.6168 

48.7049 

\ 154 

6.1279 

12.2558 

18.3837 

24.5116 

30.6395 

36.7674 

42.8953 

46.0232 

155 

6.1676 

12.3353 

18.5026 

24.6707 

30.8483 

37.0060 

43.1736 

49.3414 

156 

6.2074 

12.4149 

18.6223 

24.8269 

31.0373 

37.2448 

43.4522 

49.6597 

157 

6.2472 

12.4645 

18.7417 

24.9890 

31.2362 

87.4835 

43.7307 

49.9780 

158 

6.2870 

12.5741 

18.8611 

25.1482 

31.4352 

37.7223 

44.0093 

50.2963 

159 

6.3268 

12.6536 

18.9801 

25.3073 

31.6341 

37.9600 

44.2877 

50.6146 

160 

6.3666 

12.7332 

16.0668 

25.4665 

31.8381 

38.1997 

44.5663 

50.9329 

161 

6.4064 

12.8128 

19.2192 

25.6256 

32.0320 

38.4384 

44.8448 

51.2512 

162 

6.4462 

12.8924 

16.3386 

25.7848 

32.2310 

38.6772 

45.1234 

51.5665 

163 

6.4859 

12.9719 

16.4578 

25.6436 

32.4298 

38.9158 

45.4017 

51.8878 

164 

6.5257 

13.0515 

16.5772 

26.1030 

32.6287 

39.15-15 

45.6802 

52.2060 

165 

6.5655 

13.1311 

19.6966 

26.2622 

32.8277 

39.3933 

45.9588 

52.5243 

166 

6.6053 

13.2106 

19.8156 

26.4213 

33.0266 

39.6319 

46.2372 

52.8426 

167 

6.6450 

13.2901 

16.6351 

26.5804 

33.2254 

39.8705 

46.5155 

53.1608 

168 

6.6849 

13.3668 

20.0547 

26.7396 

33.4245 

40.1094 

46.7943 

53.4791 

166 

6.7244 

13.4488 

20.1732 

26.8977 

33.6221 

40.3465 

47.1109 

53.7974 

170 

6.7644 

13.5289 

20.2633 

27.0578 

33.8222 

40.5867 

47.3511 

54.1157 

171 

6.8042 

13.6085 

20.4127 

27.2170 

34.0212 

40.8255 

47.6297 

54.4:140 

172 

6.8440 

13.6881 

20.5321 

27.3762 

34.2202 

41.0643 

47.9083 

54.7523 

173 

6.8838 

13.7676 

20.6511 

27.5363 

34.4161 

41.3029 

48.1867 

55.0706 

174 

6.9236 

13.8472 

20.7708 

27.6945 

34.6181 

41.5417 

48.4653 

55.3889 

175 

6.6634 

13.6268 

20.8602 

27.8536 

34.8170 

41.7804 

48.7438 

55.7072 

176 

7.0032 

14.0064 

21.0066 

28.0128 

35.0160 

42.0172 

40.0224 

56.0255 

177 

7.0429 

14.0859 

21.1288 

28.1719 

35.2148 

42.2578 

49.3007 

56.3 438 

178 

7.0827 

14.1655 

21.2482 

28.3311 

35.4133 

42.4966 

46.5793 

56.6621 

176 

7.1225 

14.2451 

21.36 < 0 

28.4902 

35.6127 

42.7353 

46.8578 

56.9804 

180 

7.1623 

14.3247 

21.3870 

28.6494 

35.8117 

42.9741 

50.1359 

57.2987 

181 

7.2021 

14.4042 

21.6063 

28.8085 

36.0106 

43.2127 

554148 

57.6170 

182 

7.2419 

14.4838 

21.7257 

28.6677 

36.2066 

43.4515 

50.6934 

57.9353 

183 

7.2817 

14.5634 

21.8451 

26.1268 

36.4035 

43.6602 

50.9719 

58.2536 

184 

7.3215 

14.6130 

21.9645 

26.2860 

36.6075 

43.0290 

51.2505 

58 5720 

185 

7.3613 

14.7226 

22.0836 

26.4452 

36.8065 

44.1678 

51.5291 

58.8903 






































Pitch-Line Diasinter of Gear-Wheexs. 377 


Number 

Teeth. 

1 

8 

1 

4 

Chordal 

1 

Pitch of 

i 

Wheel 

$ 

or Piuioi 

* 

). 

7 

1 

1 inch. 

186 

7.4010 

14.8021 

22.2031 

29.6043 

37.0053 

44.4064 

51.8074 

59.2086 

187 

7.4408 

14.8817 

22.3225 

20.7635 

37.2043 

44.6452 

52.0860 

59.5269 

188 

7.4806 

14.9613 

22.4419 

29.9226 

37.4032 

44.8839 

52.3655 

59.8452 

180 

7.5204 

15.0409 

22.5613 

30.0818 

37.6022 

45.1227 

52.6431 

60.1635 

100 

7.5702 

15.1404 

22.7106 

30.2409 

37.8111 

45.3813 

52.9515 

60.4818 

101 

7.6000 

15.2000 

22.8000 

30.4001 

38.0001 

45.6001 

53.2001 

60.8001 

102 

7.6398 

15.2796 

22.9194 

30.5592 

38.1990 

45.8388 

53.4786 

61.1184 

193 

7.6795 

15.3591 

23.0386 

30.7183 

38.3978 

46.0774 

53.7569 

61.4366 

104 

7.7193 

15.4387 

23.1580 

30.8775 

38.5968 

46.3162 

54.0355 

61.7549 

105 

7.7091 

15.5183 

23.2774 

31.0366 

38.7967 

46.5549 

54.3150 

62.0732 

106 

7.7980 

15.5979 

23.3068 

31.1958 

38.9947 

46.7937 

54.5926 

62.3916 

107 

7.8387 

15.6775 

23.5162 

31.3550 

39.1937 

47.0325 

54.8712 

62.7099 

108 

7.8785 

15.7571 

23.6356 

31.5142 

39.3927 

47.2713 

55.1498 

63.0283 

100 

7.0183 

15.8366 

23.7549 

31.6733 

30.5916 

47.5099 

55.4282 

63.3466 

200 

7.0581 

15.9162 

23.8743 

31.8325 

39.7906 

47.7487 

55.7068 

63.6649 

201 

7.9979 

15.9958 

23.9937 

31.9916 

39.9805 

47.9874 

55.9853 

63.9832 

202 

8.0376 

16.0753 

24.1129 

32.1507 

40.1883 

48.2260 

56.2636 

64.3014 

203 

8.0773 

16.1547 

24.2320 

32.3094 

40.3S67 

48.4641 

56.5414 

64.6197 

204 

8.1172 

16.2345 

24.3480 

32.4690 

40.5862 

48.7035 

56.8207 

64.9380 

205 

8.1570 

16.3141 

24.4711 

32.6282 

40.7852 

48.9423 

57.0993 

65.2563 

206 

8.1968 

16.3936 

24.5904 

32.7873 

40.9841 

49.1809 

57.3777 

65.5746 

207 

8.2366 

16.4732 

24.6348 

32.9465 

41.1831 

49.4197 

57.6563 

65.8929 

208 

8.2764 

16.5528 

24.8292 

33.1056 

41.3820 

49.6584 

57.9348 

68.2112 

200 

8.3162 

16.6324 

24.9486 

33.2648 

41.5810 

49.8972 

58.2134 

66.5295 

210 

8.3559 

16.7119 

25.0678 

33.4239 

41.779S 

50.1358 

58.4917 

66.8478 

211 

8.3057 

16.7015 

25.1872 

33.5831 

41.9788 

50.3746 

58.7703 

67.1661 

212 

8.4355 

16.8711 

25.3066 

33.7422 

42.1777 

50.6133 

59.0488 

67.4844 

213 

8.4753 

16.0506 

25.4259 

33.9013 

42.3766 

50.8519 

59.3272 

67.8026 

214 

8.5151 

17.0302 

25.5453 

34.0605 

42.5756 

51.0907 

59.6058 

68.1209 

215 

8.5549 

17.1098 

25.6647 

34.2196 

42.7735 

51.3294 

59.8833 

68.4391 

216 

8.5047 

17.1894 

25.7841 

34.3789 

42.9736 

51.5683 

60.1630 

68.7574 

217 

8.6344 

17.2689 

25.9033 

34.5378 

43.1722 

51.8067 

60.3411 

69.0757 

218 

8.6742 

17.3485 

26.0227 

34.6970 

43.3712 

52.0455 

60.7197 

69.3940 

210 

8.7140 

17.4281 

26.1421 

34.8562 

43.5702 

52.2843 

60.9983 

69.7123 

220 

8.7538 

17.5076 

26.2614 

35.0153 

43.7691 

52.5229 

61.2767 

70.0306 

221 

8.7936 

17.5872 

26.3808 

35.1745 

43.9681 

52.7617 

61.5553 

70.3489 

222 

8.8334 

17.6668 

26.5002 

35.3336 

44.1670 

52.0004 

61.8338 

70.6672 

223 

8.8732 

17.7464 

26.6296 

35.4928 

44.3660 

53.2392 

62.1124 

70.9855 

224 

8.9120 

17.8259 

26.7388 

35.6519 

44.5648 

53.4778 

62.4907 

71.3038 

225 

8.0527 

17.9055 

26.8582 

35.8111 

44.7638 

53.7166 

62.6693 

71.6221 

226 

8.0925 

17.0851 

26.9776 

35.9702 

44.9627 

53.9553 

62.9478 

71.9404 

227 

0.0323 

18.0647 

27.0970 

36.1294 

45.1617 

54.1941 

63.2264 

72.2587 

228 

0.0721 

18.1442 

27.2163 

36.2885 

45.3606 

54.4327 

63.5048 

72.5770 

220 

0.1110 I 

18.2238 

27.3357 

36.4477 

45.5596 

54.6715 

63.7834 

72.8953 

230 

0.1517 

18.3034 

27.4551 

36.6068 

45.7585 

54.9102 

64.0619 

73.2136 

231 

0.1914 

18.3829 

27.5743 

36.7659 

45.9573 

55.1488 

64.3402 

73.5319 

232 

9.2312 

18.4625 

27.6937 

36.9251 

46.1563 

55.3876 

6-4.6188 

73.8502 

233 

0.2710 

18.5421 

27.8131 

37.0843 

46.3553 

55.6264 

64.8974 

74.1685 

234 

0.3108 

18.6217 

27.9325 

37.2434 

46.5542 

55.8651 

65.1759 

74.4867 

235 

9.3506 

18.7012 

28.0518 

37.4025 

46.7531 

56.1037 

65.4543 

74.8050 

236 

9.3904 

18.7808 

28.1712 

37.5617 

46.9521 

56.3425 

65.7329 

75.1233 

237 

9.4302 

18.8604 

28.2906 

37.7208 

47.1510 

56.5812 

66.0114 

75.4416 

238 

0.4700 

18.9400 

28.4100 

37.8800 

47.3500 

56.8200 

66.2900 

75.7599 

230 

0.5007 

19.0195 

28.5282 

38.0391 

47.5488 

57.0586 

66.5683 

76.0782 

240 

0.5495 

19.0991 

28.6486 

38.1983 

47.7478 

57.2974 

66.8469 

76.3965 

241 

9.5894 

19.1788 

28.7682 

38.3574 

47.9468 

57.5362 

67.1256 

76.7148 

242 

9.6201 

19.2583 

28.8874 

38.5166 

48.1457 

57.7649 

67.4040 

77.0331 

243 

9.6689 

19.3378 

29.0067 

38.6757 

48.3446 

58.0135 

67.6814 

77.3514 

24-1 

9.7087 

19.4174 

29.1261 

38.8349 

48.5436 

58.2523 

67.9610 

77.6697 

245 

9.7485 

10.4070 

29.2455 

38.9940 

48.7425 

58.4910 

68.2395 

77.9880 






































378 


Pitch-Line Diahi. . Of Gear-Wiieels. 


Number 

Teeth. 

n 

( 

n 

Chordal 

n 

?it c; i 

2 inches. 

Wheel o 

2\ 

r Pinion 

91 

Z 2 

2| 

3 inches. 

6 

2.5000 

3.0000 

2.5000 

4.0000 

4.5000 

5.0000 

5.5000 

6.0000 

7 

2.8634 

3.4602 

40170 

4.6136 

5.1903 

5.7268 

6.3035 

6.9204 

8 

3.2664 

3.9194 

4.4728 

5.2262 

5.8794 

6.5328 

7.1860 

7.8393 

9 

3.6528 

4.3856 

5.1166 

5.8476 

6.5785 

7.3056 

8.0365 

8.7714 

10 

4.0450 

4.8540 

5.6630 

6.4722 

7.2812 

8.0900 

8.8990 

9.7083 

11 

4.4364 

5.3234 

6.2106 

7.0980 

7.9852 

8.8728 

9.7600 

10.647(1 

12 

4.8296 

5.7954 

6.7614 

7.7274 

8.6933 

9.6592 

10.6251 

11.5911 

13 

5.2232 

6.2676 

7.3124 

8.3570 

9.4016 

10.4464 

11.4910 

12.5355 

14 

5.6174 

6.7410 

7.8646 

8.9880 

10.1115 

11.2348 

12.3583 

13.4820 

15 

6.0122 

7.2144 

8.4170 

9.6194 

10.8218 

12.0244 

13.2268 

14.4291 

16 

6.4074 

7.6886 

8.9702 

10.2518 

11.5332 

12.S148 

14.0962 

15.3777 

17 

0.8028 

- 8.1632 

9.5240 

10.8846 

12.2451 

13.6056 

14.9661 

16.3260 

18 

7.1984 

8.6382 

10.0780 

11.5176 

12.9573 

14.3968 

15.8365 

17.2764 

19 

7.5946 

9.1134 

10.6324 

12.1512 

13.6701 

15.1892 

16.7081 

18.2268 

20 

7.9916 

9.5886 

11.1868 

12.7850 

14.3831 

15.9832 

17.5813 

19.1775 

21 

8.3868 

10.0640 

11.7416 

13.4190 

15.0964 

16.7736 

18.4510 

20.1285 

22 

8.7832 

10.5398 

12.2966 

14.0532 

15.8098 

17.5664 

19.3230 

21.0798 

23 

9.1672 

11.0006 

12.8:142 

14.6676 

16.5010 

18.3344 

20.1678 

22.0014 

24 

9.5766 

11.4918 

13.4072 

15.3226 

17.2379 

19.1532 

21.0685 

22.9839 

25 

9.9732 

11.9678 

13.9626 

15.9574 

17.9520 

19.9464 

21.9410 

23.9361 

26 

10.3702 

12.4442 

14.5184 

16.5924 

18.6664 

20.7404 

22.8144 

24.8886 

27 

10.7672 

12.9206 

15.0742 

17.2276 

19.3810 

21.5344 

23.6878 

25.8414 

28 

11.1644 

13.3970 

15.6300 

17.8630 

20.0958 

22.3288 

24.5616 

26.7945 

29 

11.5616 

13.8738 

16.1862 

18.4986 

20.8109 

23.1232 

25.4355 

27.7479 

30 

11.9584 

14.3502 

15.7420 

19.1336 

21.5253 

23.9168 

26.3085 

28.7004 

31 

12.3556 

14.8266 

17.2976 

19.7690 

22.2401 

24.7112 

27.1823 

29.6535 

32 

12.7528 

15.3032 

17.8540 

20.4046 

22.9552 

25.5056 

28.0562 

30.6069 

33 

13.1500 

15.7800 

18.4102 

21.0402 

23.6702 

26.3000 

28.9300 

31.5603 

34 

13.5474 

16.2570 

18.9666 

21.6760 

24.3855 

27.0948 

29.8043 

32.5140 

35 

13.9446 

16.7334 

19.5224 

22.3114 

25.1003 

27.8892 

30.6781 

33.4671 

36 

14.3422 

17.2104 

20.0790 

22.9474 

25.8158 

28.6844 

31.5528 

34.4211 

37 

14.7396 

17.6874 

20.6:154 

23.5834 

26.5313 

29.4792 

32.4271 

35.3751 

38 

15.1370 

18.1624 

21.1898 

24.2192 

27.2466 

30.2740 

33.3014 

36.3288 

39 

15.5344 

18.6410 

21.7480 

24.8550 

27.9618 

31.0688 

34.1756 

37.2825 

40 

15.9318 

19.1180 

22.3044 

25.4910 

28.6774 

31.8636 

35.0500 

38.2365 

41 

16.3294 

19.6154 

22.8814 

26.1272 

29.3931 

32.6588 

35.9247 

39.1908 

42 

16.7270 

20.0724 

23.4178 

26.7632 

30.1086 

33.1540 

36.7994 

40.1448 

43 

17.1244 

20.5494 

23.9744 

27.3992 

30.8241 

34.2488 

37.6737 

41.0988 

44 

17.5222 

21.0264 

24.5310 

28.0354 

31.5398 

35.0444 

38.5488 

42.0531 

45 

17.9196 

21.5034 

25.0874 

28.6714 

32.2553 

35.8392 

39.4231 

43.0071 

46 

18.3172 

21.9806 

25.6442 

29.3076 

32.9710 

36.6344 

40.2978 

43.9614 

47 

18.7146 

22.4576 

26.2006 

29.9436 

33.6865 

37.4292 

41.1721 

44.9154 

48 

19.1122 

22.9346 

26.7572 

30.5796 

34.4020 

38.2244 

42.0468 

45.8694 

49 

19.5098 

23.4116 

27.3136 

31.2158 

35.1177 

39.0196 

42.9215 

46.8167 

50 

19.9074 

23.8890 

27.8704 

31.8520 

35.8335 

39.8148 

43.7963 

47.7780 

51 

20.3050 

24.3660 

28.4282 

32.4882 

36.5492 

40.6100 

44.6710 

48.7323 

52 

20.7026 

24.8432 

28.9838 

33.1244 

37.2649 

41.4052 

45.5457 

49.6866 

53 

21.1050 

25.3260 

29.5470 

33.7680 

37.9890 

42.2100 

46.3505 

50.6520 

54 

21.4982 

25.7976 

30.0974 

34.3970 

38.6966 

42.9964 

47.2960 

51.5955 

55 

21.8958 

26.2748 

30.6540 

35.0334 

39.4125 

43.7916 

48.1707 

52.5501 

56 

22.2936 

26.7522 

31.2110 

35.6698 

40.1285 

44.5872 

49.0459 

53.5047 

57 

22.6912 

27.2294 

31.7678 

36.3060 

40.8442 

45.3824 

49.9206 

54.4590 

58 

23.0888 

27.7064 

32.3242 

36.9422 

41.5599 

46.1776 

50.7953 

55.4133 

59 

23.4864 

28.1838 

32.8812 

37.5784 

42.2757 

46.9728 

51.6701 

56.3676 

60 

23.9842 

28.6608 

33.4378 

38.2146 

42.9914 

47.9684 

52.7452 

57.3219 

61 

24.2818 

29.1380 

33.9944 

38.8510 

43.7073 

48.5636 

53.4199 

58.2765 

62 

24.6794 

29.6154 

34,5514 

39.4872 

44.4231 

49.3588 

54.2947 

59.2308 

63 

25.0774 

30.0926 

35.1082 

40.1236 

45.1390 

50.1548 

55.1702 

60.1854 

64 

25.4748 

30.5696 

35.6646 

40.7598 

45.8547 

50.9496 

56.0445 

61.1397 

65 

25.8724 

31.0470 

36.2216 

41.3960 

46.5705 

51.7448 

56.9193 

62.0910 






























Pitch-Line .0i-vmet'ck op Gear-Wheels. 


379 


Number 

Teeth. 

n 

11 

Dhordal 

If 

Pitch of 

2 inches. 

Wheel ( 

91 

■^4 

>r Pinioi 

). 

03 

3 inches. 

66 

26.2701 

31.5242 

36.7782 

42.0322 

47.2862 

52.5443 

57.7943 

63.0483 

67 

26.6679 

32.0015 

37.3351 

42.6686 

48.0022 

53.3358 

58.6694 

64.0029 

68 

27.0655 

32.4786 

37.8917 

43.3048' 

48.7179 

54.1310 

59.5441 

64.9572 

69 

27:4631 

32.9558 

38.4484 

43.9410 

49.4330 

54.9263 

60.4189 

65.9115 

70 

27.8609 

33.4331 

39.0053 

44.5774 

50.1496 

55.7218 

61.2940 

66.8661 

71 

28.2588 

33.9105 

39.5623 

45.2140 

50.8658 

56.5175 

62.1693 

67.8210 

72 

28.6566 

34.3880 

40.1193 

45.8506 

51.5819 

5 / .3133 

63.0446 

68.7759 

73 

29.0544 

34.8653 

40.6762 

46.4870 

52.2979 

58.1088 

63.9193 

69.7305 

74 

29.4521 

35.3426 

41.2330 

47.1234 

53.0138 

58.9043 

64.7947 

70.6851 

/o 

29.8500 

35.8200 

41.7900 

47.7600 

53.7300 

59.7000 

65.6700 

71.6400 

76 

30.2477 

36.2973 

42.3468 

48.3964 

54.4459 

60.4955 

66.5450 

72.5946 

77 

30.6456 

36.7748 

42.9039 

49.0330 

55.1621 

61.2913 

67.4204 

73.5495 

78 

31.0435 

37.2522 

43.4609 

49.6696 

55.8783 

62.0870 

68.2957 

74.5044 

79 

31.4412 

37.7295 

44.0177 

50.3060 

56.5942 

62.8825 

69.1707 

75.4590 

80 

31.8391 

38.2070 

44.6048 

50.9426 

57.3104 

63.6783 

70.0461 

76.4139 

81 

32.2369 

38.6843 

45.1317 

51.5790 

58.0564 

64.4738 

70.9212 

77.3685 

82 

32.6347 

39.1617 

45.6886 

52.2156 

58.7425 

65.2695 

71.7964 

78.3234 

83 

33.0325 

39.6390 

46.2455 

52.8520 

59.4585 

66.0650 

72.6715 

79.2780 

84 

33.4304 

40.1165 

46.8026 

53.4886 

60.1747 

66.8608 

73.5469 

80.2329 

85 

33.8281 

40.5938 

47.3594 

54.1250 

60.8906 

67.6563 

74.4219 

81.1875 

86 

34.2259 

41.071 i 

47.9163 

54.7614 

61.6066 

68.4518 

75.2970 

82.1421 

87 

34.6237 

41.5485 

48.4632 

55.3980 

62.3227 

69.2475 

76.1622 

83.0970 

88 

35.0215 

42.0258 

49.0301 

56.0344 

63.0387 

70.0430 

77.0473 

84.0516 

89 

35.4192 

42.5031 

49.5869 

56.6708 

63.7546 

70.8385 

77.9223 

85.0062 

90 

35.8171 

42.9806 

50.1440 

57.3074 

64.4708 

71.6343 

78.7977 

85.9611 

91 

36.2149 

43.4579 

50.7009 

57.9438 

65.1868 

72.4298 

79.6728 

86.9157 

92 

36.6127 

43.9353 

51.2578 

58.5804 

65.9029 

73.2255 

80.5480 

87.8706 

93 

37.0105 

44.4126 

51.8147 

59.2168 

66.6189 

74.0210 

81.4231 

88.8252 

94 

37.4084 

44.8900 

52.3707 

59.8534 

67.3351 

74.8167 

82.2974 

89.7801 

95 

37.8061 

45.3674 

52.9286 

60.4898 

68.0510 

75.6123 

83.1735 

90.7347 

96 

38.2040 

45.8448 

53.4856 

61.1264 

68.7672 

76.4080 

84.0488 

91.6896 

97 

38.6017 

46.3221 

54.0424 

61.7628 

69.4831 

77.2035 

84.9238 

92.6442 

98 

38.9996 

46.7996 

54.5995 

62.3994 

70.1993 

77.9993 

85.7992 

93.5991 

99 

39.3973 

47.2768 

55.1562 

63.0358 

70.9152 

78.7947 

86.6741 

94.5537 

100 

39.7952 

47.7543 

55.7133 

63.6724 

71.6314 

79.5905 

87.5495 

95.5086 

101 

40.1930 

48.2316 

56.2702 

64.3088 

72.3474 

8013860 

88.4646 

96.4632 

102 

40.5907 

48.7089 

56.8270 

64.9432 

73.0613 

81.1795 

89.2976 

97.4178 

103 

40.9836 

49.1863 

57.3790 

65.5918 

73.7845 

81.9872 

90.1799 

98.3727 

104 

41.3814 

49.6637 

57.9360 

66.2182 

74.4905 

82.7728 

91.0451 

99.3273 

105 

41.7841 

50.1410 

58.4968 

66.8546 

75.2114 

83.5683 

91.9241 

100.2819 

106 

42.1819 

50.6183 

59.0547 

67.4910 

75.9274 

84.3638 

92.8002 

101.2365 

107 

42.5796 

51.0955 

59.6114 

68.1274 

76.6433 

85.1592 

93.6751 

102.1911 

108 

42.9773 

51.5728 

60.1682 

68.7 638 

77.3592 

85.9547 

94.5501 

103.1457 

109 

43.3751 

52.0502 

60.7252 

69.4002 

78.0752 

86.7503 

95.4253 

104.1003 

110 

43.7729 

52.5275 

61.2821 

70.0366 

78.7912 

87.5458 

96.3004 

105.0549 

111 

44.1707 

53.0049 

61.8390 

70.6732 

79.5073 

88.3415 

97.1766 

106.0098 

112 

44.5687 

53.4825 

62.3962 

71.3100 

80.2237 

89.1375 

98.0212 

106.9650 

113 

44.9666 

53.9600 

62.9533 

71.9166 

80.9399 

89.9333 

98.9266 

107.9199 

114 

45.3645 

54.4374 

63.5103 

72.5832 

81.6561 

90.7290 

99.8019 

108.8748 

115 

45.7625 

54.9150 

64.0675 

73.2200 

82.3725 

91.5250 

100.6775 

109.8300 

116 

46.1604 

55.3925 

64.6246 

73.8566 

83.0887 

92,5208 

101.5559 

110.7849 

117 

46.5582 

55.8699 

65.1815 

74.4932 

83.8048 

93.1165 

102.4281 

111.7398 

118 

46.9562 

56.3475 

65.7387 

75.1300 

84.5212 

93.9125 

103.3037 

112.6950 

119 

47.3541 

56.8250 

66.2958 

75.7666 

85.2374 

94.7083 

104.1793 

113.6499 

120 

47.7520 

57.3024 

66.8528 

76.4032 

85.9536 

95.5040 

105.0544 

114.6048 

121 

48.1497 

57.7797 

67.4096 

77.0396 

86.6695 

96.2995 

105.9294 

115.5594 

122 

48.5476 

58.2572 

67.9667 

77.6762 

87.3857 

97.0953 

106.8048 

116.5143 

123 

48.9455 

58.7346 

68.5237 

78.3128 

88.1016 

97.8910 

107.6801 

117.4692 

124 

49.3434 

59.2121 

69.0808 

78.9494 

88.8181 

98.6868 

108.5555 

118.4241 

125 

49.7412 

59.6895 

69.6377 

79.5860 

89.534 *2 

99.4825 

109.4307 

119.3790 





























380 Pitch-Line Diami o. ( Iear-Wiieels. 


Number 

Teeth. 

n 

C 

n 

hordal F 

If 

itch of 1 

2 inches. 

Vheel oi 

2J 

Pinion. 

2 h 

2f 

3 inohes. 

12G 

50.1390 

60.1668 

70.1946 

80.2224 

90.2502 

100.27S0 

110.30o8 

120.3336 

127 

50.5369 

60.6443 

70.7517 

80.8590 

90.9664 

101.1808 

111.1812 

121.2885 

128 

50.9347 

61.1217 

71.3086 

81.4956 

91.5825 

101.8695 

112.0564 

122.2434 

129 

51.3325 

61.5990 

71.8655 

82.1320 

92.3985 

102.6650 

112.9315 

123.1980 

130 

51.7304 

62.0765 

72.4226 

82.7678 

93.1147 

103.4608 

113.8069 

124.1529 

131 

52.1282 

62.5539 

72.9795 

83.4052 

93.8308 

104.2565 

114.6821 

125.1078 

132 

52.5260 

63.0612 

73.5364 

84.0416 

94.5468 

105.0520 

115.5572 

126.0624 

133 

52.9239 

63.5087 

74.0945 

84.6782 

95.2630 

105.8478 

116.4326 

127.0173 

134 

53.3217 

63.9861 

74.6504 

85.3148 

95.9791 

106.6435 

117.3078 

127.9722 

135 

53.7196 

64.4636 

75.2075 

85.9514 

96.6953 

107.1393 

118.1832 

128.9271 

136 

54.1173 

64.9408 

75.7842 

86.5878 

97.4112 

108.1347 

119.0781 

129.8817 

137 

54.5152 

65.4183 

76.3213 

87.2244 

98.1274 

108.9305 

149.9335 

130.8366 

138 

54.9131 

65.8958 

76.8784 

87.8610 

98.8436 

109.8263 

120.8089 

131.7915 

139 

55.3122 

66.3747 

77.4371 

88.4996 

99.5620 

110.7245 

121.6869 

l: 12.7494 

140 

55.7089 

66.8507 

77.9925 

89.1342 

100.2760 

111.4178 

122.5596 

133.7013 

141 

56.1067 

67.3281 

78.5494 

89.7708 

100.9921 

112.2135 

123.4348 

134.6562 

142 

56.5046 

67.8056 

79.1665 

90.4074 

101.7083 

113.0093 

124.3702 

135.6111 

143 

56.9002 

68.2803 

79.6603 

91.0404 

102.4204 

113.8005 

125.1805 

136.5606 

144 

57.3002 

68.7603 

80.2203 

91.6804 

103.1404 

114.6005 

126.0605 

137.5206 

145 

57.6981 

69.2378 

80.7774 

92.3170 

103.8566 

115.3963 

126.9359 

138.4755 

146 

58.0960 

69.7152 

81.3344 

92.9536 

104.5728 

116.1920 

127.8112 

139.1304 

147 

58.4939 

70.1827 

81.8915 

93.5902 

105.2890 

116.9878 

128.6866 

140.3853 

148 

58.8917 

70.6701 

82.4484 

94.2268 

106.0051 

117.7835 

129.5618 

141.3402 

149 

59.2896 

71.1476 

83.0055 

94.8634 

106.7213 

118.5793 

130.4372 

142.2951 

150 

59.6875 

71.6250 

83.5625 

95.5000 

107.4375 

119.3750 

131.3125 

143.2500 

151 

60.0854 

72.1025 

84.1196 

96.1366 

108.1537 

120.1708 

132.1879 

144.2049 

152 

60.4832 

72.5799 

84.6765 

96.7732 

108.8698 

120.9665 

133.0631 

145.1598 

153 

60.8811 

73.0574 

85.2336 

97.4098 

109.5860 

121.7623 

133.9385 

146.1147 

154 

61.2790 

73.5348 

85.7906 

98.0464 

110.3022 

122.5580 

134.8138 

147.0696 

155 

61.6767 

74.0121 

86.3474 

98.6828 

111.0181 

123.3535 

135.6888 

148.0242 

156 . 

62.0746 

74.4896 

86.9045 

99.3194 

111.7343 

124.1493 

136.5642 

148.9791 

157 

62.4725 

74.9670 

87.4615 

99.9560 

112.4505 

124.9450 

137.4395 

149.9340 

158 

62.8704 

75.4445 

88.0186 

100.5926 

113.1667 

126.7408 

138.3149 

150.8889 

159 

63.2682 

75.9219 

88.5755 

101.2292 

113.8828 

126.5365 

139.1901 

151.8438 

160 

63.6661 

76.3994 

89 1326 

101.8658 

114.6990 

127.3323 

140.0655 

152.7987 

161 

64.0640 

76.8768 

89.6896 

102.5024 

115.3152 

128.1280 

140.9408 

153.7536 

162 

64.4619 

77.3543 

90.2467 

103.1390 

116.0314 

128.9238 

141.8162 

154.7085 

163 

64.8597 

77.8317 

90.8036 

103.7756 

116.7475 

129.7195 

142.6914 

155.6634 

164 

65.2575 

78.3090 

91.3605 

104.4120 

117.4635 

130.5150 

143.5665 

156.6180 

165 

65.6554 

78.7865 

91.9176 

105.0486 

118.1597 

131.3108 

144.4419 

157.5729 

166 

66.0532 

79.2639 

92.4745 

105.6852 

118.8958 

132.1065 

145.3171 

158.5278 

167 

66.4509 

79.7412 

93.0313 

106.3216 

119.6117 

132.9020 

146.1921 

159.4824 

168 

66.8489 

80.2187 

93.5885 

106.9582 

120.3280 

133.6978 

147.0676 

160.4373 

169 

67.2462 

80.6951 

94.1439 

107.5948 

121.0436 

134.4925 

147.9413 

161.3922 

170 

67.6446 

81.1735 

94.7024 

108.2314 

121.7603 

135.2892 

148.8181 

162.3471 

171 

68.0425 

81.6510 

95.2595 

108.8680 

122.4765 

136.0850 

149.6935 

163.3020 

172 

68.4404 

82.1285 

95.8166 

109.5046 

123.1927 

136.8808 

150.5689 

164.2569 

173 

68.8382 

82.6059 

96.3735 

110.1412 

123.9088 

137.6765 

151.4441 

165.2118 

174 

69.2361 

83.0834 

96.9306 

110.7778 

124.6250 

138.4723 

152.3195 

166.1667 

175 

69.6340 

83.5608 

97.4876 

111.4144 

125.3412 

139.2680 

153.1948 

167.1216 

176 

70.0319 

84.0383 

98.0427 

112.0510 

126.0574 

140.0638 

154.0182 

108.0765 

177 

70.4297 

84.5157 

98.6016 112.6876 

126.7735 

140.8595 

154.9454 

169.0314 

178 

70.8276 

84.9932 

99.1587 

113.3242 

127.4897 

141.6553 

155.8208 

169.9863 

179 

71.2255 

85.4706 

99.7157 

113.9608 

128.2059 

142.4510 

156.6961 

170.9412 

ISO 

71.6234 

85.9481 

100.2728 

114.5974 

128.9221 

143.2468 

157.5715 

171.8961 

181 

72.0212 

86.4255 

100.8297 

115.2340 

129.6382 

144.0425 

158.4467 

172.8510 

182 

72.4191 

86.9030 

101.3868 115.8706 

130.3544 

144.8383 

159.3221 

173.8059 

183 

72.8170 

87.3804 

101.9438 116.5072 

131.0706 

145.6840 

160.1974 

174.7608 

184 

73.2150 

87.8580 

102.5010 

117.1540 

131.7970 

146.4400 

161.0830 

175.7160 

185 

73.6129 

88.3355 

103.0581 

117.7806 

132.5032 

147.2258 

161.9484 

176.6709 










































PlTCH-LlNE 1 I.\ )• KR OF GEAR-WHEELS. 381 


Number 
Teeth. 

n 

H 

Chordal 

It 

Pitch of 

2 inches 

Wheel ( 

2* 

)r Pinior 

2* 

>. 

2# 

3 inches. 

186 

74.0107 

88.8129 

103.6150 

118.4172 

133.2193 

148.0215 

162.8236 

177.6258 

187 

74.4086 

89.2903 

104.1721 

119.0538 

133.9355 

148.8173 

163.6990 

178.5807 

188 

74.8065 

89.7678 

104.7291 

119.6904 

134.6517 

149.6130 

164.5743 

179.5356 

189 

75.2044 

90.2453 

105.2862 

120.3270 

135.3679 

150.4088 

165.4497 

180.4905 

190 

75.6222 

90.7227 

105.8631 

120.9636 

136.1040 

151.2045 

166.3449 

181.4448 

191 

76.0001 

91.2002 

106.4002 

121.6002 

136.8002 

152.0003 

167.2003 

182.4003 

192 

76.3980 

91.6776 

106.9572 

122.2368 

137.5164 

152.7960 

168.0756 

183.3552 

193 

76.7957 

92.1549 

107.5140 

122.8732 

138.2323 

153.5955 

168.9506 

184.3098 

194 

77.1936 

92.6324 

108.0711 

123.5098 

138.9485 

154.3873 

169.8260 

185.2647 

195 

77.5915 

93.1098 

108.6281 

124.1464 

139.6647 

155.1830 

170.7013 

186.2196 

196 

77.9895 

93.5874 

109.1853 

124.7832 

140.3811 

155.9790 

171.5769 

187.1748 

197 

78.3874 

94.0649 

109.7424 

125.4198 

141.0973 

156.7748 

172.4523 

188.1297 

198 

78.7854 

94.5425 j 110.2996 

126.0566 

141.8137 

157.5708 

173.3279 

189.0849 

199 

79.1832 

95.0199 110.8565 

126.6932 

142.5298 

158.3665 

174.2031 

190.0398 

200 

79.5811 

95.5974 

111.4136 

127.3298 

143.2460 

159.1623 

175.0785 

190.9947 

201 

79.9790 

95.9748 

111.9706 

127.9664 

143.9622 

159.9580 

175.9538 

191.9496 

202 

80.3772 

96.4521 

112.5274 

128.6028 

144.6881 

160.7535 

176.8288 

192.9042 

203 

80.7744 

96.9291 

113.0838 

129.2394 

145.3941 

161.5488 

177.7035 

193.8591 

204 

81.1725 

97.4070 

113.6415 

129.8760 

146.1105 

162.3450 

178.5795 

194.8140 

205 

81.5704 

97.8845 

114.1986 

130.5126 

146.8267 

163.1408 

179.4549 

195.7689 

206 

81.9682 

98.3619 

114.7555 

131.1492 

147.5428 

163.9365 

180.3301 

196.7238 

207 

82.3661 

98.8394 

115.3126 

131.7858 

148.2590 

164.7323 

181.2055 

197.6787 

208 

82.7640 

99.3168 

115.8696 

132.4224 

148.9752 

165.5280 

182.0808 

198.6336 

209 

83.1619 

99.7943 

116.4267 

133.0590 

149.6914 

166.3238 

182.9662 

199.5885 

210 

83.5597 

100.2717 

116.9836 

133 6956 

150.4075 

167.1195 

183.8314 

200.5434 

211 

83.9576 

100.7492 

117.5407 

134.3322 

151.1237 

167.9153 

184.7068 

201.4983 

212 

84»d5oo 

101.2266 

118.0977 

134.9688 

151.8399 

168.7110 

185.5821 

202.4532 

213 

84.7532 

101.7039 

118.6545 

135.6052 

152.5558 

169.5065 

186.4571 

203.4078 

214 

85.1511 

102.1814 

119.2116 

136.2418 

153.2720 

170.3023 

187.3325 

204.3627 

215 

85.5489 

102.6587 

119.7685 

136.8782 

153.9880 

171.0978 

188.2076 

205.3173 

216 

85.9468 

103.1363 

120.3257 

137.5148 

154.7042 

171.893f 

189.0831 

206.2722 

217 

86.3446 

103.6135 

120.8824 

138.1514 

155.4203 

172.6892 

189.9581 

207.2271 

218 

86.7425 

104.0910 

121.4395 

138.7880 

156.1365 

173.4850 

190.8335 

208.1820 

219 

87.1404 

104.5685 

121.9906 

139.4246 

156.8527 

174.2808 

191.7089 

209.1369 

220 

87.5382 

105.0159 

122.5535 

140.0612 

157.5688 

175.0765 

192.5841 

210.0918 

221 

87.9361 

105.5234 

123.1106 

140.6978 

158.2850 

175.8723 

193.4595 

211.0467 

222 

88.3340 

106.0008 

123.6676 

141.3344 

159.0012 

176.6680 

194.3348 

212.0016 

223 

88.7319 

106.4783 

124.2247 

141.9710 

159.7174 

177.4638 

195.2102 

212.9565 

224 

89.1297 

106.9557 

124.7816 

142.6076 

160.4335 

178.2595 

196.0854 

213.9114 

225 

89.5276 

107.4332 

125.3387 

143.2442 

161.1497 

179.0553 

196.9608 

214.8663 

226 

89.9255 

107.9106 

125.8957 

143.8808 

161.8659 

179.8510 

197.8361 

215.8224 

227 

90.3234 

108.3881 

126.4528 

144.5174 

162.5821 

180.6468 

198.7115 

216.7761 

228 

90.7212 | 

108.8655 

127.0097 

145.1540 

163.2982 

181.4425 

199.5867 

217.7310 

229 

91.1191 

109.3430 

127.5668 

145.7906 

164.0144 

182.2383 

200.4621 

218.6859 

230 

91.5170 

109.8204 

128.1238 

146.4272 

164.7306 

183.0340 

201.3374 

219.6408 

231 

91.9148 

110.2978 

128.6807 

147.0638 

165.4467 

183.8297 

202.2126 

220.5957 

232 

92.3127 

110.7753 

129.2378 

147.7004 

166.1629 

184.6255 

203.0880 

221.5506 

233 

92.7106 

111.2528 

129.7949 

148.3370 

166.8791 

185.4213 

203.9634 

222.5055 

234 

93.1084 

111.7301 

130.3518 

148.9734 

167.5951 

186.2168 

204.8385 

223.4601 

235 

93 5062 

112.2075 

130.9087 

149.6100 

168.3112 

187.0125 

205.7137 

224.4150 

236 

93.9041 ; 

112.6850 

131.4658 

150.2466 

169.0274 

187.8083 

206.5891 

225.3699 

237 

94.3020 | 

113.1624 

132.0228 

150.8832 

169.7436 

188.6040 

207 4644 

226.3248 

238 

'7)4.7999 

113.6399 

132.5799 

151.5198 

170.4998 

189.3998 

208.3398 

227.2797 

239 

95.0977 ! 

114.1173 

133.1368 

152.1564 

171.1759 

190.1955 

209.2150 

228.2346 

240 

95.4956 

114.5948 

133.6939 

152.7930 

171.8921 

190.9913 

210.0904 

229.1895 

241 

95.8936 : 

115.07221 

134.2510 

153.4296 

172.6084 

191.7870 

210.9658 

230.1444 

242 

96.2914 

115.5997 

134.7980 

154.0662 

173.3245 

192.5828 

211.8311 

231.0993 

243 

96.6892 

116.0271 

135.3649 

154.7028 

174 0406 

193.3785 

212.7163 

232.0542 

244 

97.0871 

116.50461 

135.9220 

155.3394 

174.7568 

194.1643 

213.5917 

233.0091 

245 

97.4850 

1 

116.9820 

136.4790 

155.9760 

1 / o*4 / o0 

194.9700 

214.4670 

233.9640 









































382 Pitch-Line Diam Gear-Wheels. 


Number 

Teeth. 

Q1 

( 

4 inches. 

Chordal 1 

4* 

’itch of 

5 inches. 

vVheel c 

5§ 

r Pinioi 

6 inches. 

i. 

6* 

7 inches. 

6 

7.0000 

8.0000 

9.0000 

10.0000 

11.0000 

12.0000 

13.0000 

14.0000 

7 

8.0738 

9.2272 

10.3806 

11.5340 

12.6874 

13.8408 

14.9942 

16.1476 

8 

9.1458 

10.4524 

11.7589 

13.0655 

14.3720 

15.6786 

16.9851 

18.2917 

y 

10.2333 

11.6952 

13.1571 

14.6190 

16.0809 

17.5528 

19.0147 

20.4666 

10 

11.3263 

12.9444 

14.5624 

16.1805 

17.7985 

19.4166 

21.0346 

22.6527 

n 

12.4215 

14.1960 

15.9705 

17.7450 

19.5195 

21.2940 

23.0685 

24.8430 

12 

13.5229 

15.4548 

17.3866 

19.3185 

21.2503 

23.1822 

25.1140 

27.0459 

13 

14.6247 

16.7140 

18.8032 

20.8925 

22.9817 

25.0710 

27.1602 

29.2495 

14 

15.7290 

17.9760 

20.2230 

22.4700 

24.7170 

26.9640 

29.2110 

31.4580 i. 

15 

16.8339 

19.2388 

21.6436 

24.0485 

26.4533 

28.8582 

31.2630 

33.5679 

16 

17.9406 

20.5036 

23.0665 

25.6295 

28.1924 

30.7554 

33.3183 

35.8813 

17 

19.0480 

21.7692 

24.4903 

27.2115 

29.9326 

32.6538 

35.3749 

38.0961 

18 

20.1558 

23.0352 

25.9146 

28.7940 

31.6934 

34.5528 

37.4322 

40.3116 

19 

21.2646 

24.3024 

27.3402 

30.3780 

33.4158 

36.4536 

30.4914 

42.5292 

20 

22.3737 

25.5700 

28.7662 

31.9625 

35.1587 

38.35.S0 

41.5512 

44.7475 

21 

23.4832 

26 8380 

30.4327 

33.o4 / o 

36.9022 

40.2570 

43.6117 

46.9665 

22 

24.5931 

28.1064 

31.6197 

35.1330 

38.6463 

42.1596 

45.6729 

49.1862 

23 

25.6683 

29.3352 

33.0021 

36.6690 

40.3359 

44.0028 

47.6697 

51.3366 

24 

26.8145 

30.6452 

34.4758 

38.3065 

42.1371 

45.9678 

49.7984 

53.6291 

25 

27.9254 

31.9148 

35.9041 

39.8935 

43.8828 

47.8722 

51.8615 

55.8509 

2(5 

29.0367 

33.1848 

37.3329 

41.4810 

45.6291 

49.7772 

53.9253 

58.0754 

27 

30.1483 

34.4552 

38.7621 

43.0690 

47.3759 

51.6828 

55.9897 

60.2966 

28 

31.2602 

35.7260 

40.1917 

44.6575 

49.1232 

53.5890 

58.0547 

62.5205 

29 

32.3725 

36.9972 

41.6218 

46.2465 

50.8711 

55.4958 

60.1204 

G4.7451 

30 

33.4838 

38.2672 

43.0506 

47.8340 

52.6174 

57.4008 

62.1842 

66.9676 

31 

34.5957 

39.5380 

44.4802 

49.4225 

54.3647 

59.3070 

64.2492 

69.1915 

32 

35.7081 

40.8092 

45.9014 

51.0115 

56.1127 

61.2138 

66.3150 

71.4161 

33 

36.8203 

42.0804 

47.3404 

52.6005 

57.8605 

63.1206 

68.3806 

73.6407 

34 

37.9330 

43.3520 

48.7710 

54.1900 

59.6090 

65.0280 

70.4470 

75.8660 

35 

39.0449 

44.6228 

50.2006 

55.7785 

61.3568 

66.9342 

72.5120 

78.0899 

30 

40.1579 

«1.8948 

51.6316 

57.3685 

63.1053 

68.8422 

74.5790 

80.3159 

37 

41.2709 

47.1668 

53.0626 

58.9585 

64.8543 

70.7502 

76.6460 

82.5419 

38 

42.3836 

48.4384 

54.4932 

60.5480 

66.5928 

72.6576 

78.7124 

84.7672 

39 

43.4962 

49.7100 

55.9237 

62.1375 

68.3512 

74.5650 

80.7787 

86.9925 

40 

44.6093 

50.9820 

57.3548 

63.7275 

70.1003 

76.4730 

82.8458 

89.2185 

41 

45.7226 

52.2544 

58.7862 

65.3180 

71.8498 

78.3816 

84.9134 

91.4452 

42 

46.8356 

53.5264 

60.2172 

66.9080 

73.5988 

80.2896 

86.9804 

93.6712 

43 

47.9486 

54.7984 

61.6482 

68.4980 

75.3478 

82.1976 

89.0474 

95.8972 

44 

49.0619 

56.0708 

63.0796 

70.0885 

77.0973 

84.1062 

91.1150 

98.1229 

45 

50.1749 

57.3428 

64.5106 

71.6785 

78.8463 

86.0242 

93.1920 

100.3499 

4G 

51.2883 

58.6152 

65.9421 

73.2690 

80.5959 

87.9228 

95.2497 

102.5766 

47 

52.4013 

59.8872 

67.3731 

74.8590 

82.3449 

89.8308 

97.3167 

104.8026 

48 

53.5143 

61.1592 

68.8041 

76.4490 

84.0939 

91.7388 

99.3837 

107.0286 

49 

54.6206 

62.4316 

70.2355 

78.0158 

85.8197 

93.6474 

101.4513 

109.2553 

50 

55.7410 

63.7040 

71.6670 

79.6300 

87.5930 

95.5560 

103.5190 

111.4820 

51 

56.8543 

64.9764 

73.0984 

81.2205 

89.3425 

97.4646 

105.5866 

113.7087 

52 

57.9677 

66.2488 

74.5299 

82.8110 

91.0921 

99.3732 

107.6543 

115.9354 

53 

59.0940 

67.5360 

75.9780 

84.4200 

92.8620 

101.3040 

109.7460 

118.1880 

54 

60.1947 

68.7940 

77.3932 

85.9925 

94.5917 

103.1910 

111.7802 

120.3895 

55 

61.3034 

70.0668 

78.8251 

87.5835 

96.3418 

105.1002 

113.8595 

122.6169 

50 

62.4221 

71.3396 

80.2570 

89.1745 

98.0919 

107.0294 

115.9468 

124.8443 

57 

63.5355 

72.6120 

81.6885 

90.7650 

99.8415 

108.9180 

117.9945 

J27.0710 

58 

64.6488 

73.8844 

83.1199 

92.3555 

101.5910 

110.8266 

119.0621 

"129.2977 

59 

65.7622 

75.1568 

84.5514 

93.9460 

103.3406 

112.7352 

122.1298 

131.5244 

60 

66.875.0 

76.4292 

85.9828 

95.5365 

105.0901 

114.6438 

124.1974 

133.7511 

61 

67.9892 

77.7020 

87.4147 

97.1275 

106 8402 

116.5530 

126.2657 

135.9785 

02 

69.1026 

78.9744 

88.8462 

98.7180 

108.5898 

118.4616 

128.3334 

138.2052 

63 

70.2163 

80.2472 

90.2781 

100.3090 

110.3399 

120.3708 

130.4017 

140.4326 

04 

71.3296 

81.5196 

91.7095 

101.8995 

112.0894 

122.2794 

132.4693 

142.6593 

65 

72.4430 

S2.7920 

93.1410 

103.4900 

113.8390 

124.1880 

134.5370 

144.8860 

































!s T uml 

Teet 

66 

67 

68 

69 

70 

71 

72 

78 

74 

75 

76 

77 

78 

79 

80 

81 

82 

83 

84 

85 

86 

87 

88 

89 

90 

91 

92 

93 

94 

95 

96 

97 

98 

99 

100 

101 

102 

103 

104 

105 

106 

107 

108 

109 

110 

111 

112 

113 

114 

115 

116 

117 

118 

119 

120 

121 

122 

123 

124 

125 


Pitch-Link Dia ter C ' Gear-Wheels. 


383 


Chordal Piich of Wheel or Pinion. 




4 inches. 


84.0644 

85.3872 

86.6096 

87.8820 

89.1548 

90.4280 

91.7012 

92.9740 

94.2468 

95.5200 

96.7928 

98.0660 

99.3392 

100.6120 

101.8S5S 

103.1580 

104.4312 

105.7040 

106.9772 

108.2500 

109.5228 

110.7960 

112.00S8 

113.3416 

114.6148 

115.8876 

117.1608 

118.4336 

119.7068 

120.9796 

122.2528 

123.5256 

124.7988 

126.0716 

127.3448 

128.6352 

129.8904 

131.1636 

132.4364 

133.7092 

134.9820 

136.2548 

137.5276 

138.8004 

140.0732 

141.3464 

142.6200 

143.8932 

145.1664 

146.4600 

147.7132 

148.9864 

150.2600 

151.3332 

152.8064 

154.0792; 

155.3524, 

156.6956 

157.89881 

159.1720; 


4* 


5 incheB 


94.5725; 105.0805 
96.0044] 106.6715 
97.43581108.2020 


98.8673 

100.2992 

101.7315 

102.1639 

103.5958 

105.0277 

107.4600 

108.8919 

110.3243 

111.7566 

113.1885 

114.6209 

116.0528 

117.4851 

118.9170 


109.8525 

111.4435 

113.0350 

114.6265 

116.2175 

117.8085 

119.4400 

120.9910 

122.5825 

124.1740 

125.7650 

127.3565 

128.9475 

130.5390 

132.1560 


120.3494 133.7215 
121.7813 135.3125 
122.9132 136.9035 
124.6455 138.4950 
126.0774 140.0860 
127.5093 141.6770 
12S.9417 143.2685 
130.3736 144.8595 
131.8059 146.4510 
133.2378 148.0420 
134.6701 149.6335 
136.1021 151.2245 
137.5844 152.8160 
138.9663 154.4070 
140.3987 155.9985 


141.8305 

143.2629 


157.5895 

159.1810 


144.7124 160.7720 


146.1267 

147.5590 

148.9910 

149.4229 

151.8548 

153.2866 


162.3630 

163.9545 

165.5455 

167.1365 

168.7275 

170.3185 


154.7185 171.9095 
156.1505 173.5005 
157.58241175.0915 
159.0147 176.6830 
160.4475 178.2750 
161.8799] 179.7265 
163.3122 181.4580 
164.7650 183.0500 


166.1774 

167.6097 

169.0425 

170.2749 

171.9072 

173.8391 

174.7715 

176.2038 

177.6362 

179.0685 


184.6415 

186.2330 

187.8250 

189.4165 

191.0080 

192.5990 

194.1905 

195.7820 

197.3735 

198.9650 


5b 


115.5886 ! 
117.3387 
119.0882 ; 
120.8378] 
122.5879 
124.3385 
. 120.0892 
127.8393 
129.5894 
131.3800 
133.0901 
134.8408 
136.5914 
138.3415 
140.0922 
141.8423 
143.5929 
145.3690 
147.0937 
148.8438! 
150.5939 
152.3445 
154.0946 
155.8447 
157.5954 
159.3455 
161.0961 
162.8462 
164.596S 
166.3470 
168.0976 
169.8877 
171.5984 
173.3484 
175.0991 
176.8492 
178.5993 
180.3499 
181.9001 
183.8502 
185.6003 
187.3503 
189.1004 
190.8506 
192.60071 
194.3513 


199.6038 

201.3550 

203.1057 
204.8563 
206.6075j 
208.3572; 

210.1058 
211.8989 
213.6096 
215.3602 
217.1109 
218.8615 


6 inches. 


126.0966 

128.0058 

129.9144 

131.8230 

133.7322 

135.6420 

137.5518 

139.4610 

141.3702 

143.2800 

145.1892 

147.0990 

149.0088 

150.9180 

152.8278 

154.7370 

156.6468 

158.5560 

160.4658 

162.3750 

164.2842 

166.1940 

168.1032 

170.0124 

171.9222 

173.8314 

175.7412 

177.6504 

179.5602 

181.4694 

183.3792 

185.2884 

187.1982 

189.1074 

191.0172 

192.9264 

194.8356 

196.7454 

198.6546 

200.5638 

202.4730 

204.3822 

206.2914 

208.2006 


219.6600 

221.5698 

223.4796 

225.3900 

227.2998 

229.2096 

231.1188 

233.0286 

234.9384 

236.8482 

238.7580 




136.6047 
138.6730 
140.7406 
142.8083 
144.8766 
146.9455 
149.0135 
151.0828 
153.1511 
155.2200 
157.2883 
159.3573 
161.4262 
163.4945 
165.5635 
167.6318 
169.7007 
171.7690 
173.8380 
175.9063 
177.9746 
180.0435 
182.1118 
184.1801 
186.2491 
188.3174 
190.3863 
192.4546 
194.5235 
196.5919 
198.6608 
200.7291 
202.7981 
204.8663 
206.9353 
209.0036 
211.0719 
213.1408 
215.2092 
217.2775 
•219.3458 
221.4140 
223.4823 
225.5507 
227.6190 
229.6879 
231.7575] 
233.8265] 
235.8954 
237.9650 
240.0340 
242.1029 
244.1725 
246.2415 
248.3104 
250.3787 
252.4477 
254.5166 
256.5856 
258.6545 


210.1098 
212.0196 
196.1025; 213.9300 
197.7132| 215.8398 
217.7496 


7 inches. 


147.1127 

149.3401 

151.5668 

153.7935 

156.0209 

158.2490 

160.4771 

162.7045 

164.9319 

167.1600 

169.3874 

171.6155 

173.8436 

176.0710 

178.2991 

180.5265 

182.7546 

184.9820 

187.2101 

189.4375 

191.0649 

193.8930 

196.1204 

198.3478 

200.5759 

202.8233 

205.0314 

207.2588 

209.4869 

211.7143 

213.9424 

216.1698 

21S.3979 

220.6253 

222.8534 

225.0808 

227.3082 

229.5363 

231.7637 

233.9911 

236.2185 

238.4459 

240.6733 

242.9007 

245.1281 

247.3562 

249.5850 

251.8131 

254.0412 

256.2700 

258.4981 

260.7262 

262.9550 

265.1831 

267.4112 

269.6386 

271.8667 

274.0948 

276.3229 

278.5510 
















































384 


Pitch-Line Dia vet;-: a op Gear-Wheels. 


Number 


Chordal Pitch of Wheel or Pinion. 


Teeth. 


4 inches. 

4* 

5 inches. 

5£ 

6 inches. 

65 

7 inches. 

126 

140.3892 

160.4448 

180.50041 

200.5560 

220.6116 

240.6672 

260.7228 

280.7784 

127 

141.5033 

161.7180 

181.9328 

202.1475 

222.3623 

242.5770] 

262.7918 

283.0065 

128 

142.6173 

162.9912 

183.3651 

203.7390 

224.1129 

244.4868 

264.8607 

285.2346 

129 

143.7310 

164.2640 

184.7970 

205.3300 

225.8630 

246.39601 

266.9290J 

287.4620 

130 

144.8451 

165.5372 

186.2294 

206.9215 

227.6137 

248.30581 

288.9980 

289.6901 

131 

145.9591 

166.8104 

187.6617 

208.5130 

229.3643 

250.21561 

271.0669 

291.9182 

132 

147.0728 

168.0832 

189.0936 

210.1040 

231.1144 

2o2.1248| 

273.1352 

294.1456 

133 

148.1869 

169.3564 

190.5260 

211.6955 

232.8651 

2.34.0346 

275.2042 

296.3737 

134 

149.3009 

170.6296 

191.9583 

213.2870 

234.6157 

255.9444 

277.2731 

298.6018 

135 

150.4150 

171.9028 

193.3907 

214.8785 

236.3664 

257.8.342 

279.3421 

300.8299 

136 

151.5286 

173.1756 

194.8225 

•216.4695 

238.1164 

259.7634) 

281.4103 

303.0573 

137 

152.6427 

174.4488 

196.2549 

218.0610 

239.8671 

261.6732 

283.4793 

305.2854 

138 

153.7568 

175.7220 

197.6873 

219.6525 

241.6178 

263.5830 

285.5483 

307.5135 

139 

154.8743 

176.9992 

199.124! 

221.2490 

243.3739 

265.4988 

287.6237 

309.7486 

140 

155.9849 

178.2684 

200.5520 

222.8355 

245.1191 

267.4026 

289.6862 

311.9697 

141 

157.0989 

179.5416 

201.9843 

224.4270 

246.8697 

269.3124 

291.7551 

314.1878 

142 

158.2130 

180.8148 

203.4167 

226.0185 

248.6204 

271.2222 

293.8241 

316.4259 

143 

159.3207 

182.0808 

204.8409 

227.601-0 

250.3611 

273.1212 

295.8813 

318.6414 

144 

160.4407 

183.3608 

206.2809 

229.2010 

252.1211 

275.0412 

297.9613 

320.8814 

145 

161.5548 

1846340 

207.7133 230.7925 

253.8718 

276.9510 

300.0303 

3213.1095 

146 

162.6688 

185.9072 

209.1456 

232.3640 

255.6224 

278.8608 

302.0992 

325.3376 

147 

163.7829 

187.0804 

210.4780 

233.9755 

257.3731 

280.7706 

304.1682 

327.5657 

148 

164.8969 

1SS.4536 

212.0103 

235.5670 

259.1237 

282.6804 

306.2371 

329.7938 

149 

166.0110 

189.7268 

213.4427 

237-1585 

260.8744 

284.5902 

308.3061 

332.0219 

150 

167.1250 

191.0000 

214.8750 

238.7500 

262.6250 

286.5000 

310.3750 

334.2500 

151 

168.2391 

192.2732 

216.3074 

240.3415 

264.3757 

288.4098 

312.4440 

336.4781 

152 

169.3531 

193.5464 

217.7397 

241.9330 

266.1263 

290.3196 

314.5129 

338.7062 

153 

170.4672 

1948196 

219.1721 

243.5245 

267.8770 

292.2294 

316.5819 

340.9243 

154 

171.5812 

196.0928 

220.6044 

245.1460 

269.6276 

294.1392 

318.6508 

343.1624 

155 

172.6949 

197.3656 

222.0363 

246.9070 

271.5777 

296.0484 

320.7191 

345.3898 

156 

173.8090 

198.6388 

223.4687 

248.2985 

233.1284 

297.9582 

322.7881 

347.6179 

157 

174.9230 

199.9120 

224.9010 

249.8900 

274.8790 

299.8680 

324.8570 

319.8460 

158 

176.0371 

201.1852 

226.3334 

251.4815 

276.6297 

301.7778 

326.9260 

352.0741 

159 

177.1511 

202.4584 

227.7657 

2o-101 • »0 

278.3803 

303.6876 

328.9949 

354.3022 

160 

178.2652 

203.7316 

229.1981 

254.6645 

•280.1310 

305 5974 

331.0639 

356.5303 

161 

179.3892 

205.0048 

230.6304 

256.2560 

281.8816 

307.5072 

333.1328 

358.7584 

162 

180.4933 

206.2780 

232.0628 

257.8475 

283.6323 

309.4170 

335.2018 

360.9865 

163 

181.6073 

207.5512 

253.4951 

259.4390 

286.3829 

311.3268 

337.2707 

363.2146 

164 

182.7210 

208.8240 

234.9270 

261.0300 

287.1330 

313.2360 

339.3390 

365.4420 

165 

183.8351 

210.0972 

236.3594 

262.6215 

288.8837 

315.1458 

341.4080 

367.6701 

166 

181.9491 

211.3704 

237.7917 

264.2130 

290.6343 

317.0556 

343.4769 

369.8982 

167 

186.0628 

212.6432 

239.2236 

265.8040 

292.3844 

319.9648 

345.5452 

372.1256 

168 

187.1769 

213.9164 

240.6560 

267.39.55 

294.1351 

320.8746 

347.6142 

374.3537 

169 

188.2899 

215.1896 

242.0873 

268.9870 

295.8847 

322.7844 

349.6821 

376.5818 

170 

189.4049 

216.4628 

243.5206 

270.5785 

297.6363 

324.6942 

351.7520 

378.8099 

171 

190.5190 

217.7360 

244.9530 

272.1700 

299.3870 

326.6040 

353.8210 

380.0380 

172 

191.6331 

219.0092 

246.3854 

273.7615 

301.1377 

328.5138 

355.8900 

383.2661 

173 

192.7471 

220.2824 

247.8177 

275.3530 

302.8883 

330.4248 

357.9601 

385.4942 

174 

193.8612 

221.5556 

249.2501 

276.9445 

304.6390 

332.3334 

360.0279 

387.7223 

175 

194.9752 

222.8288 

250.6824 

278.5360 

303.3896 

334.2432 

362.0968 

389.9504 

176 

196.0893 

224.1020 

252.1148 

280.1275 

308.1403 

336.1530 

364.1658 

392.1785 

177 

197.2033 

225.3652 

253.5371 

281.7190 

309.8909 

333.0628 

366.2347 

394.4066 

178 

198.3174 

226.6484 

254.9795 

283.3105 

311.6416 

339.9726 

368.3037 

396.6347 

179 

199.4314 

227.9216 

i 256.4118 

281.9020 

313.3922 

341.8824 

370.3726 

398.8628 

180 

200.5455 

229.1948 

257.8442 

286.4935 

315.1429 

343.7822 

372.4316 

401.0909 

181 

201.6595 

230.4680 

259.2765 

288.0850 

316.8935 

345.7020 

374.5105 

403.3190 

182 

202.7736 

231.7412 

260.7089 

289.6765 

318.6442 

347.6128 

376.5705 

405.5471 

183 

203.8876 

233.0144 

262.1412 

291.2680 

320.3948 

349.5216 

378.6484 

407.7752 

184 

205.0020 

234.2880: 263.5740 

292.8600 

322.1460 

351.4320 

380.7581 

410.0040 

185 

206.1161 

235.5612 

265.0064 

1 

294.4515 

323.8967 

353.341b 

j 382.7870 j 412.2321 























































Pitch-Line Diami tf.h ok Gear-Wheels. 385 


Number 

Teeth. 

f 

3| 

( 

4 inches. 

<—< 

'P 

O 

5 itch of 

5 inches. 

Wheel o 

r Pinion 

6 inches. 


7 inches. 

1S6 

207.2301 

236.8344 

266.4387 

296.0430 

325.0473 

o55.2f>16 

384.8549 

414.4602 

187 

208.3442 

238.1076 

267.8711 

297.0345 

327.3980 

357.1614 

366.9249 

416.6883 

188 

209.4582 

239.4208 

269.3434 

299.2260 

329.1486 

:159.0712 

388.9938 

418.9164 

189 

210.5723 

240.6540 

270.7538 

300.8175 

330.8993 

360.9810 

391.0628 

421.1445 

190 

211.6857 

241.9272 

272.1681 

302.4090 

332.6499 

362.8908 

393.1317 

423.3726 

191 

212.8004 

243.2004 

273.6005 

304.0005 

334.4006 

364.8006 

395.2007 

425.6007 

192 

213.9144 

244.4736 

275.0328 

305.5920 

336.1512 

366.7104 

397.2696 

427.8288 

193 

215.0281 

245.7464 

276.4647 

307.1830 

337.9013 

368.6196 

399.3379 

430.0262 

194 

216.1422 

247.0196 

277.8971 

308.7745 

339.6520 

370.5294 

401.4069 

432.2843 

195 

217.2562 

248.2928 

279.3294 

310.3660 

341.4026 

372.4396 

403.4762 

434.5124 

19G 

218.3706 

249.5664 

280.7622 

311.9580 

343.1538 

374.3496 

405 5454 

436.7412 

197 

219.4847 

250.8396 

282.1946 

313.5495 

344.9045 

376.2595 

407.6145 

438.9693 

198 

220.5991 

252.1132 

283.6274 

315.1415 

346.6557 

S/8.1698 

409.6840 

441.1981 

199 

221.7131 

253.3864 

285.0597 

316.7330 

348.4063 

380.0796 

411.7529 

443.4262 

200 

222.8272 

254.6596 

286.4921 

318.3245 

350.1570 

381.9894 

413.8219 

445.6543 

201 

223.9412 

255.9328 

287.9244 

319.9160 

351.9076 

383.8992 

415.8908 

417.8824 

202 

225.0549 

257.2056 

289.3563 

321.5070 

353.6577 

385.8084 

417.9591 

450.1098 

203 

226.1685 

258.4788 

290.7882 

323.0985 

355.4079 

387.7182 

420.0276 

452.3379 

204 

227.2830 

259.7520 

292.2210 

324.6900 

357.1590 

389.6280 

422.0970 

454.5660 

205 

228.3971 

261.0252 

293.6534 

326.2815 

358.9097 

391.5378 

424.1660 

456.7941 

206 

229.5111 

262.2984 

295.0857 

327.8730 

360.6603 

393.4476 

426.2349 

459.0122 

207 

230.6252 

263.5716 

296.5181 

329.4645 

362.4110 

395.3574 

428.3039 

461.2503 

208 

231.7392 

264.8448 

297.9504 

331.0560 

364.1616 

397.2672 

430.3728 

463.47S4 

209 

232.8533 

266.1180 

299.3828 

332.6480 

365.9128 

397.1770 

432.4418 

465.7065 

210 

233.9673 

267.3912 

300.8151 

334.2390 

367.6629 

401.0868 

434.5107 

467.9346 

211 

235.0814 

268.6644 

302.2475 

335.8305 

369.4136 

402.9966 

436.5797 

470.1627 

212 

236.1954 

269.9376 

303.6798 

337.4220 

371.1642 

404.9064 

438.6486 

472.3908 

213 

237.3091 

271.2104 

305.1117 

339.0130 

372.9143 

406.8156 

440.7169 

474.6182 

214 

238.4232 

272.4836 

306.5441 

340.6045 

374.6650 

408.7254 

442.7859 

476.8463 

215 

239.5369 

273.7564 

307.9760 

342.1955 

376.4151 

410.6346 

444.8542 

479.0737 

216 

240.6511 

275.0296 

309.4085 

343.7870 

378.1659 

412.5444 

446.9233 

481.3018 

217 

241.7649 

276.3028 

310.8406 

345.3785 

379.9163 

414.4542 

448.9920 

483.5299 

218 

•242.8790 

277.5760 

312.2730 

346.9700 

381.6670 

416.3640 

451:0610 

485.7580 

219 

243.9931 

278.8492 

313.7054 

348.5615 

383.4177 

418.2738 

453.1300 

487.9861 

220 

245.1071 

280.1224 

315.1377 

350.1530 

385.1683 

420.1836 

455.1989 

490.2142 

221 

246.2212 

281.3956 

316.5701 

351.7445 

386.9190 

422.0934 

457.2679 

492.3423 

222 

247.3352 

282.6688 

318.0024 

353.3360 

388.6696 

424.0032 

459.3868 

494.6704 

223 

248.4493 

283.9420 

319.4:848 

354.9275 

390.4203 

425.9130 

461.4058 

496.8985 

224 

249.5633 

285.2152 

320.8671 

356.5190 

392.1709 

427.8228 

463.4747 

499.1266 

225 

250.6774 

286.4884 

322.2995 

358.1105 

393.9216 

429.7326 

465.5436 

501.3547 

226 

251.7926 

287.7616 

323.7318 

359.7020 

395.6722 

431.6424 

467.6126 

503.5828 

| 227 

252.9055 

289.0348 

325.1642 

361.2935 

397.4229 

433.5522 

469.6816 

505.8109 

228 

254.0195 

290.3080 

326.5965 

362.8850 

399.1735 

435.4620 

471.7505 

508.0390 

229 

255.1336 

291.5812 

328.0289 

364.4765 

400.9242 

437.3718 

473.8195 

510.2671 

230 

256.2476 

292.8544 

329.4612 

366.0680 

402.6748 

439.2816 

475.8884 

512.4952 

231 

257.3616 

294.1276 

330.8935 

367.6585 

404.4244 

441.1914 

477.9573 

514.8233 

232 

258.4757 

295.4008 

332.3259 

369.2510 

406.1761 

443.1012 

480.0263 

516.9514 

233 

259.5898 

296.6740 

333.7583 

370.8415 

407.9258 

445.0110 

482.0953 

519.1795 

234 

260.7035 

297.9468 

335.1902 

372.4335 

409.6769 

446 9202 

4841636 

521.4069 

235 

261.8175 

299.2200 

336.6225 

374.0250 

411.4275 

448.8300 

486.2325 

523.6350 

236 

262.9316 

300.4932 

338.0549 

375.6165 

413.1782 

450.7398 

488.3015 

525.8631 

237 

264.0456 

301.7664 

339.4872 

377.2080 

414.9288; 

452.6496 

490.3704 

528.0912 

238 

265.1597 

303.0396 

340.9196] 

378.7995 

416.6795 

454.5594 

492.4394 

530.2193 

239 

266.2737 

304.1128 

342.1519 

380.2910 

418.3301 

456.4692 

494.5083 

532.5474 

240 

267.3878 

305.5860 

343.7843 

381.9825 

420.1808 

458.3790 

496.5773 

534.7755 

241 

268.5018 

306.8592 

345.2166 

383.5740 

421.9314 

460.2888 

498.6462 

537.0036 

242 

269.6159 

308.1324 

346.6490 

385.1655 

423.6821 

462.1986 

500.7152 

539.2317 

243 

270.7299 

309.4056 

348.0813 

386.7570 

425.4327 

464.1084 

502.7841 

531.4598 

244 

271.8440, 

310.6788 

349.5137 

388.3485 

427.1834: 

466.0182 

504 8531 

533.6879 

245 

272.95801 

311.9520 

350.9460j 389.9400 

428.9340 j 467.9280 

506.9220 

535.9160 


25 











































386 


Pi V A WTCJ. 


DYNAMICS. 


ALGEBRAICAL AND GEOMETRICAL EXPRESSIONS OF THE 
FUNDAMENTAL PRINCIPLES OF DYNAMICS. 


Elements. 
Force = F. 

Space = S. 


Functions. 
Power P=F V. 

Velocity ~ • 


Time= T. 
Mass = M. 


F:M=V: T. 


Work K=FV T. 
Work K=%M V 2 . 

F:M=i V 2 : S. 


Momentum. 

F T - M V. 


Work. 


FS = 


MV 2 

_ ,• 

2 


These are the fundamental principles in Mechanics. 


♦ 


Dynamics is that branch of mechanics which t reats of forces in motion, 
producing power and work. It comprehends the action of all kinds of ma¬ 
chinery, manual and animal labor in the transformation of physical work. 

Quantity is that which can be increased or diminished by addition or 
subtraction of homogeneous parts, and which can be expressed by a num¬ 
ber. 

Element is that which cannot be resolved into two or more different 
things. 

A Function is composed of two or more different elements. 

A function is resolved by dividing it into one or more of its elements. 

Force, Space, and Time are simple physical elements. 

Power, Velocity, and Work are functions of those elements. 


Force (F). 

Force is any action which can be expressed simply by weight, w-ithout 
regard to motion, time, power, or work ; it is an ultimate thing which can¬ 
not be resolved into two or more things, and is therefore a simple physical 
element, corresponding with length in geometry. 

Force is expressed by a great variety of terms, such as attraction, repulsion, 
gravity, pressure, tension, compression, cohesion, adhesion, resistance, inertia, strain, 
stress, strength, thrust, burden, load, squeeze, pull, push, pinch, punch , etc., the mag- 
nit ude of which can be expressed by any established unit of weight. 

Motive force is t hat which produces motion, but otherwise it is the same 
as static force, and is denoted by the letter F. Force is the first element in 
mechanics. 

Motion. 

Motion is a continuous change of position in regard to assumed fixed 
objects. Motion or rest is only relative; that is to say, when two bodies 
change their relative position, either one of them can be considered at rest 
and the other in motion. Therfe is no absolute rest known in the universe. 

Motion is expressed by the following terms: move, going, walking, passing, 
transit, involution, evolution, run, locomotion, flux, rolling, flow, sweep, wander, shi/t, 
flight, current, etc. 

Velocity (F). 

Velocity is speed or rate of motion; it is the space passed over in the 
unit of time in uniform motion, or in variable motion is the space which 
would be passed over in a unit of time if the velocity were rendered con¬ 
stant at any instant. 










I'm* v.:cs 


387 


Velocity 


o ( 
T 


Velocity or rate of motion is expressed by many terms. 


Quick Motion. 

Speed, swiftness, rapidity, fleetness, 
speediness, quickness, haste,'hurry, race, 
forced march, gallop, trot, run, rush, 
scud, dash, spring, etc. 


Slow Motion. 

Slowness, tardiness, dilatoriness, slack¬ 
ness, drawl, retardation, hobbling, creep¬ 
ing, lounging, linger, sluggish, crawl, 
loiter, glide, languid, drowsy, etc. 


Angular velocity is the curvilinear velocity of a point at a unit’s dis¬ 
tance lrom the axis around which a body turns this point, turning with the 
body. 

Time ( T ). 

Time implies a continuous perception, recognized as duration. 

Chronology is the science of time. 

Instant and moment are points of time. 

Fpocli is the beginning of any time marked with some remarkable events 
and recorded by historians or ciironologists. Era is nearly the same as epocii, 
except that it is generally fixed by nations or denominations, as the Chris¬ 
tian era. 

Time is expressed by a great variety of units—namely, millennium, a thou¬ 
sand years; century , one hundred years; score, twenty years; year, season, 
month, fortnight, week, day, hour, minute, and second. 

Time is an ultimate tiling which cannot be resolved into two or more dif¬ 
ferent things, and is therefore a simple physical element. 

Power (P= F W). . 

Power is the product of force and velocity, and is therefore a function. 

A force multiplied by the velocity with which it is acting is the power in 
operation. 

The English unit for measuring power is a force of one pound acting with 
a velocity of one foot per second, called foot-pound. 

Man-power is a unit of power established by Morin to be equivalent to 
50 foot-pounds of power, or 50 effects ; that is to say, a man turning a crank 
with a force of 50 pounds and with a velocity of one foot per second is a 
standard man-power, or a force of 25 pounds by two feet per second is a 
man-power. 

An ordinary workingman can exert this power eight hours per day with¬ 
out overstraining himsell. 

Horse-power is a unit of power established by James Watt to be equiv¬ 
alent to a force of 33,000 pounds acting with a velocity Of one foot per minute, 
which is the same as a force of 550 pounds acting with a velocity of one foot 
per second. 

That is to say, one horse-power is 550 foot-pounds per second, or 11 man¬ 
power. 

The product of any force in pounds and its velocity in feet per second di¬ 
vided by 550 gives the horse-power. 

Power is the differential of work or any action which produces work, 
whether mental or physical. 

Power multiplied by the time of action is work; work divided by time is 
power. Writers on dynamics have heretofore assumed “power is the work done 
in a unit of time." 

The number which expresses the work done in a unit of time is equal to 
the number which expresses the power in operation. 

When we say “in a certain time,” which is equivalent to the expression 
“per unit of time,” we divide by the time. 

Work is the product of the two elements Force and Space. When we 
divide work by the time of its operation, the result is power, which is the 
product of force and velocity. 

Power may be expressed by the following terms: 

Traction, propulsion, impulsion, capability, puissance, labor, haul, drug, draw, 
heave, occupation, activity, vigor, energy, etc., or any action which implies force 
and motion with regard to time. 











388 


DYH titled. 


Space (S=VT). 

Space in dynamics means linear space; it is an element. 

Space is herein denoted by 

VT ; 

which means that the space S ,, expressed in linear feet, is the product obtainei 
by multiplying together the velocity V and time T. 

Mr. Nystrom says velocity is an element and space is a function of velocit; 
and time. 

There does not appear to he any difficulty in conceiving space to be an ele 
ment apart from time or velocity.—W. D. M. 

Geometrical spaces are magnitudes of three different kinds—namely, lineal 
superficial, and voluminous. 

Linear space is that generated by the motion of a point. 

Superficial space is that generated by the lateral motion of a line. 

Voluminous space is that generated by the lateral motion of a plane. 

Space in dynamics means the generation of that, space by velocity and time. 
A line of any kind cannot be drawn without velocity and time. 

A locomotive running with a uniform velocity of 30 miles per hour will 
make 2610 feet per minute or 44 feet per second; and if we diminish the 
spaces and time to infinitely small values, or, say, absolutely nothing, the 
velocity is still constant when passing that time and space reduced to a 
point. 

Work (JT=* FS). 

Work is the product obtained by multiplying together the elements force 
F and space S. 

Work may also be expressed by K=PT, or the product of power and time. 

The work of a steam-engine operating with a constant power will bedirectly 
as the time of operation, and so with all labor, whether it be mechanical or 
manual. 

Moment of a Force (FI). The moment of a force is its lever arm at 
right angles to its direction of action multiplied by its intensity in pounds 
or tons. 

Momentum (J/ V). 

The momentum of a moving body is the intensity of that constant force 
which, resisting its movement, will bring it to rest in one second. 

jir_ weight 
32.2 * 

V =* velocity in feet per second. 

Moment of Inertift (M Vr). 

The moment of inertia of a rotating body is the moment of its momen¬ 
tum, and is equal to its momentum M V multiplied by its radius of oscil¬ 
lation r. 

It is the universal custom to consider the angular velocity as unity, and we 
tints obtain a mathematical expression which is the comparative measure of 
the moment of inertia only. 

Nevertheless, the true moment of inertia is a real thing, and can be ex¬ 
pressed as the statical moment of a constant force acting for one second. 

Many able writers on mechanics do not seem to have a clear physical con¬ 
ception of its true meaning.—W. D. M. 

Virtual Velocities. 

By the theorem of virtual velocities is meant the instantaneous equality 
of elementary quantities of power transmitted. 

The reason for its name does not appear clearly. It is used in older works 
on mechanics and higher forms of analytical mechanics. 












Dynamics. 389 


! 


Radius of Oscillation. 

The radius of oscillation is the mean lever-arm of the momentum of a re¬ 
volving body. It is equal to the moment of inertia divided by the momentum 
of the revolving body. 


Radius of Gyration. 

The square of the radius of gyration of an oscillating body is equal to the 
product of the radius of oscillation and of the'distance of the centre of 
gravity of the suspended body from its point of suspension. 

The intensity of the force of momentum is proportional to the distance of 
the centre of gravity from the axis of suspension, and the mean leverage 
of the momentum is the radius of oscillation. The square of the “radius 
of gyration,” then, is a convenient product of these two quantities, as in¬ 
cluding both, and therefore giving them in a convenient mathematical 
form. If a straight rod be balanced at its middie, we are obliged to consider 
each half separately and add them together. 

While we can locate both centres of gravity and of oscillation, we cannot 
locate a centre of gyration, nor has it an actual physical existence, being a 
product of two quantities only. 

This quantity has proved a constant stumbling-block to students of 
mechanism, but a little reflection and the solution of a few examples will 
make its nature clear.—W. D. M. 


Units of Work.—Foot-Pouml. 

The English unit of work is assumed to be that accomplished by a force 
of one pound raising an equal weight one foot, high, which unit is called a 
foot-pound. Then a force of 6 pounds working through a space of 4 feet 
is equivalent to 24 foot-pounds of work. 

This unit is very convenient for small amounts of work, but it is too 
small for many purposes in practice. 

Foot-Ton. 

English ordnance officers have adopted a larger unit for work—namely, 
foot-ton, which is used for expressing work of heavy ordnance. It means 
the work of lifting one ton one foot high. 

Workman day. 

A laborer working eight hours per day can exert a power of 50 foot-pounds 
per second. A day’s work will then be 50 X 8 X 60 X 60 =1,440,000 foot-pounds 
of work, which may be termed a workmanday. 

All kinds of heavy work can be estimated in workmandays, such as the 
building of a house, a bridge, a steamboat, canal and railroad excavations 
and embankments, loading or unloading a ship, powder and steam-boiler ex¬ 
plosions, the capability of heavy ordnance, etc. 

The magnitude of the unit workmanday is easily conceived, because it is 
that amount of work which a laborer can accomplish in one day. Work 
expressed in foot-pounds, divided by 1,440,000, gives the work in workman- 
days. 

A work of 20 workmandays can be accomplished by 20 men in 1 day, by 
one man in 20 days, by 4 men in 5 days, or by 10 men in 2 days. 

Work clone is expressed by the following terms: 

Hauled, dragged, raised, heaved, cultivated, tilted, broken , crushed, thrown, 
wrought, fermented, labored, embroidered, etc., or any expression which implies 
the three simple elements force, velocity, and time. 

Power is the differential of work. 

Work is the integral of power. 








390 


I 


DYNAMICAL. FORMULAS. 


Force or Pressure in Pounds. 


F— 

P 

V ’ * 

• • • • *• 

• 

1^1=0 

II 

« • 

• • 

3. 

F= 

550 IP 

V ’ 

• • • • 2. 

Fam JL 

r y X* * 

• • 

• • 

4. 


Velocity in Feet per Second. For Uniform Motion. 


V r= 

5 

T‘ * 


550 IP 

V ~ F * 

• • 

• • 

7. 

v= 

P 

F' * 

• • • • 6» 

V-K 

V FT' * 

• • 

• • 

8. 


Time 

of Action in Seconds. For Uniform Motion. 


T = 

S 

V 


m FS 

550 IP * 

• • 

• • 

11. 

T= 

FS 

P ' * 

. . . . 10. 

r - K 

• • 

• • 

12. 



Power in Effects. 

Work per Second. 



P = 

FV . 

. . . . 13. 

P = 550 IP. 

• • 

• • 

15. 

P = 

FS 

rp * * 

. . . . 14. 

P-K 

1 r * 

• • 

• • 

1G. 


Space Passed Through in the Time T. 


S=VT. . 

• 

. 17. 

uo\j l 

o — # * • • 

F 

. 19. 

PT 

S = ~. . . . 

• 

. 18. 

• 

• 

• 

• 

II 

. 20. 



Horse- 

Power. 


P 



FS 




. 21. 

II — . . . . 

. 23. 

550 * 



5o0 T 


FV 



K 


IP = ~r. • 


. 22. 

TP 

. 24. 

550 



550 T 



Work in Foot-Pounds. 


K=FVT in time T. 

• 

. 25. 

K — FS. .... 

. 27. 

K=P T in time T. 

• 

. 26. 

K — 550 IP T in time T. 

CO 

<M 

• 


It will be observed in the preceding formulas that an element is never 
divided by an element, but a function is divided by an element only when 
that function contains that element. 

Power divided by velocity gives force, because power contains the elements 
force and velocity; but power cannot be divided by time, because time is 
not a constituent element of power. 

Work can be divided by either one or two of its three constituent factors. , 
When work is divided by either two of its elements, the product will be the ; 
third element. 

Different, elements or functions cannot be added to or subtracted from one 
another. Power or space cannot be added to or subtracted from work. Force, 
velocity, or time cannot be added to or subtracted from space. 

























I>\ s y. rrs. 


S91 




When a formula contains several terms, all the terms must be of the same 
kind; for instance: 

Work AT= T {f V+ P— * ). 

The terms within the parentheses are all power, which multiplied by time 
gives work. 

Mistakes in dynamical formulas are easily detected by the above rules. 

No element can be converted into an element of a different kind. 

Different. Kinds of Foot-Poumls. 

There are two different kinds of foot-pounds in mechanics—namely, 

1st. A foot-pound of static moment, which is force in pounds multiplied by 
by its lever of action in feet. 

2d. A foot-pound of work is force in pounds multiplied by space in feet. 

It will be observed that foot-pounds of static moment and foot-pounds of 
work are both the product of force and linear space, from which it would 
appear that these two functions are substantially alike; but they are of 
entirely different nature. 

Static moment is force multiplied by the geometrical element length, with¬ 
out regard to velocity and time; in which case the force has nothing to do 
with the generation of that length. 

Work is force multipied by space. 


EXAMPLES CORRESPONDING WITH THE FORMULAS. 


Force or Pressure in Pounds. 

Example 1. A power P=6400 effects is operating with a velocity of V= 12 
feet per second. Required the force F? 


F- 


6400 

12 


= 533 pounds. 


Example 2. The piston of a steam-engine of IP — 24 horses is moving at 
the rate of F=8 feet per second. Required the force F? 


F = 


550 IP 550 X 24 
V~ ~ 8 


1650 pounds. 


Example 3. A work of K = 3266 foot-pounds is accomplished in a space 
S = 16 feet. Required the force F'l 


K 

S 


3266 

" 16 


= 204 pounds. 


Example 4. A work of K= 183600 foot-pounds was accomplished with a 
velocity V— 18 feet per second in a time of 3 minutes, or T— 3 X 60 = 180 
seconds. Required the force F? 

„ K 183600 „ e . 

F= ~ — = vt;—~ rr = 56.6 pounds. 

VT 18X180 1 


















392 


I \’A.tfICS. 


Velocity in Jb’eet per Second. 

Example 5. A body moves through a space of S = 160 feet in a time o 
T — 40 seconds, Required the velocity V? 

V — = 4 feet per second. 

Example 6. A power of P = 4266 effects is operating with a force F= 76 
pounds. Required the velocity VI 

P 4266 

V = —=- = „ = 5.6 feet per second. 

A /bO 

Example 7. The cylinder of a steam-engine of IP — 160 horse-power is 2 4 
inches in diameter, and tlie effective steam-pressure is 30 pounds to tin 
square inch. Required the velocity of the steam-piston? 

The area of the piston is 452.39 square inches, which multiplied by 3C 
pounds to the square inch will be a force of 

F = 13570.8 pounds. 

Tr 550 IP 550 X 160 _ _ , . , 

F 13o/0.8 

Example 8. A work of E= 864360 foot-pounds is accomplished with a force 
of F— 68 pounds in a time of 5 minutes. Required the velocity K? 

The time T = 5 X 60 = 300 seconds. 

' 42 .4 feet per second. 


V = 


FT 


68-X 300 


Time of Action In Seconds. 


Example 9. A space of S = 2896 feet is generated with a velocity of F= 25 
feet per second. Required the time T1 

„ S 2896 

T = —= llo.84 seconds. 

V Zt) 

A force of ^=4596 pounds is working through a space 
What time is required for the force to generate a power of 


Example 10. 

S = 960 feet. 

P = 840680 effects ? 


T = 


FS 

P 


4596 X 960 


5.25 seconds. 


S40680 

Example 11a. The stroke of a steam-piston is four feet, and the effective 
pressure of steam is F — 46360 pounds. The power of the engine is 
IP = 500 horse-power. What time is required of the engine to make 64 
double strokes? 

The space <5=4X2X64 = 512 feet. 

_ FS 46360 X 512 

550 IP 550 X 500 — 86 seeonds ' 

Example 116. What time is required to raise a weight of 200 tons to a 
height of S= 50 feet with an engine of IP = 8 horse-power? 

200 X 2240 = 448000 pounds. 

T = - = 509 seconds 

550 IP 550 X 8 seconds, 

or 8 minutes and 29 seconds. 

Example 12. What time is required to accomplish a work of K = 96236000 
• foot-pounds, with a force F= 88 pounds, moving with a velocity of F= 1.5 
feet per second? 

_ 96236000 

T = — -——j-r- = /29066 seconds, 

oo X 

or 202 hours 31 minutes and 6 seconds. 

Assuming a workmanday to be 1,440,000 foot-pounds, it would require 
about 67 such units to accomplish the work; that is to say, one man could 
do the work in 67 days, or 67 men could accomplish it in one day. 






















Dyn \ m res 


393 


Power in Effects or Foot-Pounds. 

Example. 13. A weight of five tons is raised vertically at the rate of 
inches per second. Required the power P? 

The force F= 5X 2240 = 11200 pounds. 

Velocity F= 0.125 feet per second. 

P = 11200 X 0.125 = 1400 ft.-lbs. per second. 

One man-power is 50 effects, and it would require 1400:50 = 28 men to 
raise five tons with a velocity of 11 inches per second at continued work. 

One horse-power is 550 effects, and it would require 1400 : 550 = 2.55 horse¬ 
power for the same work. 

Example, 14. What power is required to lift a weight of three tons a space 
of S = 5 feet in a time of 10 minutes? 

P -= —jr = —— = 56 ^.-lbs. per second. 

Example 15. How many effects are there in 2P= 30 horse-power? 

P = 550 X 50 = 16500 effects, or ft.-lbs. per second. 

Example 16. What power is required to do a work of K= 186000 foot¬ 
pounds in one minute? T= 60. 

186000 


P = 


60 


= 31000 effects. 


Space Passed Through in the Time T. 

Example 17. A body moving with a velocity of F=960 feet per second for 
a time of T= 5 seconds. Required the space passed through ? 

X = F T = 4800 feet. 

Example 18. A power of P= 6500 effects is operating for a time of T = 12 
seconds with a force F— 240 pounds. Required the space passed through? 

PT 6500 X 12 


S = 


325 feet. 


F 240 

Example 19. To what height can a steam-engine of IP - 
a weight of 25 tons in a time of 5 minutes ? 

F— 25 X 2240 = 56000 pounds. 

T= 5 X 60 = 300 seconds. 

0 550 X IP T 550 X 8X300 

The height S— - 


■■ 6 horse-power lift 


23.6 feet. 


F 56000 

Example 20. A work of K— 7280 foot-pounds is to be accomplished by a 
force of F= 24 pounds. In what space can the force do the work ? 

S== fr = ^r =304feet - 

_r Z*± 


Example 21. 


Horse-Power. 

How many horse-power are there in P— 56680 effects? 
P 56680 


IP = = 


103 horse-power. 


550 . 550 

Example 22. A weight of three tons is to be raised with a velocity of 
F= 6 feet per second. RequiredMhe horse-power ? 


FV 3 X 2240 X 6 


IP = —- = 


= 73.3 horse-power. 


550 550 

Example 23. A steam-crane is to be constructed to lift 30 tons 12 feet high 
in 5 minutes. Required the horse-power? 

Force P= 30 X 2240 = 67200 pounds. 

Time T= 5 X 60 = 300 seconds. 

FS 67200 X 12 


IP = 


550 T 550 X 300 


= 5 horse-power, nearly. 




























394 


Hvxvmics. 


Example 24. What horse-power is required to accomplish a work of 
/l= 3460U0 foot-pounds in T=o seconds? 


IP = 


346000 


550 T 550 X 5 


12.G horse-power. 


WORK IN FOOT-POUNDS. 

Example 25. How much work is accomplished with a force of J F=2S0 
pounds, moving with a velocity of V=9 feet per second for a time of 
T =\200 seconds, or 20 minutes? 

K= F V T= 280 X 9 X 1200 = 3024000 foot-pounds. 

Example 26. How much work can be accomplished by a power of P=36 
effects during T= 4 seconds? 

A’'=PI'=36 X4 = 144 foot-pounds. 

Example 27. A weight of 25 tons is lifted S= 18 feet. Required the work ? 

K= FS = 25 X 2240 X 18 = 1008000 foot-pounds. 

Example 28. How much work is accomplished per minute by an engine 
of TP = 48 horse-power? 

K = 550 IP T= 550 X 48 X 60 = 1584000 foot-pounds. 

Dynamics of Circular or Rotary Motion. 

In circular motion it is supposed that the motive force is applied in the 
direction of the tangent to the circle of radius R in feet, like that of a belt 
or rope over a pulley or in all kinds of gearing. 

n = revolutions of the circle per minute. 

N— total number of revolutions in the time T, or for generating a definite 
circular space S, and also for accomplishing a definite work K. 

Example 33. The radius of a wheel or crank-pin is 11 = 2.5 feet, and makes 
» = 56 revolutions per minute. Required the velocity in the circumference? 




2 7T R n 
60 


= 0.10472 Rn = 0.10472 X 2.5 X 56 = 20.6 feet per second. 


Revolutions n = 


9.55 V 9.55 X 60 


R 


2.25 


254*- per minute. 


Radius R = 


9.55 V 9.55 X 50 


n 


150 


= 3.183 feet. 


Example 35. A pulley of 54 inches diameter, or R = 2.25 feet, is to run a 
belt V= 60 feet per second. Required the revolutions per minute ? 


Example 41. A pulley is to make n = 150 revolutions per minute with a 
velocity of the belt F=50 feet per second. Required the radius of the 
pulley ? 


Example 34. Find the velocity of the circumference of a pulley 27 inches 
diameter, making n = 250 revolutions per minute? 27 inches = 2.25 feet. 
R = 1.125. ♦ 

Velocity V— 0.1047 X 1.125 X 250 = 29.45 feet per second. 

Example 36. A pulley of 72 = 1.5 feet radius is to transmit ZP = 4.8 horse¬ 
power with a motive force F=G4 pounds. Required the number of revo¬ 
lutions per minute? 

5250 IP _ 5250 X 4.8 

= FR ““ “64 X 1.5 "" 


n 


262.5 per minute.. 















Dynamics, 


395 


Force F Acting in tlie Direction of tlie Tangent. 


60P 

. 29. 


„ 9.55 IC 

. 31. 

2 n Rn' ' 


Rn T ‘ ' 

9.55 P 

^ — " n • • • 

P 71 

. . 30. 


„ 5252 IP 

If - n • • • 

P 71 

. 32. 

Circumferential Velocity and 

Revolutions per 

Minute. 

2 n R n 
~ 60 * ’ 

• • 33. 


9.55 V 

71= p . . . 

. 35. 

V= 0.10472 P «. . 

. . 34 . 


5252 IP 

71 — n T , • • • 

FR 

. 36. 


Time of Operation in Seconds. 


T= 

9.55 s 

Rn ’ * 

• • • 37. 

ig 

II 

• 

• 

. 39. 

T — 

9.55 K 

. . . 38. 

FRN 



. 40. 


1< Rn' ' 

87.5 IP ' ' 

• 

• 



Radius of Revolution. 




p= 

9.55 V 

• • 

n 

. . . 41. 

5252 IP 

1\> - ZZ ' • • 

Fn 

• 

• 

. 43. 

7 ?_ 

9.55 P 

. . . 42. 

„ 9.55 K 



. 44. 

j. — 

Fn ' ’ 

Fn T ' 

• 

• 


Power Generated in Effects, or Foot-pounds per Second. 


P = 
P = 

S= 


2 n R n F 


60 
F Rn 
9.55 


2 7 x Rn T 
60 


Rn T 

^-- 9 . 55 “* ‘ 


. 45. 
. 46. 


P = 
N-- 


FRN 
9.55 T ' 
9.55 P T 
FR ’ 


Space Generated in Feet. 

Fn N 


. 49. 
. 50. 




755.625 ZP 
S = N2nR. . 


Horse-Power Generated. 


47. 

48. 

51. 

52. 


IP = 

FR n 

5252~ * * 

• 

• 

. 53. 

„ 87.5 IP T 

N = FR * 

• 

• 

. 55. 

IP — 

FRN 



. 54. 

N =-———. . 



. 56. 

87.5 T ' ’ 

• 

• 

2 irP 




Work; Accomplished in Foot-pounds in Time T. 


K=* 


FRn T 


9.55 

K= F2n R N. 


. 57. 
. 58. 


A= 
P = 


K 


F2n R' ’ 
K 


F2nN’ ' 


59. 

60. 



























396 


OBSERVE E ’ I ' OF ’OWER. 


OBSERVED RESULTS OF POWER. 


Description of Works. 

A man can raise a weight by a single fixed 

pulley,. 

A man working a crank, .... 

A man on a tread-wheel (horizontal), 

A man in a tread-wheel (axis 24° from ver- 
1 ical), ........ 

A man draws or pushes in a horizontal 

direction,. 

A man pulls up or down, .... 

A man can hear on his back, 

A horse in a horse-mill, walking moderately, 
A horse in a horse-mill, running fast, 

An ox in a horse-mill, walking moderately, 
A mule “ 

An ass “ 


<( 


On had Foot-roads, like those in 
Pern. 

A man can bear, ...... 

Llama of Peru can bear, .... 

Donkey can bear,. 

Mule can bear,. 


Work- 
lirs. per 

Force. 

Teloo’y 

Effects, 
or ft.- 

Horses 

day. 

F 

V 

lbs. per 
sec. P. 

H 

6 

50 

0.8 

40 

0.072 

8 

20 

2.5 

50 

0.090 

8 

144 

0.5 

72 

0.130 

8 

30 

2.3 

69 

0.125 

8 

30 

2 

60 

0.109 

8 

12 

3.7 

44.4 

0.080 

7 

95 

2.5 

237.5 

• • • 

8 

106 

3 

318 

0.677 

5 

72 

9 

648 

1.178 

8 

154 

2 

308 

0.558 

8 

71 

3 

213 

0.30S 

8 

33 

2.65 

87.4 

0.160 

10 

50 

3.5 

175 


10 

100 

3.5 

350 


10 

200 

3.5 

700 


10 

400 

5 

2000 



Flour Mills, 

For every 100 pounds of fine flour ground per hour, require, . . 

One pair of mill-stones of 4 feet diameter, making 120 revolutions 
per minute, can grind 5 bushels of wheat to fine flour per hour, 
One pair of mill-stones of 4 feet diameter, making 120 revolutions 
per minute, can grind 5 bushels of rye to coarse flour per hour, . 

Saw Mills, reciprocating. 

For every 20 square feet sawed per hour, in dry oak, there re¬ 
quires, .. 

Dry pine, 30 square feet per hour,. 

Circular Saw. 

A saw 2.5 feet in diameter, and making 270 revolutions per 
minute, will saw 40 square feet in oak per hour, with . . . 

In dry spruce, 70 square feet per hour,. 

Threshing Machine. 

Velocity of the feed-rollers at the circumference, 0.55 feet per 
second. Diameter of threshing-cylinder 3.5 feet and 4^ feet long, 
making 300 revolutions per minute, can thresh from 30 to 40 
bushels of oats, and from 25 to 35 bushels of wheat, per hour, 

One man with a flail can thresh half a bushel per hour (wheat), 

Rolling Mills. 

Far iron-mills. Two pair of rough rollers, two pair of finishing 
rollers, six puddle furnaces, two welding furnaces, making 10 tons 
of bar iron per 24 hours, rollers making 70 revolutions per minute, 

rc(]uirGj • • • • • • • • • # • 

Plate-mill requires about five IP per square foot of plates rolled. 
Largest size plate rollers should not make over 30 revolutions per 
minute. 


550 

2400 

1600 


650 

550 


550 

550 


2200 

70 


29000 


1.000 

4.36 

2.91 


1.000 

1.000 


1.000 

1.000 


4.000 

0.127 


52.7 






















Dredging-* h t vs. 


397 


DREDGING-MACHINES. 


Ladder-Dredge.—' The ladder-dredge consists of an endless chain upon 
Which a number of buckets are fixed and work continually like a Noria. 
This appears to be the best form of dredge for deepening harbors, but is not 
so well suited for docks, where the dipper and grapple dredges are the best. 


Formulas. 


Notation. 

T — tons of materials excavated and 
raised per hour. 

h = height in feet to which the ma¬ 
terials are raised above t he bot¬ 
tom of the excavated channel. 
k = 0.1 for hard clay with gravel. 
k = 0.07 for hard pure clay. 
k — 0.05 for common clay or sand. 
k — 0.04 for soft clay or loose sand. 
k — 0.03 for very loose materials. 

IP — horse-power required for exca¬ 
vating and raising the mate¬ 
rials. 

F— force in pounds required to feed 
the dredge ahead. 

v =s= velocity of the buckets in feet 
per second. 

Example. 1. What power is required to excavate 2*= 160 tons of hard pure 
clay per hour, and raise it up h = 25 feet above the bottom of the channel? 
For hard clay k = 0.07. 


H> = 

*(&+*) 

• • • 1# 

T — 

700 IP 

. . . 2. 


h + 700 k ’ 

F — 

550 IP 

• • 

V 

• • • 3. 

F = 

550 Tk 

• • 

V 

. . 4 . 

k = 

IP h 

T + 700 ‘ 

• • • 


£P= 160 + 0.07) = 

\ 700 / 


16.9, or 17 horses. 


Example 2. What force F=? is required to feed the dredge ahead for the 
above example when the buckets move v = l foot per second? 

„ 550 X 16.9 QOO _ , 

F = ---= 929o pounds. 

Dipper-Dredge, consisting of one scoop, worked with a triple chain 
wound on a 15-inch drum, and driven by a pair of engines 10 inches in 
diameter by 15 inches stroke of cylinders. Under ordinary work the scoop 
makes 30 to 40 dips per hour, and takes up about two cubic yards, or three 
tons, of materials each dip. 

The dipper-dredge is used in harbors and docks, and also iu railroad exca¬ 
vations. 

Grapple-Dredge, consisting of a double scoop opening in the bottom 
like a mouth, takes up about five tons of materials each grapple, it is 
■worked by a single chain wound on a drum three feet in diameter, with a 
pair of engines 14 inches diameter by 20 inches stroke of cylinders. Under 
ordinary work it makes 50 to 60 grapples per hour. 


BELTING. 

Preliminary Note by Reviser. 

The English rule for belting is, A single-thickness leather belt, one inch 
wide, running at a speed of 1000 feet per minute, will transmit about one 
horse-power. Double-thickness belts will do twice as much. However 
tightlv drawn, belts slack themselves in running, and it is not safe to over¬ 
load them or there will be trouble shortly from slipping. At high speeds 
centrifugal force greatly diminishes the adhesion of belts to pulleys. Belts 
should never be run on cast-iron pulleys at a higher speed than 8000 teet per 
minute. Formulae 13 and 26, page 399, give rational formulae for belts, taken 
from Rankine, 
















393 


Belt? n .ys. 




BELTING. 


Fig. 1, Plate II., represents a pulley hung from the points a and b by 
belt T,t, forming an angle of contact 2z on the pulley. The weight IP is b 
on the pulley for stretching the belt from a and b, in which case the tens 
TandMvillbe alike. The letters on the illustrations correspond with 
letters in the formulas, and the number of each example corresponds v 
the number of the formula used. 


:he 

ng 

fits 


sin. 45° 0.70711 


That is, 118.7928 :2 = 59.3964 pounds, the tension at each point of suspens¬ 
es and b. 

Example 2. It is found by experiment that, the tension at a is 41.36 poun 
that is T -f t = 41.36 X 2 = 82.72 pounds, half the angle of contact being 2. 
4S° 36'. Required the weight W. 

Weight W= 82.72 X sin. 48° 36' = 82.72 X 0.75011 = 61.05 pounds. 
Example 4. The suspended weight W= 86 pounds and the angle of cont: 
120°, making Z— 60°. Required the force of contact of the belt on the pulh / 

„ 86 X 3.1416X 60 _ 

C= ■ • „„„ . / -———= 104 pounds. 


180 X sin.60° 

Example 9. Fig. 2. What motive force F can be transmitted by a leath 
belt of tension T -{■ t = 450 pounds when at rest, half the angle of contact Z 
78° on a smooth cast-iron pulley of friction/= 0.35? 


„ „ 0.35 X3.1416X 78X450 _ 

Motive force F— --—- ~ -= 214.4 pounds. 


180 




\V — weight in pounds hung on the belt, 
rand t = respective tensions of the belt in pounds. 

^■=half the angle of contact of the belt on the pulley. 

C = force of contact of the belt on the pulley in pounds. 

F= motive force in pounds transmitted by the belt. 

/= friction coefficient of the surfaces in contact. 

F= ( T — t). 2'={F+t). I = (T — F). T+t=F + 2l = 2T-F. 

Fig. 2, Plate II., represents two pulleys of different sizes and connec 
by a belt T, l, in which case the smallest pulley will be in the same condit mi 
as that in Fig. 1, but the weight II' is the pressure in the journal boxes, i- 
the system is arranged for transmitting power. The greatest motive ft 
F that can be transmitted by the belt cannot exceed the product of the fo- 
of contact (?and the friction / without slipping of the belt. 

When Fz=-fC, the belt will slide. 

When F-^fC, there is no sliding. 

In good practice the motive force .F should not exceed 75 per cent, of fl 

Example 1. Fig. 1. The weight W— 84 pounds, and half the angle of c 
tact Z=Ab°. Required the sum of tensions. 

84 84 

Tension T-\-1 = <gC ; = ~ _ A _,; = 118.7928 pounds. 


This is the maximum motive force that can be applied under tbe given eo 
ditibns, but only 75 per cent, of it should be applied in practice, or 214.4 
0.75 = 160.8 pounds. 

When at rest, ( T + t) = 450 pounds, or T—t — 225 pounds, but when in m 
tion half the motive force is added to T= 250 + 107.2 = 357.2 pounds, tl 
pulling tension, and the other half of the motive force is subtracted from l 
225 —107.2= 117.8 pounds, the slack tension. This rule is, however, infli 
enced by the grade of elasticity of the belt. 

Example 10. How much tension must be given to a belt when at rest, i 
order to transmit when in motion a motive force ^’ = 500 pounds, when tl 
angle of contact is 165°, or^= 82° 30', and the friction coeliicient/= 0.4? 

m 180X 500 

T + t = pounds. 


0.4X3.1416X82.5 




The teusion of the belt should be 868 : 2 = 434 pounds, to which must b 
added |d for practical working, making the required tension 579 pounds. 
The friction coefficient is found by formulas 13 and 26. 

The formulas will answer equally well for any system of weights an< 
measures. 





















Transmission or r .h-fr -y Belting. 


399 


Formulas for Oblique Belting, 

Figs. 1 ami 2. 


(P+0 : W= 1: sin. Z. 

W 


Sum of tensions (P-f t) 


. . 1 . 


sin. Z 

Weight TF=(r-K)sin. Z..2, 

TP 


Half-angl. cont. sin. Z= 


*Force of contact C= — 


(T+t) 
WwZ 


. • • 3. 

4. 


180° sin. Z ' 

*Force of contact C— . . . 5. 

180 


Sum of tensions (T-\-1)- 


180 c 


jt.Z’ ' 


. . 6 . 


Half-ang. cont. Z= 


1 80 °C 
(T+t)’ 


Formulas for Parallel Belting 
on Pulleys of Equal Diam¬ 
eters. 


Sura of tensions (T+t) — W 


14 


Pressure in journals TF= (T+t). . 15 

IF 


Half-angl. cont. sin. Z=* 
*Force of con tact C 


(T + t ) 
Wn 


2 


1 1G. 

. 17. 


*Force of contact C= ... 18. 


Sum of tensions (T +1) 


1C 


. 19. 


7 

• • • • • 


„ r . , , „ r 180°C'siu.Z „ 

Weight suspended W= -—-. 8. 

7 TZj 

Motive force F .9- 


180° F 


10 . 


Sum of tensions (T + t) — . 

jttZ 

Greatest tension T<= t (• H. 

Slack tension (= I (jfj£=£*). • 12- 
(Weisbach’s formula.) 

550 IP 

Motive force F— t (e-f *— 1) = — — —.13. 
e = 2.7183 V— speed of belt in ft.persec. 


Half-angle cont. Z= 90°..20. 


1C 

Pressure in journals W — —. ... 21. 

7T 

Motive force .22. 

A 


Sum of tensions (T + t ) = 


1F_ 

/» 


23. 


Formulas for Oblique Belting 

(Weisbach’s formulae). 

Greatest tension T = te?f *.24. 

F 

Slack tension t — -— ...25. 

e 2 /* — 1 

Motive force F g = 900 — i (e n f — 1) 
.550 IP 

.... 26 


550 IP 

_ F~' 


* Factor of adhesion of belt to pulley. 

Friction Coefficient for Different Surfaces in Contact. 


Surface 

or 

pulley. 

C< 

Hair side 
ou pulley. 

>ndition of 

Flesh 

side 

on pulley. 

leather b( 

Wet belt. 

!lt. 

Good 

adhesive. 

India- 

rubber 

belt. 

Canvas 

belt. 

Gutta¬ 

percha 

belt. 


/ 

/ 

/ 

/ 

/ 

/ 

/ 

Rubber. 

0 50 

0.4G 

0.42 


0.43 

0.30 

0.42 

Leather. 

0.4S 

0.45 

0.50 

0.60 

0.42 

0.27 

0.40 

Wood. 

0.46 

0.40 

0.48 

0.55 

0.41 

0.23 

0.38 

Iron. 

0.40 

0.35 

0.45 

0.50 

0.38 

0.20 

0.35 
































































400 Transmission of IV.' c : r . by Belting. 


TABLE I.- 

-Motive Eorcc F, when the Pulling Tension T = 1. 


Friction coefficient/, for the surfaces in contact on the smallest pulley. 

contact. 

0.15 

0.20 

0.25 

0.30 

0.35 

0.40 

0.45 

0.50 

0.55 

0.60 

2 Z 

F 

F 

F 

F 

F 

» 

F 

F 

F 

F 

F 

60° 

0.147 

0.189 

0.231 

0.272 

0.310 

0.34G 

0.367 

0.415 

0.447 

0.462 

70° 

0.171 

0.219 

0.269 

0.316 

0.352 

0.393 

0.455 

0.467 

0.481 

0.536 

80° 

0.189 

0.245 

0.297 

0.346 

0.393 

0.436 

0.478 

0.514 

0.555 

0.598 

90° 

0.202 

0.274 

0.330 

0.3S8 

0.431 

0.477 

0.522 

0.544 

0.620 

0.648 

100° 

0.231 

0.300 

0.358 

0.415 

0.468 

0.517 

0.564 

0.608 

0.648 

0.687 

110° 

0.254 

0.325 

0.392 

0.453 

0.503 

0.542 

0.602 

0.648 

0.688 

0.731 

120° 

0.272 

0.346 

0.416 

0.478 

0.536 

0.590 

0.640 

0.687 

0.731 

0.793 

130° 

0.292 

0.373 

0.445 

0.515 

0.568 

0.624 

0.676 

0.724 

0.768 

0.810 

140° 

0.310 

0.393 

0.468 

0.536 

0.594 

0.656 

0.709 

0.758 

0.797 

0.846 

150° 

0.330 

0.418 

0.498 

0.570 

0.028 

0.687 

0.741 

0.791 

0.837 

0.880 

160° 

0.346 

0.436 

0.517 

0.591 

0.656 

0.717 

0.793 

0.822 

0.869 

0.912 

170° 

0.365 

0.461 

0.545 

0.623 

0.683 

0.745 

0.801 

0.852 

0.898 

0.942 

180° 

0.380 

0.478 

0.564 

0.640 

0.709 

0.772 

0.828 

0.880 

0.927 

0.970 

190° 

0.399 

0.499 

0.592 

0.671 

0.727 

0.797 

0.854 

0.906 

0.953 

0.997 

200° 

0.415 

0.517 

0.607 

0.687 

0.758 

0.822 

0.880 

0.932 

0.975 

1.000 

210° 

0.433 

0.539 

0.633 

0.717 

0.781 

0.846 

0.904 

0.956 

1.000 

1.000 

220° 

0.447 

0.555 

0.668 

0.731 

0.803 

0.868 

0.926 

0.979 

1.000 

1.000 

230° 

0.464 

0.571 

0.674 

0.758 

0.825 

0.890 

0.949 

1.000 

1.000 

1.000 

240° 

0.478 

0.590 

0.687 

0.772 

0.845 

0.912 

0.970 

1.000 

1.000 

1.000 

250° 

0.492 

0.612 

0.706 

0.795 

0.866 

0.932 

0.991 

1.000 

1.000 

1.000 

TABLE II. 

—Pulling Tension T, 

when the 

Motive Force F = l. 

Angle of 


Friction coefficient/ for the surfaces in contact on the smallest pulley. 

contact. 

0.15 

0.20 

0.25 

0.30 

0.35 

0.40 

0.45 

0.50 

0.55 

0.60 

2 X 

T 

T 

T 

T 

T 

T 

T 

T 

T 

T 

60° 

6.779 

5.291 

4.321 

3.680 

3.227 

2.887 

2.724 

2.410 

2.236 

2.163 

70° 

5.855 

4.558 

3.711 

3.165 

2.8:i8 

2.546 

2.198 

2.140 

2.078 

1.863 

80° 

5.274 

4:077 

3.363 

2.887 

2.546 

2.290 

2.092 

1.944 

1.802 

1.671 

90° 

4.703 

3.646 

3.028 

2.580 

2.319 

2.092 

1.915 

1.838 

1.613 

1.543 

100° 

4.321 

3.361 

2.792 

2.410 

2.136 

1.933 

1.773 

1.645 

1.542 

1.455 

110° 

3.941 

3.079 

2.548 

2.206 

1.988 

1.845 

1.660 

1.542 

1.453 

1.368 

120° 

3.680 

2.887 

2.403 

2.091 

1.864 

1.693 

1.561 

1.455 

1.368 

1.261 

130° 

3.421 

2.683 

2.24G 

1.942 

1.759 

1.601 

1.479 

1.381 

1.302 

1.234 

140° 

3.227 

2.546 

2.136 

1.864 

1.699 

1523 

1.411 

1.319 

1.254 

1.182 

150° 

3.032 

2.390 

2.009 

1.755 

1.591 

1.455 

1.349 

1.264 

1.195 

1.137 

16U° 

2.887 

2.290 

1.932 

1.690 

1.523 

1.395 

1.261 

1.216 

1.151 

1.097 

170° 

2.739 

2.169 

1.835 

1.581 

1.463 

1.342 

1.249 

1.174 

1.113 

1.062 

180° 

2.631 

2.091 

1.773 

1.561 

1.410 

1.296 

1.207 

1.136 

1.079 

1.030 

190° 

2.506 

2.004 

1.688 

1.490 

1.374 

1.253 

1.170 

1.103 

1.049 

1.003 

200° 

2.410 

1.932 

1.646 

1.455 

1.319 

1.216 

1.136 

1.073 

1.026 

1.000 

210° 

2.307 

1.853 

1.580 

1.395 

1.280 

1.182 

1.106 

1.046 

1.000 

1.000 

220° 

2.236 

1.802 

1.495 

1.368 

1.244 

1.151 

1.080 

1.021 

1.000 

1.000 

230° 

2.153 

1.730 

1.4S9 

1.318 

1.212 

1.123 

1.053 

1.000 

1.000 

1.000 

240° 

2.091 

1.693 

1.455 

1.296 

1.196 

1.098 

1.030 

1.000 

1.000 

1.000 

250° 

2.023 

1.633 

1.416 

1.257 

1.155 

1.073 

1.009 

1.000 

1.000 

1.000 

It is assumed in tiie above fables that the friction 

gripe on the smallest 

pulley just balances the motive force, for which allowance must be made to 

prevent slipping. 









The slack tension £ 

is the difference between the pulling tension Tand the 

motive force 1- 

’or t — 

= T — F. 







When 

the friction 

and angle of contact are great., the pulling: tension is 

equal to the motive lorce, 

and uo slack tension is then required. 

























































Transmit.' o:: Powhr. 


401 

— 


TABLE III.—Pressure P in the Shaft Journals, wlieu the Mo¬ 


tive Force F = 1 and the System in Motion. 


Angle of 

Friction 

coefficient / for the surfaces in 

contact on the smallest pulley. 

contact. 

0.15 

0.20 

0.25 

0.30 

0.35 

0.40 

0.45 

0.50 

0.55 

0.60 

2 Z 

P 

P 

P 

P 

P 

P 

P 

P 

P 

P 

60° 

6.779 

5.291 

4.325 

3.680 

3.227 

2.887 

2.724 

2.410 

2.236 

2.163 

70° 

6.570 

5.082 

4.110 

3.484 

3.109 

2.774 

2.414 

2.308 

2.135 

1.989 

80° 

6.495 

4.956 

4.038 

3.352 

2.988 

2.658 

2.339 

2.214 

2.030 

1.863 

90° 

6.237 

4.742 

3.868 

3.235 

2.865 

2.544 

2.294 

2.093 

1.867 

1.768 

100° 

6.088 

4.624 

3.746 

3.100 

2.740 

2.430 

2.184 

1.988 

1.829 

1.697 

110° 

5.818 

4.406 

3.536 

2.976 

2.619 

2.384 

2.081 

1.S88 

1.701 

1.602 

120° 

5.642 

4.268 

3.430 

2.890 

2.497 

2.200 

1.972 

1.788 

1.637 

1.451 

130° 

5.388 

4.051 

3.259 

2.708 

2.376 

2.0S9 

1.868 

1.691 

1.547 

1.424 

140° 

5.185 

3.906 

3.135 

2.(i24 

2.257 

1.983 

1.772 

1.600 

1.477 

1.342 

150° 

4.925 

3.685 

2.949 

2.456 

2.142 

1.879 

1.674 

1.510 

1.377 

1.264 

160° 

4.717 

3.541 

2.836 

2.359 

2.030 

1.778 

1.513 

1.431 

1.297 

1.191 

170° 

4.465 

3.329 

2.664 

2.157 

1.922 

1.681 

1.496 

1.351 

1.224 

1.122 

180° 

4.262 

3.182 

2.546 

2.122 

1.824 

1.592 

1.414 

1.272 

1.158 

1.060 

190° 

4.000 

3.000 

2.371 

1.976 

1.745 

1.504 

1.339 

1.169 

1.097 

1.005 

200° 

3.777 

2.836 

2.261 

1.896 

1.628 

1.425 

1.268 

1.134 

1.050 

1.000 

210° 

3.525 

2.648 

2.120 

1.763 

1.541 

1.352 

1.205 

1.101 

1.000 

1.000 

220° 

3.323 

2.507 

1.930 

1.692 

1.458 

1.284 

1.150 

1.020 

1.000 

1.000 

230° 

3.090 

2.323 

1.886 

1.576 

1.384 

1.223 

1.066 

1.000 

1.000 

1.000 

240° 

2.890 

2.200 

1.788 

1.513 

1.340 

1.170 

1.032 

1.000 

1.000 

1.000 

250° 

2.676 

2.037 

1.681 

1.421 

1.254 

1.120 

1.015 

1.000 

1.000 

1.000 

TABLE IV. 

—Pressure P in the Shaft Journals, when the Mo- 


tive Force F = 1 and the System at Best. 


Angie of 

Friction coefficient / for the surfaces iu contact on 

the smallest pulley. 

contact. 

0.15 

0.20 

0.25 

0.30 

0.35 

0.40 

0.45 

0.50 

0.55 

0.60 

2 Z 

P 

P 

P 

P 

P 

P 

P 

P 

P 

p 

60° 

5.779 

4.291 

3.325 

2.680 

2.227 

1.8S7 

1.724 

1.410 

1.236 

1.163 

70° 

5.570 

4.082 

3.110 

2.484 

2.109 

1.774 

1.414 

1.308 

1.135 

0.989 

80° 

5.495 

3.956 

3.038 

2.352 

1.988 

1.658 

1.339 

1.214 

1.030 

0.862 

90° 

5.237 

3.742 

2.803 

2.235 

1.865 

1.544 

1.294 

1.093 

0.867 

0.768 

100° 

5.088 

3.624 

2.746 

2.100 

1.740 

1.430 

1.184 

0.988 

0.829 

0.697 

110° 

4.818 

3.406 

2.536 

1.976 

1.619 

1.384 

1.081 

0.888 

0.701 

0.602 

120° 

4.642 

3.268 

2.430 

1.890 

1.497 

1.200 

0.972 

0.788 

0.637 

0.451 

130° 

4.388 

3.051 

2.259 

1.708 

1.376 

1.089 

0.868 

0.691 

0.547 

0.424 

140° 

4.185 

2.906 

2.135 

1.624 

1.257 

0.9S3 

0.772 

0.600 

0.477 

0 342 

150° 

3.925 

2.685 

1.949 

1.456 

1.142 

0.879 

0.674 

0.510 

0.377 

0 264 

160° 

3.717 

2.511 

1.836 

1.359 

1.030 

0.778 

0.513 

0.431 

0.297 

0.191 

170° 

3.465 

2.329 

1.664 

1.157 

0.922 

0.681 

0.496 

0.351 

0.224 

0.122 

180° 

3.262 

2.182 

1.546 

1.122 

0.824 

0.592 

0.414 

0.272 

0.158 

0.060 

190° 

3.000 

2.000 

1.371 

0.976 

0.745 

0.504 

0.339 

0.169 

0.097 

0.005 

200° 

2.777 

1.836 

1.261 

0.896 

0.628 

0.425 

0.268 

0.134 

0.050 

0.000 

210° 

2.525 

1.648 

0.120 

0.763 

0.541 

0.352 

0.205 

0.101 

0.000 

0.000 

220° 

9. 323 

1.507 

0.930 

0.692 

0.458 

0.284 

0.150 

0.020 

0.000 

0.000 

230° 

2.090 

1.323 

0.886 

0.576 

0.384 

0.223 

0.066 

0.000 

0.000 

0.000 

240° 

1.890 

1.200 

0.788 

0.513 

0.340 

0.170 

0.032 

0.000 

0.000 

0.000 

250° 

1.676 

1.037 

0.681 

0.421 

0.254 

0.120 

0.015 

0.000 

0.000 

0.000 

The belt should be tightened to the pressure pin the journals when the 

svstem is at rest, to enable 

it to transmit the 

motive 

force Pwlien the sys- 

tem is in motion. The friction gripe on the smallest pulley 

will then just 

balance the motive force, oi 


’=P- 

p. The pressure p should therefore 

be made 

| greater for safe working. 







26 



























































402 


Mo: ' .: ~e. 


TABLE V.- 

-Motive Force F 

when the Slack Tension t 

= 1. 

Angle 


Friction Coefficient / for the Surfaces in Contact. 


Contact. 












0.15 

0.20 

0.35 

0.30 

0.35 

0.40 

0.45 

0.50 

0.55 

0.60 

2 ^ 

F' 

F' 

F' 

F> 

F' 

F' 

F’ 

F' 

F’ 

F> 

00 ° 

.1780 

.2330 

.3011 

.3731 

.4404 

.5319 

.5800 

.7092 

0.8091 

0.800 

70 ° 

.2000 

.2810 

.3689 

.4619 

.5441 

.(>408 

. 8 : >47 

.8792 

0.928 

1.158 

80 ° 

.2340 

.3250 

.4232 

.5299 

.6468 

.7752 

.9157 

1.059 

1.247 

1.490 

90 ° 

.2700 

.3779 

.4931 

.6329 

.7581 

.9157 

1.093 

1.293 

1.031 

1.842 

100 ° 

.3011 

.4230 

.5580 

.7092 

.8803 

1.072 

1.294 

1.550 

1.845 

2.198 

110 ° 

.3400 

.4810 

.6400 

.8292 

1.012 

1.183 

1.515 

1.845 

2.336 

2.717 

120 ° 

.3731 

.5299 

.7127 

.9106 

1.157 

1.443 

1.782 

2.198 

2.717 

3.831 

130 ° 

.4130 

.5942 

.8026 

1.001 

1.317 

1.004 

2.087 

2.625 

3.311 

4.273 

140 ° 

.4490 

.0408 

.8803 

1.157 

1.495 

1.912 

2.433 

3.135 

3.937 

5.494 

150 ° 

.4921 

.7194 

.9911 

1.324 

1.692 

2.198 

2.865 

3.788 

5.128 

7.299 

160 ° 

.5299 

.7740 

1.073 

1.419 

1.912 

2 A 31 

4.831 

4.629 

6.(522 

10.31 

170 ° 

.5750 

.8598 

1.198 

1.721 

2.100 

2.934 

4.010 

5.747 

8.849 

16.13 

180 ° 

.0131 

.9106 

1.294 

1.782 

2.415 

3.378 

4.831 

7.353 

12.60 

33.33 

190 ° 

.0040 

.9900 

1.453 

2.041 

2.073 

3.952 

5.882 

9.709 

20.41 

333.3 

200 ° 

.7092 

1.073 

1.502 

2.198 

3.135 

4.629 

7.353 

13.70 

38.46 

0.000 

210 ° 

.7051 

1.172 

1.724 

2.532 

3.571 

5.494 

9 . 4:44 

21.74 

0.000 

0.000 

220 ° 

.8091 

1.247 

2.020 

2.717 

4.098 

6.622 

12.5 

47.62 

0.000 

0.000 

2 : i 0 ° 

.8073 

1.370 

2.045 

3.144 

4.717 

8.130 

18.87 

0.000 

0.000 

0.000 

240 ° 

.9106 

1.443 

2.198 

3.378 

5.102 

10.20 

33.33 

o.ooo 

0.000 

o.ooo 

250 ° 

.9775 

1.580 

2.404 

3.891 

0.451 

13.70 

111.1 

0.000 

0.000 

0.000 

TABLE 

VII. 

—Slack Tension 

t when the Motive Force F= 1. 

Angle 


Friction Coefficient / for the Surfaces in Contact. 


Contact. 












0.15 

0.30 

0.35 

0.30 

0.35 

0.40 

0.45 

0.50 

0.55 

0.60 

2 ^ 

t ' 

f 

t' 

t 

t ' 

t ' 

t 

f 

t r 

t' 

00 ° 

5.779 

4.291 

3.321 

2.680 

2.227 

1.887 

1.724 

1.410 

1.236 

1.163 

70 ° 

4.855 

3.558 

2.711 

2.165 

1.838 

1.546 

1.198 

1.140 

1.078 

0.803 

80 ° 

4.274 

3.077 

2.363 

1.887 

1.540 

1.290 

1.092 

0.944 

0.802 

0.671 

90 ° 

3.703 

2.040 

2.028 

1.580 

1.319 

1.092 

0.915 

0.773 

0.013 

0.543 

100 ° 

3.321 

2.304 

1.792 

1.410 

1.136 

0.933 

0.773 

0.645 

0.542 

0.455 

110 ° 

2.943 

2.079 

1.518 

1.206 

0.988 

0.845 

0.600 

0.512 

0.428 

0.308 

120 ° 

2 . GS 0 

1.887 

1.403 

1.091 

0.804 

0.693 

0.561 

0.455 

0.308 

0.261 

180 ° 

2.421 

1.683 

1.246 

0.942 

0.759 

0.001 

0.479 

0.381 

0.302 

0.234 

140 ° 

2.227 

1.546 

1.136 

0.864 

0.609 

0.523 

0.411 

0.319 

0.254 

0,182 

150 ° 

2.032 

1.390 

1.009 

0.755 

0.591 

0.455 

0.349 

0.264 

0.195 

0.137 

100 ° 

1.887 

1.290 

0.932 

0.090 

0.523 

0.395 

0.261 

0.216 

0.151 

0.097 

170 ° 

1.739 

1.169 

0 . 8:15 

0.581 

0.463 

0.342 

0.249 

0.174 

0.113 

0.002 

180 ° 

1.631 

1.091 

0.773 

0.561 

0.414 

0.290 

0.207 

0.136 

0,079 

0.030 

190 ° 

1.500 

1.004 

0.688 

0.490 

0.374 

0.253 

0.170 

0.103 

0.049 

0.003 

200 ° 

1.410 

0.932 

0.640 

0.455 

0.319 

0.216 

0.136 

0.073 

0.026 

0.000 

210 ° 

1.307 

0.853 

0.580 

0.395 

0.280 

0.182 

0.106 

0.046 

0.000 

0.000 

220 ° 

1.236 

0.802 

0.495 

0.308 

0.244 

0.151 

0.080 

0.021 

0.000 

0.000 

2 : i 0 ° 

1.153 

0.730 

0.489 

0.318 

0.212 

0.123 

0.053 

0.000 

0.000 

0.000 

240 ° 

1.091 

0.693 

0.455 

0.296 

0.190 

0.098 

0.030 

0.000 

0.000 

0.000 

250 ° 

1.023 

0.633 

0.416 

0.257 

0.155 

0.073 

0.009 

0.000 

0.000 

0.000 

1 


























































Slack Tension. 403 


— 

TABLE VII.- 

-Slack Tension t 

when the Pulling Tension T= 1. 

Angle 


Friction Coefficient / for the Surfaces in Contact. 


Contact. 












0.15 

0.20 

0.25 

0.30 

0.35 

0.40 

0.45 

0.50 

0.55 

O.GO 

2 Z 

t' 

t' 

f 

V 

l’ 

f 

V 

i' 

t ' 

t' 

60° 

.8525 

.8110 

.7686 

.7283 

.6901 

.6536 

.6329 

.5851 

.5528 

.5376 

70° 

.8292 

.7806 

.7505 

.6840 

.6477 

.6072 

.5449 

.5328 

.5187 

.4634 

80° 

.8104 

.7547 

.7027 

.6536 

.6072 

.5634 

.5219 

.4826 

.4450 

.4017 

90° 

.7874 

.7257 

.6698 

.6124 

.5688 

.5219 

.4778 

.4361 

.3799 

.3518 

100 ° 

.7(18(3 

.7027 

.6418 

.5851 

.5319 

.4826 

.4361 

.3923 

.3514 

.3127 

110 ° 

.7463 

.6752 

.6075 

.5467 

.4970 

.4581 

.3976 

.3515 

.2997 

.2691 

120 ° 

.7283 

.6536 

.5838 

.5219 

.4636 

.4095 

.3594 

.3127 

.2691 

.2070 

130° 

.7077 

.6273 

.5549 

.4850 

.4316 

.3756 

.3241 

.2762 

.2320 

.1900 

140° 

.6901 

.6072 

.5319 

.4636 

.4010 

.3435 

.2912 

.2417 

.2025 

.1540 

150° 

.6702 

.5817 

.5023 

.4303 

.3716 

.3127 

.2587 

.2088 

.1633 

.1202 

100 ° 

.6536 

.5634 

.4826 

.4085 

.3434 

.2832 

.2070 

.1778 

.1312 

.0883 

170° 

.6349 

.5391 

.4550 

.3765 

.3164 

.2552 

.1993 

J483 

.1014 

.0581 

180° 

.6200 

.5219 

.4361 

.3594 

.2909 

.2283 

.1718 

.1202 

.0730 

.0296 

100 ° 

.6010 

.5010 

.4078 

.3288 

.2725 

.2024 

.1154 

.09354 

.0468 

.0026 

200 ° 

.5851 

.4826 

.3925 

.3127 

.2417 

.1777 

.1202 

.06807 

.0251 

.0000 

210 ° 

.5666 

.4604 

.3672 

.2833 

.2187 

'.1542 

.09597 

.04374 

.0000 

.0000 

220 ° 

.5528 

.4452 

.3314 

.2691 

.1964 

.1314 

.07305 

.02052 

.0000 

.0000 

230° 

.5356 

.4219 

.3286 

.2417 

.1749 

.1094 

.05089 

.0000 

.0000 

.0000 

240° 

.5219 

.4095 

.3127 

.2283 

.1543 

.08826 

.02969 

.0000 

.0000 

.0000 

25(1° 

.5058 

.3879 

.2938 

.2045 

.1341 

.06793 

.00927 

.0000 

.0000 

.0000 

TABLE VIII. 

—Pulling Tension Twken tlie Slack Tensiont= 2. 

Angle 


Friction Coefficient / for the Surfaces in Contact. 


Contact. 












0.15 

0.20 

0.25 

0.30 

0.35 

0.40 

0.45 

0.50 

0.55 

O.GO 

2 Z 

T' 

T' 

T' 

T’ 

T' 

T' 

T' 

T’ 

T > 

T’ 

60° 

1.173 

1.233 

1.301 

1.373 

1.449 

1.530 

1.580 

1.709 

1.809 

1.860 

70° 

1.206 

1.281 

1.369 

1.462 

1.544 

1.617 

1.835 

1.877 

1 928 

2.157 

8 (i° 

1.234 

1.325 

1.423 

1.530 

1.647 

1.775 

1.916 

2.072 

2.247 

2.489 

00 ° 

1.270 

1.378 

1.493 

1.633 

1.758 

1.916 

2.093 

2.293 

2.632 

2.842 

100 ° 

1.301 

1.423 

1.558 

1.709 

1.880 

2.072 

2.293 

2.549 

2.846 

3.198 

110 ° 

1.340 

1.481 

1.646 

1.829 

2.012 

2.183 

2.515 

2.845 

3.237 

3.716 

120 ° 

1.373 

1.530 

1.713 

1.916 

2.157 

2.442 

2.782 

3.198 

3.716 

4.831 

130° 

1.413 

1.594 

1.802 

2.062 

2.317 

2.662 

3.085 

3.621 

4.311 

5.263 

140° 

1.449 

1.647 

1.880 

2.157 

2.494 

2.911 

3.434 

4.138 

4.939 

6.492 

150° 

1.492 

1.719 

1.991 

2.324 

2.691 

3.198 

3.866 

4.788 

6.124 

8.319 

100 ° 

1.530 

1.77o 

2.072 

2.448 

2.912 

3.531 

4.832 

5.624 

7.619 

11.33 

170° 

1.575 

1.85o 

2.198 

2.656 

3.160 

3.919 

5.018 

6.742 

9.866 

17.20 

180° 

1.613 

1.916 

2.293 

2.782 

3.437 

4.381 

5.822 

8.317 

13.70 

33.78 

190° 

1.664 

1.996 

2.452 

3.041 

3.670 

4.941 

6.877 

10.69 

21.70 

386.1 

200 ° 

1.709 

2.072 

2.548 

3.198 

4.138 

5.625 

8.317 

14.69 

39.77 


210 ° 

1.764 

2.172 

2.723 

3.530 

4.573 

6.484 

10.42 

22.86 



220 ° 

1.809 

2.246 

2.846 

3.716 

5.091 

7.612 

13.69 

48 72 



230° 

1.867 

2.370 

3.043 

4.137 

5.716 

9.141 

19.65 

0.000 



240° 

1.916 

2.442 

3.198 

4.380 

6.481 

11.33 

33.68 

0.000 



250° 

1.977 

2.578 

3.404 

4.890 

7.457 

14.72 

107.8 

0.000 




























































404 


Transmission o ■ r by Belts. 


CONE PULLEYS. 

The illustration Fig. 2, Plate II., represents the largest and smallest diam¬ 
eters of a pair of cone pulleys, of which the letters denote as follows: 

72= radius of the largest step or pulley. 
r = radius of the smallest step or pulley. 
a = distance between the centres of rotation of the pulleys. 

6 = distance or length of the belt between the tangential points on the 
two pulleys. 

Z/= whole length of the belt.. 

x — distance from the centre of the small pulley' to the vertex o'. 

Z? = half the angle of contact of the belt on the large pulley. 

J?=half the angle of contact of the belt on the small pulley. 

For open belts the sum of the whole angles of contact is always 3G0°, and 
Z' + Z== 180°, omitting the slack of belt. 

Z = -^(72.Z , +r.Z) + 2& . .4. 

Three-step Pulley, Fig. 3, Plate II. 

The object is to make two cone pulleys cast from one pattern, and to have 
three steps on each pulley. Having given the diameters of the largest and 
smallest steps, the problem is to find the diameter of the middle step, so pro¬ 
portioned that the belt will have the same tension on all the three steps. 

7> = diameter of the middle step. 

77 = 72 + r + .5. 

ira 

Four-step Pulley, Fig. 4, Plate II. 

Having given the radii of the largest and smallest pulleys 1 and 4, the 
problem is to find the diameters of the inner pulleys 2 and 3. 
d = diameter of the pulley 3, next to the smallest. 
d'= diameter of the pulley 2, next to the largest pulley. 

d = f (72 + 2r) + .6. 

d'=§(272 + r)+^^ 2 .7. 


b = y/a- — (72 — r) 2 . . . . 1. 

Sin. Z=>~ .2. 

a 


Five-step Pulley, Fig. 5, Plate II. 


Cone pulleys of five steps are constructed as follows: The largest and 
smallest pulleys, 1 and 5, are assumed to be given from the first start. The 
diameter of the middle step 3 is calculated by formula 5, and the problem 
remains to find the diameters of the steps 2 and 4. 

72'= radius of the middle step 3; d and d' = diameters of the respective 
steps 4 and 2. 

d = 72' + r + r ^ 2 .8. 

7r a 

d'= 72 + 72'+^-—^.9. 

ira 

It is supposed in the above that each pair of cone pulleys is cast from one 
pattern. 

To find llic Proper Distance between Centres. 


Having given two equal cone pulleys, to find at what distance they ought 
to be placed to make the belt of equal tension on all the steps. 

_(72-rP_ 

a it (71 — 72 — r) . 


10 . 


This formula can he used only for pulleys of an odd number of steps, of 
which 71= diameter of the middle step, and 72 and r are the respective radii 
of the largest and smallest pulleys. 























Trans? . t Tower. 


405 


Cone Pulleys of any Number of Steps and Proportions, Figs. 

6 and 7, Plate II. 

The conoids a, 5, cc£Z? and A deb a are drawn alike and inverted to one 
another, as represented by Fig. 6. The centre lines A B can be divided into 
any number of steps, and the conoids determine the corresponding diameters 
on each pulley. The pulleys can thus be made of widely different sizes, as 
required in many cases, particularly in foot-lathes. 

The construction of the conoid is represented by Fig. 7. Draw the centre¬ 
line A B, in the middle of which erect the mean diameter D, which will be 
alike on each pulley. The largest radius R is determined by the formula 


In' 2 a na 

/i== \~T +7mjD_ T* 


Divide the centre line A B into four equal parts and draw the diameters d 
and d\ the lengths of which are determined by the following formulas ; 

d = iD + *—.12. 

na 

+ + .13. 

na 

Draw a regular curve through the points n, b, c, d and B which form the 
sides of the conoid. Draw also the inverted conoid as shown by the dotted 
lines. Any line drawn through the conoids at right angles to A B determines 
the diameters of t he corresponding steps on each cone pulley. The diameter 
b o on one pulley corresponds to the diameter m n on the other pulley. It is 
not intended that the conoid should determine the width of the steps, as may 
be inferred from Fig. 6, for it only determines the diameters of the steps. 

In practice the conoids should be laid out on a full-size scale to make the 
measurements correct. The length of the centre-line A B should not be less 
than the diameter IJ, but better to make it 2 D, for a less inclination of the 
conoid makes sharper measurements for the diameters of the steps. 

The whole length of the belt will be, 

L = nD + 2 a .14. 


For different lengths of the belt and diameter 1), with the same distance a 
between the shafts, the conoid a, b , c, d, B will be of more or less curvature. 
When a — oo, the conoid becomes a straight line, and when a — D, the radius 
i2 = 0.8Z>, which forms the greatest curvature of the conoid. 


For a given length of belt the diameter of the middle step should be 


D 


L — 2 a 

n 


15. 


The diameter of the middle step can also be obtained by assuming a def¬ 
inite value for the greatest radius R— namely : 

R 2 


T) = R + 

The length of the belt will then be- 


na 


1G. 


R 2 

LnR -f- -f- 2ct. 

a 


17 . 


Having given the greatest radius R and middle diameter D, the distance 
between Ihe shafts should be— 

n(B-R) . 






















Poweu . . : PINO. 


406 


Dimensions, Strength, and Power of Belts. 

The strain on a belt is its pulling tension T, and not only the motive force 
F, as is often considered. 

The motive force, under some conditions, is only a small fraction of the 
pulling tension, as seen in the Tables I. and II., page 400. 

The strength of the belt must, therefore, be in proportion to the pulling 
tension T. 

The following formula 1 is more correct for calculating the breadth of belts 
than the formulas on page 407. The difference is, however, very small. 

S= maximum strain in pounds per inch of width of belt, which should not 
exceed the safety strength given in the accompanying tables, but may 
be made less. 

B — breadth of belt iu inches. T— pulling tension in pounds. 

T 

Breadth of belt B = -.1. 

O 

Pulling tension T — BS. .2. 

T 

Strain per inch S = ^ .3. 

India-rubber belts are best in wet or damp places where leather belts can¬ 
not be used. 


Dimensions and Strength of India-rubber Beits. 


Number of plies. 

Wt. per. sq. ft. 
Pounds. 

Thickness. 

Inches. 

Ult. strength. 
Lbs. per iu. 

Safety 

strength. 

2 plv. 

1.25 

=35 0.1 8 l 0 

625 

104 

3 ply. 

1.66 

2 \ = 0.2083 

830 

138 

4 ply. 

2 

A = 0.3125 

1000 

166 

5 ply. 

2.4 

yi, = 0.4166 

1200 

200 

6 ply. 

2.8125 

— 0.4375 

1400 

233 


Thickness in Incites,and Strength in Pounds,of Beits. 


Kind of material in belts. 

Thickness. 

Strei 

Break. 

igth. 

Safety. 

Oak-tanned leather... 

0.25 

1000 

166 

a a 

0.1875 

780 

130 

u u 

0.125 

560 

95 

Ordinary tanned leather. 

0.25 

740 

125 

u a u 

0.1875 

560 

95 

it (< H 

0.125 

290 

50 

Paw hide, best quality. 

• •• 

1250 

225 

“ “ ordinary. 

• •• 

1100 

185 

Horse-skin. 

... 

800 

135 

Calf s-sk in.. 

... 

360 

60 

Sheep-skin. 

• •• 

322 

54 

Cowhide. 

... 

790 

130 

Cotton duck. 

• •• 

200 

66 

Flax, woven belt. 

... 

1250 

200 


The above data are for new belts, and cannot be trusted for old and worn- 
out belts. 

Care should be taken to prevent animal oil or l'at from coming in contact 
with the working surfaces of the belt and pulleys, for it reduces the friction 
and if it once permeates the leather it is difficult to get rid of. 


I 












































































































































































I 
































1.1 


















407 


Brea - r R 0F RopES - 


HORSE-POWER AND BREADTH OF LEATHER BELTS. 

B — breadth of belt in inches. 

IP = horse-power transmitted by the belt. 

V — velocitv in feet per second of the belt. 
d = diameter in inches ) . th smallest pulley . 

n = revolutions per minute J 1 J 

F = motive force in pounds transmitted by the belt. 

Z= half angle of contact of belt on the small pulley. 

S = safe working strength in pounds per inch of width of belt, whicn for 
oak-tanned leather £ inch thick, cemented and riveted joints, can be 
taken at 100 pounds, and less in proportion for weaker belts. 


IP - 


IP 


IP = 


dnF _ 60 Z? V 
126050 — 1000 
tli ick ness. 

BdnZS 
15000000* * 

B VZS 
130000 ' ' 


for 


single 

. 1 . 


3. 


B = 


15000000 IP 
d n ZiS 


■“ 2 S ‘ 


F- 


126050 IP 
d n 


550 IP 
V * 


6 . 


Example. A leather belt is to transmit IP = 75 horse-power over a pulley 
d = 36 inches in diameter, making n = 80 revolutions per minute; angle of 
contact Z= 85°, and the safe working strength S — 100 pouuds per inch of 
width. Required the width of the belt? 

„ , , „ T . a . . 15000000 X 75 ,. . . 

Ibnmda 1. W,<lth 6 = 36 x 80 x M x 100 = 40 

Ho rse-power of Iron and Steel Ropes. 


Iron. 


Diatn. 

10 

30 

30 

40 

50 

60 

TO 

SO 

90 

100 

Diani. 

Inches. 

IP 

IP 

IP 

IP 

IP 

IP 

IP 

IP 

IP 

IP 

Inches. 

i 

4 

8 

12 

16 

20 

24 

28 

32 

36 

40 

3 

T6 

A 

0.250 

12.50 

18.75 

25.00 

31.25 

37.50 

43.75 

50.00 

56.25 

62.50 

1 

4 

t 

9.000 

18 

27 

36 

45 

54 

63 

72 

81 

90.00 

6 

IS 


12.25 

24.5 

36.75 

49.00 

61.25 

73.50 

85.75 

98.00 

101.2 

122.5 

3 

S 

h 

10.00 

32 

48 

64 

80 

96 

112 

128 

144 

160.0 

3 

V, 

A 

20.25 

40.5 

60.75 

81.00 

101.2 

121.5 

141.7 

162.0 

182.2 

202.5 

7 

T5 

5 

25.00 

50 

75 

100 

125 

150 

175 

200 

225 

250.0 

h 

« 

30.25 

60.50 

90.75 

121 

151.2 

181.5 

211.7 

242 

272.2 

302.5 

5 

TG 

i 

36.00 

72.00 

108.0 

144.0 

180.0 

216.0 

252.0 

288.0 

324.0 

360.0 

9 

TG 

a 

42.25 

84.5 

126.7 

169 

211.2 

253.5 

295.7 

338 

380.2 

422.5 

6 

8 

i 

49.00 

98 

147 

196 

245 

294 

343 

392 

441 

490.0 

\k 

it 

56.25 

112.5 

168.7 

225.0 

281.2 

337.5 

393.7 

450 

506.2 

562.5 

3 

'4 

1 in. 

64 

128 

192 

256 

320 

384 

448 

512 

576 

640.0 

1 3 

16 

11 

81 

162 

243 

324 

405 

486 

567 

648 

729 

810.0 

7 

S 

11 

100 

200 

300 

400 

500 

600 

700 

800 

900 

1000 

1 in. 

H 

121 

242 

363 

484 

605 

726 

847 

968 

1089 

1210 

n 


144 

288 

432 

576 

720 

864 

1008 

1152 

1296 

1440 

1A 

If 

169 

338 

507 

676 

845 

1014 

1183 

1352 

1521 

1690 

H 

If 

196 

392 

588 

784 

980 

1176 

1372 

1568 

1764 

1960 

n 

H 

225 

450 

675 

900 

1125 

1350 

1575 

1800 

2025 

2250 

H 

2 in. 

256 

512 

768 

1024 

1280 

1536 

1792 

2048 

2304 

2560 

1A 

21 

324 

648 

974 

1296 

1620 

1944 

2268 

2602 

2916 

3240 

if 

21 

400 

800 

1200 

1600 

2000 

2400 

2800 

3200 

3600 

4000 

2 in. 

2f 

484 

968 

1452 

1936 

2420 

2904 

3388 

3882 

4356 

4840 

2A 

3 in. 

576 

1152 

1728 

2304 

2880 

3456 

4032 

4608 

5184 

5760 

2 * 


Steel. 















































408 


Breadth .’.elts, 


\ 


Breadth of Belts in Indies for Different Motive Forces and 

Angles of Contact. 


Motive 


Whole Angle of Contact 2 Z. 


Force. 

00° 

•4 

C 

o 

80° 

| yo° 

100° 

110° 

1(40° 

130° 

140° 

1.50* 

F lbs. 

B. in. 

B. in. 

B. in. 

B. in. 

B. in. 

B. in. 

B. in. 

B. in. 

B. i n. 

B. in. 

10 

0.424 

0.372 

0.331 

0.300 

0.275 

0.254 

0.238 

0.223 

0.211 

0.200 

20 

0.847 

0.743 

0.662 

0.599 

0.549 

0.509 

0.475 

0.426 

0.421 

0.400 

30 

1.271 

1.115 

0.993 

0.S98 

0.824 

0.763 

0.713 

0.670 

0.632 

0.600 

40 

1.695 

1.4S7 

1.324 

1.198 

1.099 

1.018 

0.950 

0.893 

0.842 

0.800 

50 

2.119 

1.859 

1.655 

1.497 

1.374 

1.272 

1.188 

1.116 

1.053 

1.000 

60 

2.542 

2.230 

1.987 

1.796 

1.648 

1.526 

1.425 

1.339 

1.263 

1.200 

70 

2.966 

2.602 

2.318 

2.095 

1.923 

1.780 

1.663 

1.562 

1.474 

1.400 

80 

3.390 

2.974 

2.648 

2.396 

2.198 

2.036 

1.900 

1.786 

1.684 

1.600 

90 

3.813 

3.345 

2.980 

2.695 

2.472 

2.290 

2.138 

2.009 

1.895 

1.800 

100 

4.237 

3.717 

3.311 

2.994 

2.747 

2.544 

2.375 

2.232 

2.105 

2.000 

120 

5.0S4 

4.460 

3.974 

3.592 

3.296 

3.052 

2.850 

2.678 

2.526 

2.400 

140 

5.932 

5.204 

4.636 

4.190 

3.846 

3.560 

3.326 

3.124 

2.94S 

2.800 

160 

6.780 

5.948 

5.296 

4.792 

4.396 

4.072 

3.800 

3.572 

3.368 

3.200 

180 

7.626 

6.690 

5.960 

5.390 

4.944 

4.580 

4.276 

4.018 

3.790 

3.600 

200 

S.474 

7.434 

6.622 

5.988 

5.494 

5.088 

4. / 50 

4.464 

4.210 

4.000 

220 

9.321 

8.177 

7.284 

6.586 

6.043 

5.596 

5.225 

4.910 

4.631 

4.400 

240 

10.17 

8.920 

7.948 

7.184 

6.592 

6.104 

5.700 

5.356 

5.052 

4.800 

260 

11.02 

9.663 

8.610 

7.783 

7.141 

6.613 

6.175 

5.800 

5.473 

5.200 

280 

11 .8G 

10.41 

9.272 

8.380 

7.692 

7.120 

6.652 

6.248 

5.896 

5.600 

300 

12.71 

11.15 

9.933 

8.982 

8.241 

7.632 

7.125 

6.696 

6.315 

6.000 

320 

13.56 

11.90 

10.59 

9.584 

8.692 

8.141 

7.600 

7.144 

6.736 

6.400 

310 

14.41 

12.64 

11.21 

10.18 

9.241 

8.652 

8.075 

7.590 

7.157 

6.800 

360 

15.25 

13.38 

11.92 

10.78 

9.988 

9.160 

8.552 

8.036 

7.580 

7.200 

380 

16.10 

14.12 

12.58 

11.38 

10.53 

9.669 

9.027 

8.482 

8.001 

7.600 

400 

16.95 

14.87 

13.24 

11.98 

10.99 

10.18 

9.500 

8.928 

8.420 

8.000 

420 

17.80 

15.61 

13.90 

12.58 

11.54 

10.69 

9.975 

9.374 

8.841 

8.400 

440 

18.64 

16.35 

14.57 

13.17 

12.09 

11.19 

10.45 

9.S20 

9.262 

8.800 

460 

19.49 

17.09 

15.23 

13.77 

12.64 

11.70 

10.93 

10.27 

9.683 

9.200 

480 

20.34 

1(5.84 

15.90 

14.37 

13.18 

12.21 

11.40 

10.71 

10.10 

9.600 

500 

21.19 

18.59 

16.55 

14.97 

13.74 

12.72 

11.88 

11.16 

10.53 

10.00 

600 

25.42 

22.30 

19.87 

17.96 

16.48 

15.26 

14.25 

13.39 

12.63 

12.00 

700 

29.66 

26.02 

23.18 

20.95 

19.23 

17.80 

16.63 

15.62 

14.74 

14.00 

800 

33.90 

29.74 

26.48 

23.96 

21.98 

20.36 

19.00 

17.86 

16.84 

16.00 

900 

38.13 

33.45 

29.80 

26.95 

24.72 

22.90 

21.38 

20.09 

18.95 

18.00 

1000 

42.37 

37.17 

33.11 

29.94 

27.47 

25.44 

23.75 

22.32 

21.05 

20.00 

1100 

46.61 

40.89 

36.42 

32.93 

29.22 

27.98 

26.12 

24.55 

23.15 

22.00 

1200 

50.84 

44.60 

39.74 

35.92 

32.96 

30.52 

28.50 

26.78 

25.26 

24.00 

1300 

55.08 

48.32 

40.05 

38.91 

35.71 

33.06 

30.87 

29.01 

27.36 

26.00 

1400 

59.32 

52.04 

46.36 

41.90 

38.56 

35.60 

33.26 

31.24 

29.48 

28.00 

1500 

63.56 

55.76 

49.67 

44.S9 

41.31 

38.14 

36.63 

33.47 

31.58 

30.00 

1600 

67.80 

59.48 

52.96 

47.92 

43.96 

40.72 

38.00 

35.72 

33.68 

32.00 

1700 

72.04 

63.20 

56.27 

50.91 

46.71 

43.26 

40.38 

37.95 

35.78 

34.00 

1800 

76.26 

66.90 

59.60 

53.90 

49.44 

45.80 

42.76 

40.18 

37.90 

36.00 

1900 

80.50 

70.62 

62.91 

56.89 

52.19 

48.34 

45.14 

42.41 

40.01 

38.00 

2000 

84.74 

74.34 

66.22 

59.88 

54.94 

50.88 

47.50 

44.64 

42.10 

40.00 

2100 

88.98 

78.06 

69.53 

62.87 

57.69 

53.42 

49.88 

46.87 

44.21 

42.00 

2200 

93.21 

81.77 

72.84 

65.86 

60.43 

55.96 

52.25 

49.10 

46.31 

44.00 

2300 

97.45 

85.49 

76.15 

68.85 

63.18 

58.50 

54.63 

51.33 

48.42 

46.00 

2400 

101.7 

89.20 

79.48 

71.84 

65.92 

61.04 

57.00 

53,56 

50.52 

48.00 

2500 

105.5 

92.95 

82.75 

74.85 

68.70 

61.50 

59.40 

55 80 

52.65 

50.00 

































Breadth of Belts. 


409 


Breadth of Belts in Inches for different Motive Forces and 

Angles of Contact. 


Motive 




Whole 

Angle of Contact 5 

Z. 



h orce. 

1G0° 

170° 

180° 

100° 

200° 

210° 

320° 

230° 

240° 

350° 

F lbs. 

B. in. 

B. in. 

B. in. 

B. in. 

B. in. 

B. in. 

B. in. 

B. in. 

B. in. 

B. in. 

10 

0.190 

0.182 

0.175 

0.163 

0.162 

0.157 

0.152 

0.148 

0.144 

0.140 

20 

0.381 

0.365 

0.350 

0.337 

0.325 

0.314 

0.304 

0.295 

0.287 

0.280 

‘60 

0.571 

0.547 

0.525 

0.505 

0.487 

0.472 

0.457 

*0,442 

0.431 

0.420 

40 

0.762 

0.730 

0.700 

0.673 

0.649 

0.629 

0.609 

0.591 

0.574 

0.560 

50 

0.952 

0.912 

0.875 

0.841 

0.811 

0.786 

0.761 

0.738 

0.718 

0.700 

60 

1.143 

1.094 

1.051 

1.010 

0.974 

0.943 

0.913 

0.884 

0.862 

0.840 

70 

1.333 

1.276 

1.226 

1.178 

1.136 

1.100 

1.065 

1.032 

1.005 

0.980 

80 

1.524 

1.459 

1.401 

1.346 

1.298 

1.258 

1.218 

1.182 

1.149 

1.120 

90 

1.714 

1.842 

1.576 

1.515 

1.461 

1.415 

1.370 

1.326 

1.292 

1.260 

100 

1.905 

1.824 

1.751 

1.683 

1.623 

1.572 

1.522 

1.477 

1.436 

1.400 

120 

2.286 

2.188 

2.102 

2.020 

1.948 

1.886 

1.826 

1.768 

1.723 

1.680 

140 

2.667 

2.553 

2.452 

2.357 

2.273 

2.200 

2.130 

2.063 

2.010 

1.960 

160 

3.048 

2.918 

2.802 

2.692 

2.596 

2.516 

2.436 

2.264 

2.298 

2.240 

180 

3.429 

3.283 

3.152 

3.029 

2.921 

2.830 

2.740 

2.659 

2.585 

2.520 

200 

3.810 

3.648 

3.502 

3.366 

3.246 

3.144 

3.044 

2.954 

2.872 

2.800 

220 

4.191 

4.013 

3.852 

3.703 

3.571 

3.458 

3.348 

3.249 

3.159 

3.080 

240 

4.572 

4.376 

4.204 

4.040 

3.896 

3.772 

3.652 

3.536 

3.446 

3.260 

260 

4.953 

4.741 

4.554 

4.377 

4.221 

4.086 

3.956 

3.831 

3.733 

3.540 

280 

5.334 

5.105 

4.904 

4.714 

4.546 

4.400 

4.260 

4.126 

4.020 

3.S20 

300 

5.715 

5.472 

5.253 

5.049 

4.869 

4.716 

4.566 

4.421 

4.308 

4.200 

320 

6.096 

5.836 

5.604 

5.384 

5.192 

5.032 

4.872 

4.528 

4.596 

4.480 

340 

6.477 

6.201 

5.954 

5.720 

5.516 

5.346 

5.176 

4.823 

4.883 

4.760 

360 

6.858 

6.564 

6.306 

6.060 

5.944 

5.658 

5.478 

5.304 

5.169 

5.040 

380 

7.239 

6.929 

6.606 

6.397 

6.269 

5.972 

5.782 

5.599 

5.456 

5.320 

400 

7.620 

7.296 

7.004 

6.732 

6.732 

6.288 

6.088 

5.908 

5.744 

5.600 

420 

8.001 

7.661 

7.354 

7.068 

6.816 

6.602 

6.492 

6.203 

6.031 

5.880 

440 

8.382 

8.026 

7.704 

7.406 

7.142 

6.916 

6.696 

6.498 

6.318 

6.160 

460 

8.763 

8.391 

8.054 

7.743 

7.466 

7.230 

7.000 

6.793 

6.605 

6.440 

480 

9.144 

8.752 

8.408 

8.080 

7.792 

7.544 

7.304 

7.072 

6.892 

6.520 

500 

9.525 

9.120 

8.755 

8.415 

8.115 

7.860 

7.610 

7.385 

7.180 

7.000 

600 

11.43 

10.94 

10.51 

10.10 

9.738 

9.432 

9.132 

8.842 

8.616 

8.400 

700 

13.33 

12.76 

12.26 

11.78 

11.36 

11.00 

10.65 

10.32 

10.04 

9.800 

800 

15.24 

14.59 

14.01 

13.46 

12.98 

12.58 

12.18 

11.82 

11.49 

11.20 

900 

17.14 

16.42 

15.76 

15.15 

14.61 

14.15 

13.70 

13.26 

12.92 

12.60 

1000 

19.05 

18.24 

17.51 

16.83 

16.23 

15.72 

15.22 

14.77 

14.36 

14.00 

1100 

20.95 

20.06 

19.26 

18.51 

17.85 

17.29 

16.74 

16.25 

15.80 

15.40 

1200 

22.86 

21.88 

21.02 

20.20 

19.48 

18.86 

18.26 

17.68 

17.23 

16.80 

1300 

24.76 

23.70 

22.77 

21.88 

21.10 

20.43 

19.78 

19.16 

18.67 

18.20 

1400 

26.66 

25.52 

24.52 

23.56 

22.72 

22.00 

21.30 

20.64 

20.08 

19.60 

1500 

28.56 

27.34 

26.27 

25.24 

24.34 

23.57 

22.82 

22.12 

21.52 

21.00 

1600 

30.48 

29.18 

28.02 

26.92 

25.96 

25.16 

24.36 

22.64 

22.98 

22.40 

1700 

32.38 

30.90 

29.77 

28.60 

27.58 

26.73 

25.88 

24.12 

24.42 

23.80 

1800 

34.29 

32.83 

31.52 

30.29 

29.21 

28.30 

27.40 

26.59 

25.85 

25.20 

1900 

36.19 

34.65 

33.27 

31.97 

30.83 

29.87 

28.92 

28.06 

27.28 

26.60 

2000 

38.10 

36.48 

35.02 

33.66 

32.46 

31.44 

30.44 

29.54 

28.72 

28.00 

2100 

40.00 

38.30 

36.77 

35.34 

34.08 

33.01 

31.96 

31.02 

30.15 

29.40 

2200 

41.90 

40.12 

38.52 

37.02 

35.60 

34.58 

33.48 

32.50 

31.60 

30.80 

2300 

43.80 

41.94 

40.27 

38.70 

37.22 

36.15 

35.00 

33.98 

33.04 

32.20 

2400 

45.72 

43.76 

42.04 

40.40 

38.96 

37.72 

36.52 


34.46 

32.60 

2500 

47.62 

44.78 

43.79 

41.08 

40.58 

39.29 

38.04 

36.84 

35.90 

34.00 












































410 


Vet.ocity. 


Velocity in Feet per Secon<l of Delts, Wire Ropes, or of 
Circumference of Revolving Wiieels or Pulleys. 


Diam. 


Revolutions per Minute of Wheel 

or Pulley. 


Pulley. 

10 

20 

RO 1 

40 

50 | 

00 

70 | 

80 

90 

100 

Inches. 

V 

V 

V 

V 

V 

V 

V 

V 

V 

V 

1 

.04363 

.08727 

.130901 

.17453 

.21817 

.26180 

.30543 

.34906 

.39270 

.43633 

2 

.08727 

.17453 

.261801 

.34906 

.43633 

.52360 

.61086 

.69813 

.78540 

.87266 

3 

.13090 

.26180 

$9270 

.52360 

.65450 

.78540 

.91630 

1.0472 

1.1781 

1.3090 

4 

.17453 

.34906 

.52360 

.69813 

.87266 

1.0472 

1.2217 

1.3963 

1.5708 

1.7453 

5 

.21817 

.43633 

.65450 

.87266 

1.0908; 

1.3090 

1.5272 

1.7453 

1.9635 

2.1817 

6 

.26180 

.52360 

.78540 1 

1.0472 

1.3090 

1.5708 

1.8326 

2.0944 

2.3562 

2.6180 

7 

.30543 

.61086 

.91630, 

L.2217 

1.5271! 

1.8326 

2.1380 

2.4434 

2.7489 

3.0543 

8 

.34906 

.69813 

1.0472 

1.3963 

1.7453 

2.0944 

2.4434 

2.7926 

3.1416 

3.4906 

9 

.39270 

.78540 

1.1781 

1.5708 

1.9635 

2.3562 

2.7489 

3.1416 

3.5343 

3.9270 

10 

.43633 

.87266 

1.3090 

1.7453 

2.1817 

2.6180 

3.0543 

3.4906 

3.9270 

4.3633 

11 

.47996 

.95992 

1.4398 

1.9198 

2.399S 

2.8798 

3.3597 

3.8396 

4.3194 

4.7996 

12 

.52360 

1.0472 

1.5708 

2.0944 

2.6180 

3.1416 

3.6652 

4.1888 

4.7124 

5.2360 

13 

.56723 

1.1345 

1.7017 

2.2690 

2.8361 

3.4034 

3.9706 

4.5380 

5.1051 

5.6723 

14 

.61086 

1.2217 

1.8326 

2.4434 

3.0543 

3.6652 

4.2760 

4.8868 

5.4978 

6.1086 

15 

.65450 

1.3090 

1.9635 

2.6180 

3.2725 

3.9270 

4.5S15 

5.2360 

5.8905 

6.5450 

16 

.69813 

1.3963 

2.0944 

2.7926 

3.4906 

4.18S8 

4.8869 

5.5852 

6.2832 

6.9813 

18 

.78540 

1.5708 

2.3562 

3.1416 

3.9270 

4.7124 

5.4978 

6.2832 

7.0686 

7.8540 

20 

.8726(5 

1,7453 

2.6180 

3.4906 

4.3633 

5.2360 

6.1086 

6.9813 

7.8540 

8.7266 

21 

.91630 

1.8326 

2.7489 

3.6652 

4.5815 

5.4978 

6.4141 

7.3304 

S.'_’4t>7 

9.1630 

24 

1.0472 

2.0944 

3.1416 

4.1888 

5.2360 

6.2832 

7.3304 

8.3776 

9.424S 

10.472 

27 

1.1781 

2.3562 

3.5343 

4.7124 

5.8905 

7.06S6 

8.2467 

9.4248 

10.603 

11.781 

30 

1.3090 

2.0180 

3.9270 

5.2360 

6.5450 

7.8540 

9.1630 

10.742 

11.781 

13.090 

33 

1.4398 

2.8798 

4.3194 

5.7596 

7.1994 

8.6388 

10.079 

11.519 

12.958 

14.398 

36 

1.5708 

3.1416 

4.7124 

6.2832 

7.8540 

9.4248 

10.996 

12.566 

14.137 

15.708 

39 

1.7017 

3.4034 

5.1051 

6.8068 

8.5084 

10.210 

11.912 

13.614 

15.315 

17.017 

42 

1.8326 

3.6652 

5.4978 

7.3304 

9.1630 

10.996 

12.828 

14.661 

16.493 

18.326 

45 

1.9635 

3.9270 

5.8905 

7.8540 

9.8175 

11.781 

13.744 

15.708 

17.671 

19.635 

48 

2.0941 

4.1888 

6.2832 

8.3776 

10.472 

12.566 

14.661 

16.755 

18.850 

20.944 

51 

2.2253 

4.4506 

6.6759 

8.9012 

11.126 

13.352 

15.577 

17.802 

20.028 

22.253 

54 

2.3562 

4.7124 

7.0686 

9.4248 

11.781 

14.137 

16.493 

18.850 

21.206 

23.562 

60 

2.6180 

5.2360 

7.8540 

10.472 

13.090 

15.708 

18.326 

20.944 

23.562 

26.180 

66 

2.8798 

5.7596 

8.6394 

11.519 

14.398 

17.279 

20.159 

23.038 

25.918 

28.798 

72 

3.1416 

6.2832 

9.4248 

12.566 

15.708 

18.850 

21.991 

25.132 

28.274 

31.416 

78 

3.4034 

6.8068 

10.21(1 

13.614 

17.017 

20.420 

23.824 

27.228 

30.630 

34.034 

84 

3.6652 

7.3304 

10.996 

14.661 

18.326 

21.992 

25.6o6 

29.322 

32.988 

36.652 

90 

3.9270 

7.8540 

11.781 

15.708 

19.635 

23.562 

27.489 

31.416 

35.343 

39.270 

96 

4.1883 

8.3776 

12.566 

16.755 

20.944 

25.132 

29.322 

33.510 

37.698 

41.888 

102 

4.4506 

18.9012 

13.352 

17.802 

22.253 

26.704 

31.154 

35.604 

40.056 

44.506 

108 

4.7124 

9.4248 

14.137 

18.850 

23.562 

28.274 

32.987 

37.700 

42.411 

47.124 

114 

4.9742 

9.9484 

14.923 

19.897 

24.871 

29.846 

34.819 

39.794 

44.769 

49.742 

120 

5.2360 

10.472 

15.708 

20.944 

26.180 

31.416 

36.652 

'41.888 

47.124 

52.360 

126 

5.4978 

10.995 

16.493 

21.990 

27.489 

32.986 

38.485 

43.980 

49.479 

54.978 

132 

5.7596 

11.519 

17.279 

23.038 

28.798 

34.558 

40.317 

46.076 

51.837 

57.596 

138 

6.0214 

12.043 

18.064 

24.1(86 

30.107 

36.128 

42.150 

48.172 

54.192 

60.214 

144 

6.2832 

12.566 

18.850 

25.132 

31.416 

37.700 

43.982 

50.264 

56.650 

62.832 

150 

6.5450 

13.090 

19.635 

26.180 

32.725 

39.270 

45.815 

52.360 

58.905 

65.450 

160 

6.9813 

13.963 

20.944 

27.926 

34.906 

41.888 

48.869 

55.852 

62.832 

69.813 

180 

7.8540 

15.708 

23.562 

31.416 

39.270 

47.124 

54.978 

62.832 

70.686 

78.540 

200 

8.7266 

17.453 

26.180 

34.906 

43.633 

52.360 

61.086 

69.813 

78.540 

87.266 

240 

10.472 

20.944 

31.416 

41.888 

52.360 

62.832 

73.304 

83.776 

94.248 

104.72 































Velocity, 


411 


Velocity in Feet, per Second of Bells, Wire Ropes, or of 
Circumference of Revolving Wheels or Pulleys. 


Diam. 


Revolutions pe 


Pulley. 

110 

130 

130 

140 

150 

100 

170 

180 

190 

0 

0 

Inches. 

V 

V 

V 

V 

V 

V 

V 

V 

V 

V 

1 

.47996 

.52360 

.56723 

.61086 

.65450 

.69813 

.74176 

.78540 

.82903 

.87266 

2 

.95993 

1.0472 

1.1345 

1.2217 

1.3090 

1.3963 

1.4835 

1.5708 

1.6581 

1.7453 

3 

1.4399 

1.5708 

1.7017 

1.8326 

1.9635 

2.0944 

2.2253 

2.3562 

2.4870 

2.6180 

4 

1.9199 

2.0944 

2.2689 

2.4435 

2.6180 

2.7925 

2.9671 

3.1416 

3.3160 

3.4906 

5 

2.3998 

2.6 ISO 

2.8362 

3.0543 

3.2725 

3.4907 

3.7088 

3.9270 

4.1451 

4.3633 

6 

2 .SHOO 

3.1416 

3.4034 

3.6652 

3.9270 

4.1888 

4.4506 

4.7124 

4.9742 

5.2360 

7 

3.3597 

3.6652 

3.9706 

4.2760 

4 . 5 S 15 

4.8869 

5.1924 

5.4978 

5.8032 

6.1086 

8 

3.8397 

4.1888 

4.5378 

4.8869 

5.2360 

5.5851 

5.9341 

6.2832 

6.6322 

6.9813 

9 

4.3196 

4.7124 

5.1051 

5.4978 

5.8905 

6.2832 

6.6759 

7.0686 

7.4612 

7.8540 

10 

4.7996 

5.2360 

5.6723 

6.1086 

6.5450 

6.9813 

7.4177 

7.8540 

8.2903 

8.7266 

11 

5.2796 

5.7596 

6.2395 

6.7195 

7.1990 

7.6794 

8.1994 

8.6394 

9.1193 

9.5992 

12 

6.7596 

6.2832 

6.8068 

7.3304 

7.8540 

8.3776 

8.9012 

9.4248 

9.9483 

10.472 

13 

6.2395 

6.8000 

7.3740 

7.9412 

8.5085 

9.0757 

9.6429 

10.210 

10.777 

11.345 

14 

6.7195 

7.3304 

7.9412 

8.5521 

9.1630 

9.7738 

10.385 

10.996 

11.606 

12.217 

15 

7.1995 

7.8540 

8.5085 

9.1630 

9.8175 

10.472 

11.126 

11.781 

12.435 

13.090 

16 

7.6794 

8.3776 

9.0757 

9.7738 

10.472 

11.170 

11.868 

12.566 

13.264 

13.963 

18 

8.6394 

9.4248 

10.210 

10.996 

11.781 

12.566 

13.352 

14.137 

14.422 

15.708 

20 

9.5993 

10.472 

11.345 

12.217 

13.090 

13.963 

14.835 

15.708 

16.580 

17.453 

21 

10.079 

10.966 

11.912 

12.828 

13.744 

14.661 

15.577 

16.493 

17.409 

18.326 

24 

11.519 

12.566 

13.613 

14.661 

15.709 

16.755 

17.802 

18.850 

19.897 

20.944 

27 

12.959 

14.137 

15.315 

16.493 

17.671 

18.850 

20.027 

21.206 

22.384 

23.502 

30 

14.399 

15.708 

17.017 

18.326 

19.635 

20.944 

22.253 

23.562 

24.871 

26.180 

33 

15.839 

17.278 

18.718 

20.159 

21.597 

23.038 

24.478 

25.918 

27.358 

28.798 

36 

17.280 

18.850 

20.420 

21.991 

23.562 

25.133 

26.704 

28.274 

29.845 

31.416 

39 

18.719 

20.420 

22.122 

23.824 


27.227 

28.929 

30.631 

32.332 

34.034 

42 

20.159 

21.992 

23.824 

25.656 

27.489 

29.322 

31.154 

32.987 

34.819 

36.652 

45 

21.599 

23.562 

25.525 

27.489 

29.452 

31.416 

33.379 

35.343 

37.306 

39.270 

48 

23.039 

25.132 

27.227 

29.322 

31.416 

33.510 

35.605 

37.699 

39.792 

41.888 

51 

24.579 

26.704 

28.929 

31.154 

33.379 

35.605 

37.830 

40.055 

42.279 

44.506 

54 

26.019 

28.274 

30.621 

32.987 

oo * t > 4 t 3 

37.699 

40.055 

42.411 

44.767 

47.124 

60 

28.800 

31.416 

34.034 

36.652 

39.270 

41.888 

44.506 

47.124 

49.742 

52.360 

66 

31.678 

34.008 

37.437 

40.317 

43.197 

46.077 

48.956 

51.836 

54.716 

57.590 

72 

34.557 

37.700 

40.841 

43.982 

47.124 

50.265 

53.407 

56.549 

59.690 

62.832 

78 

37.437 

40.840 

44.244 

47.647 

51.050 

54.454 

57.858 

61.261 

64.664 

68.068 

84 

40.317 

43.984 

47.647 

51.313 

54.980 

58.643 

62.308 

65.973 

69.639 

73.304 

DO 

43.197 

47.125 

51.051 

54.978 

58.905 

62.832 

66.759 

70.686 

74 . 613 , 

78.540 

96 

46.076 

50.264 

54.454 

58.643 

62.830 

67.021 

71.209 

75.398 

79.587 

83.776 

102 

48.956 

53.404 

57.858 

62.308 

66.760 

71.209 

75.660 

80.111 

84.561 

89.012 

108 

51.836 

56.548 

61.261 

65.973 

70.685 

75.398 

80.111 

84.823 

89.535 

94.248 

114 

54.716 

59.692 

64.664 

69.638 

74.615 

79.587 

84.561 

89.535 

94.509 

99.484 

120 

57.595 

62.832 

68.068 

73.304 

78.540 

83.776 

89.012 

94.248 

99.484 

104.72 

126 

60.476 

65.972 

71.471 

76.969 

82.465 

87.965 

93.462 

98.960 

104.46 

109.95 

132 


69.116 

74.875 

SO .634 

86.395 

92.153 

97.913 

103.67 

109.43 

115.19 

138 

66.235 

72.256 

78.278 

84.299 

90.320 

96.342 

102.36 

108.38 

114.41 

120.43 

144 

69.115 

75.400 

81.681 

87.965 

94.250 

100.53 

106.81 

113.10 

119.38 

125.66 

150 

71.995 

78.540 

85.085 

91.630 

98.175 

104.72 

111.26 

117.81 

124.35 

130.90 

160 

76.794 

83.776 

90.757 

97.738 

104.72 

111.70 

118.68 

125.66 

132.64 

139.63 

180 

86.394 

94.248 

102.10 

109.96 

117.81 

125.66 

133.52 

141.37 

149.22 

150.08 

200 

87.266 

104.72 

113.45 

122.17 

130.90 

139.63 

148.35 

157.08 

165.80 

174.53 

240 

115.19 

126.66 

136.13 

146.61 

157.08 

167.55 

178.02 

188.50 

198.97 

209.44 





































412 


V KLflCITY. 




Velocity in Feet per Second of Belts, Wire Ropes, or of 
Circumference of Revolving Wheels or Pulleys. 


Diara. 

Pulley. 

310 j 

R 

330 

evolutions pe 
330 j 340 

r Minu 

350 

te of V\ 
300 

'heel o 
310 

r Pulle 

380 

y- 

290 

300 

Inches. 

V 1 

V 

V 

V 

V 

V 

V 

V 

V 

V 

1 

.91630 

95993 

1.0036 

1.0472 

1.0908 

1.1345 

1.1781 

1.2218 

1.2654 

1.3090 

2 

1.8326 

1.9199 

2.0071 

2.0944 

2.1816 

2.2689 

2.3562 

2.4435 

2.5307 

2.6180 

3 

2.7489 

2.8798 

3.0107 

3.1416 

3.2724 

3.4034 

3.5343 

3.6652 

3.7961 

3.9270 

4 

3.6652 

J.8397 

4.0112 

4.1888 

4.3633 

4.5378 

4.7124 

4.8869 

5.0614 

5.2360 

5 

4.5814 

4.7997 

5.0178 

5.2360 

5.4580 

5.6723 

5.8905 

6.1086 

6.3268 

0 

6 

5.4977 

5.7596 

6.0214 

6.2832 

6.5450 

6.8068 

7.0686 

7.3304 

7.5921 

7.8540 

7 

6.4140 

6.7195 

7.0249 

7.3304 

7.6335 

7.9412 

8.2-167 

8.5521 

8.8575 

9.1630 

8 

7.3303 

7.6794 

8.0285 

8.3776 

8.7265 

9.0757 

9.4248 

9.7738 

10.123 

10.472 

9 

8.2466 

8.6394 

9.0320 

9.4248 

9.8175 J 

10.210 

10.603 

10.995 

11.388 

11.781 

10 

9.1610 

9.5993 

10.036 

10.472 

10.908 

11.345 

11.781 

12.218 

12.654 

13.090 

11 

10.079 

10.559 

11.039 

11.519 

11.999 

12.479 

12.959 

13.439 

13.919 

14.398 

12 

10.996 

11.519 

12.043 

12.566 

13.090 

13.613 

14.137 

14.660 

15.184 

15.708 

13 

11.912 

12.479 

13.040 

13.615 

14.180] 

14.748 

15.315 

15.882 

16.450 

17.017 

14 

12.828 

13.439 

14.050 

14.662 

15.271 

15.882 

16.493 

17.104 

17.715 

18.326 

15 

13.744 

14.399 

15.052 

15.709 

16.362 

1 

17.017 

17.671 

18.326 

18.980 

19.635 

16 

14.661' 

15.359 

16.054 

16.776 

17.453 

18.151 

18.850 

19.548 

20.240 

20.944 

18 

16.498 

17.279 

18.061 

18.851 

19.635 

20.420 

21.206 

21.991 

22.770 

23.562 

20 

18.326 

19.199 

20.071 

20.944 

21.816 

22.689 

23.562 

24.435 

25.307 

26.180 

21 

19.242 

20.159 

21.748 

21.991 

22.907 

23.824 

24.790 

25.656 

26.573 

27.489 

24 

21.991 

23.088 

24.085 

25.133 

26.180 

27.227 

23.274 

29,321 

30.369 

31.416 

27 

24.740 

25.918 

27.096 

28.274 

29.452 

30.630 

31.809 

32.987 

34.165 

35.343 

30 

27.489 

28.798 

30.107 

31.416 

32.725 

34.034 


36.652 

37.961 

39.270 

33 

30.238 

31.678 

33.117 

342558 

35.997 

37.437 

38.877 

40.317 

41.757 

43.194 

36 

32.987 

84.557 

36.128 

37.699 

39.270 

40.841 

42.412 

43 982 

45.553 

47.124 

39 

35.735 

«_> / ,4«3 i 

39.138 

-10.841 

42.542 

44.24-1 

45.946 

47.647 

49.349 

51.051 

42 

38.485 

40.317 

42.149 

43.982 

45.815 

47.647 

49.480 

51.313 

53.145 

54.978 

45 

41.233 

43.197 

45.160 

47.124 

49.085 

51.051 

53.014 

54.978 

56.941 

58.905 

48 

43.982 

46.077 

48.171 

50.266 

52.860 

54.454 

56.549 

58.643 

60.737 

62.S32 

51 

46.731 

48.956 

51.182 

53.407 

55.630 

57.857 

60.083 

62.308 

64.533 

66.759 

54 

49.480 51.836 

54.192 

56.549 

58.905 

61.281 

63.617 

65.973 

68.329 

70.GS6 

60 

54.978 

57.596 

60.214 

62.832 

65.450 

68.06S 

70.686 

73.304 

75.922 

78.540 

66 

60.476 

63.356 

66.235 

! 69.115 

71.990 

7-4.874 

77.755 

80.634 

83.514 

86.394 

72 

65.973 

69.116 

72.256 

175.399 

58.540 

81.681 

84.823 

87.964 

91.106 

94.248 

78 

71.471 

74.876 

78.278 

81.682 

58.085 

88.488 

91.892 

95.295 

98.698 

102.10 

84 

76.969 

80.035 

84.299 

87.965 

91.630 

95.294 

98.960 

102.63 

106.29 

109.96 

90 

82.466 

86.395 

90.320 

94.248 

98.175 

102.10 

106.03 

109.55 

113.88 

117.81 

96 

87.964 

92.154 

96.342 

100.53 

104.72 

108.91 

113.10 

117.29 

121.47 

125.66 

102 

93 462 

97.914 

102.36 

106.81 

111.26 

115.71 

120.17 

124.61 

129.07 

133.52 

108 

98.960 

103.67 

108.38 

113.10 

117.61 

122.52 

127.23 

131.95 

136.66 

141.37 

114 

104.46 

109.43 

114.40 

119.38 

124.35 

129.33 

134.30 

139.28 

144.25 

149.23 

120 

109.96 

115.19 

120.43 

1125.66 

130.90 

136.13 

141.37 

146.66 

151.84 

157.08 

126 

115.45 

120.95 

126.45 

131.95 

137.44 

142.94 

148.44 

153.94 

159.44 

164.93 

132 

120.95 

126.67 

132.47 

138.23 

143.99 

149.80 

155.51 

161.27 

167.03 

172.79 

138 

126.45 

132.47 

138.49 

144.51 

150.53 

156.60 

162.58 

168.60 

174.62 

180.64 

144 

131.94 

138.23 

144.51 

150.80 

157.09 

163.41 

169.65 

175.93 

182.21 

188.50 

150 

137.44 

143.99 

150.52 

157.09 

163.62 

170.17 

176.71 

183.26 

189.80 

196.35 

160 

146.61 

153.59 

160.54 

167.70 

174.5c 

181.51 

188.5fc 

195.48 

202.46 

209.44 

180 

161.93 

172.79 

180.01 188.51 

191.35 

204.20 

212.OC 

219.91 

227.76 

235.62 

200 

183.26 

191.99 

200.71 

209.44 

218.16 

226.89 

247.9( 

244.35 

253.07 

261.80 

240 

219.91 

230.38 

240.85 251.33 

261.8C 

272.27 

282.74 293.21 

303.69 

314.16 












































Velocity. 


413 


Velocity in Feet per Second of Belts, Wire Ropes, or of 
Circumference of Revolving Wheels or Pulleys. 

Diam. 

Pulley. 

310 

320 

Eievolu 

330 

lions p 

340 

er Min 
350 

ute of 

360 

Wheel 

370 

or Pul 
380 

ley. 

390 

400 

Inches. 

V 

V 

V 

V 

V 

V 

V 

V 

V 

V 

1 

1.3526 

1.3963 

1.4399 

1.4835 

1.5271 

1.5708 

1.6144 

1.6581 

1.7017 

1.7454 

2 

2.7052 

2.7925 

2.8798 

2.9761 

3.0543 

3.1416 

3.2289 

3.3162 

3.4034 

3.4906 

3 

4.0579 

4.888 

4.3197 

4.4506 

4.5815 

4.7124 

4.8433 

4.9743 

5.1051 

5.2360 

4 

5.4105 

5.5850 

5.7596 

5.9341 

6.1080 

6.2832 

6.4577 

6.6324 

6.8068 

6.9813 

5 

6.7632 

6.9813 

7.1995 

7.4176 

7.6360 

7.8540 

8.0722 

8.2905 

8.5085 

8.7266 

6 

8.115S 

8.3776 

8.6394 

8.9012 

9.1630 

9.4248 

9.6866 

9.948 

10.210 

10.472 

7 

9.4G84 

9.7738 

10.079 

10.385 

10.690 

10.996 

11.301 

11.606 

11.912 

12.217 

8 

10.821 

11.170 

11.519 

11 868 

12.217 

12.566 

12.915 

13.265 

13.613 

13.963 

9 

12.174 

12.566 

12.959 

13.352 

13.744 

14.137 

14.530 

14.922 

15 315 

15.708 

10 

13.526 

13.963 

14.399 

14.835 

15.271 

15.708 

16.144 

16.581 

17.017 

17.453 

11 

14.879 

15.359 

15.839 

16.319 

16.798 

17.279 

17.759 

18.239 

18.718 

19.198 

12 

16.232 

16.755 

17.279 

17.802 

18.326 

18.850 

19.373 

19.897 

20.420 

20.944 

13 

17.584 

18.151 

18.719 

19.286 

19.853 

20.420 

20.988 

21.555 

22.122 

22.690 

14 

18.937 

19.548 

20.159 

20.769 

21.384 

21.991 

22.602 

23.213 

23.824 

24.434 

15 

20.289 

20.944 

21.599 

22.253 

22.907 

23.562 

24.216 

24.872 

25.525 

26.180 

1G 

21.652 

22.340 

23.038 

23.736 

24.434 

25.133 

25.831 

26.530 

27.227 

27.296 

18 

24.347 

25.133 

25.918 

26.704 

27.489 

28.274 

29.060 

29.846 

30.630 

31.416 

20 

27.052 

27.925 

28.798 

29.671 

30.543 

31.416 

32.289 

33.162 

34.034 

34.906 

21 

2S.405 

29.321 

30.238 

31.154 

32.071 

32.987 

33.903 

34.820 

35.735 

36.652 

24 

32.4G3 

33.510 

34.558 

35.605 

36.652 

37.699 

38.746 

39.795 

40.841 

41.888 

27 

36.521 

37.699 

38.867 

40.055 

41.284 

42.412 

43.590 

44.769 

45.946 

47.124 

30 

40.579 

41.888 

43.187 

44.506 

45.815 

47.124 

48.433 

49.743 

51.051 

52.360 

33 

44.G37 

46.077 

47.507 

48.956 

50.395 

51.836 

53.276 

54.717 

56.156 

57.596 

30 

48.695 

50.265 

51.826 

53.407 

54.980 

56.549 

58.119 

59.691 

61.261 

62.832 

39 

52.753 

54.454 

56.146 

57.858 

59.560 

61.261 

62.963 

64.666 

66.366 

68.068 

42 

56.810 

58.643 

60.466 

62.308 

64.140 

65.974 

67.806 

69.640 

71.471 

73.304 

45 

60.868 

62.832 

64.786 

66.759 

68.720 

70 686 

72.649 

74.614 

76.576 

78.540 

48 

64.926 

67.020 

69.105 

71.209 

73.305 

75.398 

77.493 

79.589 

81.681 

83.776 

51 

68.984 

71.209 

73.425 

75.660 

77.885 

80.111 

82.336 

84.563 

86.787 

89.012 

54 

73.042 

75.398 

77.745 

80.111 

82.465 

84.823 

87.179 

89.537 

91.S92 

94.248 

GO 

81.158 

83.776 

86.394 

89.012 

91.630 

94.24S 

96.866 

99.486 

102.10 

104.72 

GG 

89.274 

92.153 

95.033 

97.913 

100.79 

103.67 

106.55 

109.43 

112.31 

115.19 

72 

97.389 

100.53 

103.67 

106.81 

109.95 

113.10 

116.24 

119.38 

122.52 

125.66 

78 

105.51 

108.91 

112.31 

115.72 

119.12 

122.52 

125.93 

129.33 

132.73 

136.14 

84 

113.62 

117.29 

120.95 

124.62 

120.28 

131.95 

135.61 

139.28 

142.94 

146.61 

90 

121.74 

125.66 

129.59 

133.52 

137.44 

141.37 

145.30 

149.22 

153.15 

157.08 

9G 

129.85 

134.04 

138.23 

142.42 

146.61 

150.80 

154.99 

159.17 

163.36 

167.55 

102 

137.97 

142.42 

146.87 

151.32 

155.72 

160.22 

164.67 

169.12 

173.57 

178.02 

108 

146.08 

150.80 

155.51 

160.22 

164.93 

163.65 

174.36 

179.07 

183.78 

188.50 

114 

154.20 

159.17 

164.15 

169.12 

174.20 

179.07 

184.05 

189.02 

193.99 

198.97 

120 

162.32 

167.55 

172.79 

178.02 

183.26 

188.50 

193.73 

198.97 

204.20 

209.44 

12G 

170.31 

175.93 

181.43 

186.92 

192.42 

197.92 

203.42 

208.92 

214.41 

219.90 

132 

178.55 

184.31 

190.07 

195.83 

201.58 

207.35 

213.10 

218.87 

224.62 

230.38 

138 

186.66 

192.68 

198.71 

204.73 

210.75 

216.77 

222.79 

228.82 

234.83 

240.86 

144 

194.78 

201.06 

207.35 

213.63 

219.91 

226.19 

232.48 

238.76 

245.04 

251.32 

150 

202.89 

209.44 

215.99 

222.53 

229.07 

235.62 

242.16 

248.72 

255.25 

261.80 

160 

216.52 

223.40 

230.38 

237.36 

244.34 

251.33 

258.31 

265.30 

272.27 

279.26 

180 

243.47 

251.33 

259.18 

267.04 

274.89 

282.74 

290.60 

298.46 

306.30 

314.16 

200 

270.52 

279.25 

287.98 

296.71 

305.43 

314.16 

322.89 

331.62 

340.34 

349.06 

240 

284.05 

335.10 

345.58 

356.05 

366.52 

376.99 

387.46 

397.95 

408.41 

418.88 






































414 Motive Fokce. 


Motive Force in Pounds per Horse-Power transmitted in tlie 
Periphery of llevolving Wheels or Pulleys. 


Diaru. 


Revolutions n per Minute of 

Whee 

or Pulley. 


Pulley. 

10 

20 

30 

40 

50 

GO 

70 

80 

90 

100 

Inches. 

F 

F 

F 

F 

F 

F 

F 

F 

F 

F 

1 

1260.5 

6302.5 

4201.6 

3151.2 

2521.0 

2100.8 

1S00.7 

1578.6 

1400.6 

1260.5 

2 

6302.5 

3151.2 

2100.8 

1578.6 

1260.5 

1050.4 

900.35 

789.30 

700.30 

630.25 

.3 

4201.6 

2100.8 

1400.6 

1050.4 

840.33 

700.30 

600.23 

525.20 

466.86 

430.16 

4 

3151.2 

1575.6 

1050.4 

789.30 

630.25 

525.20 

450.17 

394.65 

350.15 

315.12 

5 

2521.0 

1260.5 

840.33 

630.25 

504.20 

420.16 

360.14 

318.12 

280.12 

252.10 

6 

2100.8 

1050.4 

700.30 

525.20 

420.16 

350.13 

300.12 

263.10 

233.10 

210.10 

7 

1800.7 

900.35 

600.23 

450.17 

360.14 

300.12 

257.24 

225.10 

200.08 

180.07 

8 

1578.6 

789.30 

525.20 

394.65 

315.72 

263.10 

225.51 

197.32 

175.40 

157.86 

9 

1400.6 

700.30 

466.86 

350.15 

280.12 

233.43 

200.10 

175.07 

155.62 

140.06 

10 

1260.51630.25 

1 

420.16 

315.12 

252.10 

210.08 

180.07 

157.56 

140.05 

126.05 

11 

1146.0 

573.00 

382.00 

286.50 

229.20 

191.00 

163.43 

133.25 

127.11 

114.60 

12 

1050.4 

525.20 

350.13 

262.60 

210.08 

175.07 

150.06 

131.30 

116.81 

105.04 

13 

969.61 

484.80 

323.20 

242.40 

193.92 

161.60 

138.51 

121.20 

107.73 

96.961 

14 

900.35 

450.17 

300.11 

225.09 

180.07 

150.06 

128.62 

112.32 

100.04 

90.035 

15 

840.33 

420.16 

280.11 

210.08 

168.06 

140.05 

120.05 

105.04 

93.37 

84.033 

16 

789.30 

394.65 

263.10 

197.57 

157.86 

131.55 

112.75 

98.66 

87.700 

78.930 

18 

700.30 

350.15 

233.43 

175.08 

140.06 

116.71 

100.04 

87.537 

78.812 

70.030 

20 

630.25 

315.12 

210.08 

157.56 

126.05 

105.04 

90.039 

78.781 

70.028 

63.025 

21 

600.23 

300.11 

200.08 

150.06 

120.05 

100.04 

85.747 

75.030 

66.692 

60.023 

24 

525.20 

262.60 

175.06 

131.55 

105.04 

87.533 

75.020 

65.650 

59.400 

52.520 

27 

466.86 

233.43 

155.62 

116.71 

93.372 

77.810 

66.694 

58.357 

51.855 

46.686 

30 

420.16 

210.08 

140.05 

105.04 

84.032 

70.027 

60.026 

52.520 

46.686 

42.016 

33 

382.00 

191.00 

127.33 

95.500 

76.400 

63.666 

54.559 

47.750 

42.444 

38.200 

36 

350.15 

175.07 

116.71 

87.54 

70.038 

58.360 

50.021 

43.769 

37.794 

35.015 

39 

323.20 

161.60 

107.73 

80.800 

64.640 

53.866 

46.171 

40.400 

35.911 

32.320 

42 

300.12 

150.06 

100.04 

75.030 

60.240 

50.020 

42.874 

37.515 

33.343 

30.012 

45 

280.12 

140.06 

93.373 

70.030 

56.024 

46.687 

40.017 

35.015 

31.123 

28.012 

48 

263.10 

131.55 

87.700 

65.775 

52.620 

43.850 

37.586 

32.885 

28.145 

26.310 

51 

247.16 

123.58 

82.386 

61.790 

49.432 

41.193 

35.310 

30.895 

27.462 

24.716 

54 

233.43 

116.71 

77.810 

58.357 

46.686 

38.905 

33.347 

29.180 

25.825 

23.343 

60 

210.08 

105.04 

70.026 

52.520 

42.016 

35.013 

30.011 

26.010 

23.342 

21.008 

06 

191.00 

95.500 

63.666 

47.750 

38.200 

31.833 

27.286 

23.875 

21.222 

19.100 

72 

175.40 

87.700 

58.466 

43.850 

35.080 

29.233 

25.057 

21.925 

19.490 

17.540 

78 

161.60 

80.800 

53.866 

40.400 

32.320 

26.933 

23.086 

20.200 

18.955 

16.160 

84 

150.06 

75.030 

50.020 

37.515 

30.012 

25.010 

21.437 

18.757 

16.682 

15.006 

90 

140.06 

70.030 

46.686 

35.015 

28.012 

23.343 

20.008 

17.507 

15.562 

14.006 

96 

131.55 

65.775 

43.850 

32.887 

26.310 

21.925 

18.793 

16.444 

14.616 

13.155 

102 

123.58 

61.790 

41.193 

30.895 

24.716 

20.597 

17.654 

15.447 

13.731 

12.358 

108 

116.71 

58.355 

38.901 

29.175 

23.342 

19.452 

16.387 

14.586 

12.968 

11.671 

114 

110.57 

55.285 

36.856 

27.642 

22.114 

18.428 

15.996 

13.821 

12.285 

11.057 

120 

105.04 

51.020 

35.013 

26.260 

21.008 

17.507 

15.006 

13.130 

11.671 

10.504 

126 

100.04 

50.020 

33.346 

25.010 

20.008 

16.673 

14.291 

12.505 

11.115 

10.004 

132 

95.500 

47.750 

31.833 

23.875 

19.100 

15.916 

13.643 

11.938 

10.611 

9.5500 

138 

90.978 

45.489 

30.326 

22.744 

18.196 

15.163 

12.997 

11.372 

10.109 

9.0978 

144 

87.533 

43.766 

29.177 

21.883 

17.506 

14.589 

12.505 

10.942 

9.7260 

8.7533 

150 

84.033 

42.016 

28.011 

21.008 

16.807 

14.005 

12.005 

10.504 

93.370 

8.4033 

160 

78.930 

39.465 

26.810 

19.742 

15.786 

13.155 

11.261 

9.8662 

8.8662 

7.8930 

180 

70.030 

35.015 

23.343 

17.507 

14.006 

11.672 

10.004 

8.7537 

7.7811 

7.0030 

200 

63.025 

31.512 

21.008 

15.756 

12.605 

10.504 

9.004 

7.8781 

7.0030 

6.3025 

240 

52.520 

26.260 

17.506 

13.130 

10.504 

8.7533 

7.5028 

6.5650 

5.8355 

5.2520 






































419 


IT ' ER. 


Horse-Power for Different. Motive Forces F and Velocities V. 


Motive 

Force. 

10 

20 

Velc 

30 

>city V 

40 

in Fe 

50 

et per 

CO 

Seconc 

70 

i. 

80 

90 

| 100 

F 1 bs. 

IP 

IP 

IP 

IP 

IP 

IP 

IP 

IP 

IP 

IP 

510 

9.2727 

18.545 

27.818 

37.091 

46.363 

55.636 

64.909 

74.182 

84.454 

92.727 

520 

9.4545 

18.909 

28.363 

37.818 

47.273 

56.727 

66.182 

75.636 

86.091 

94.545 

530 

9.6363 

19.273 

28.909 

38.545 

48.182 

57.818 

67.454 

77.091 

87.727 

96.363 

540 

9.8181 

19.636 

29.454 

39.273 

49.091 

58.909 

68.727 

78.545 

88.363 

9S.181 

550 

10.000 

20.000 

30.000 

40.000 

50.000 

60.000 

70.000 

80.000 

90.000 

100.00 

560 

10.181 

20.363 

30.545 

40.727 

50.909 

61.091 

71.273 

81.454 

91.636 

101.81 

570 

10.363 

20.727 

31.091 

41.454 

51.818 

62.182 

72.545 

82.909 

93.273 

103.63 

580 

10.545 

21.091 

31.636 

42.182 

52.727 

63.273 

73.818 

84.363 

94.909 

105.45 

590 

10.626 

21.444 

32.182 

42.909 

53.636 

64.363 

75.091 

85.818 

96.545 

106.26 

GOO 

10.909 

21.818 

32.727 

43.636 

54.545 

65.454 

76.363 

87.273 

9S.182 

109.09 

610 

11.091 

22.182 

33.273 

44.363 

55.454 

66.545 

77.636 

88.727 

99.82 

110.91 

620 

11.273 

22.545 

33.818 

45.091 

56.363 

67.636 

78.909 

90.182 

101.45 

112.73 

630 

11.454 

22.909 

34.363 

45.818 

57.273 

6S.727 

80.182 

91.636 

103.09 

114.54 

640 

11.636 

23.273 

34.909 

46.545 

58.182 

69.818 

81.454 

93.091 

104.73 

116.36 

650 

11.818 

23.636 

3o.454 

47.273 

59.091 

70.909 

82.727 

94.546 

106.26 

118.18 

660 

12.000 

24.000 

36.000 

48.000 

60.000 

72.000 

84.000 

96.000 

108.00 

120.00 

670 

12.182 

24.363 

36.545 

48.727 

60.909 

73.091 

85.273 

97.454 

109.63 

121.82 

GS0 

12.364 

24.727 

37.091 

49.454 

61.818 

74.182 

86.545 

98.909 

111.27 

123.64 

690 

12.545 

25.091 

37.636 

50.182 

62.727 

75.273 

87.818 

100.36 

112.91 

125.45 

700 

12.727 

25.444 

38.182 

50.909 

63.636 

76.363 

89.091 

101.82 

11454 

127.27 

710 

12.909 

25.818 

38.727 

51.636 

64.545 

77.454 

90.363 

103.27 

116.18 

129.09 

720 

13.090 

25.182 

39.273 

52.363 

65.454 

78.545 

91.636 

104.73 

117.82 

130.90 

730 

13.273 

25.545 

39.818 

53.091 

66.363 

79.636 

92.909 

106.18 

119.45 

132.73 

740 

13.454 

25.909 

40.363 

53.818 

67.273 

80.727 

94.182 

107.63 

121.09 

131.54 

750 

13.636 

26.273 

40.909 

54.545 

68.182 

81.818 

95.454 

109.09 

122.73 

136.36 

760 

13.818 

26.636 

41.454 

55.273 

69.091 

82.909 

96.727 

110.54 

124.36 

138.18 

770 

14.000 

28.000 

42.000 

56.000 

70.000 

84.000 

98.000 

112.00 

126.00 

140.00 

780 

14.182 

28.363 

42.545 

56.727 

70.909 

85.091 

99.273 

113.45 

127.63 

141.82 

790 

14.363 

28.727 

43.091 

57.454 

71.818 

S6.182 

100.54 

114.91 

129.27 

143.63 

800 

14.545 

29.091 

43.636 58.182 

72.727 

87.273 

101.82 

116.36 

130.91 

145.45 

810 

14.727 

29.444 

44.182 

58.909 

73.636 

88.363 

103.09 

117.82 

132.54 

147.27 

820 

14.909 

29.818 

44.727 

59.636 

74.545 

89 454 

104.36 

119.27 

134.18 

149.09 

830 

15.091 

30.182 

45.273 

60.363 

75.454 

90.545 

105.63 

120.72 

135.82 

150.91 

840 

15.273 

30.545 

45.818 

61.091 

76.863 

91.636 

106.91 

122.18 

137.45 

152.73 

850 

15.454 

30.909 

46.363 

61.818 

77.273 

92.727 

108.18 

123.63 

139.09 

154.54 

860 

15.636 

31.273 

46.909 

62.545 

78.182 

93.818 

109.45 

125.09 

140.73 

156.36 

870 

15.818 

31.363 

47.454 

63.273 

79.090 

94.909 

110.73 

126.54 

142.36 

158.18 

880 

16.000 

32.000 

48.000 

64.000 

80.000 

96.000 

112.001 

128.00 

144.00 

160.00 

890 

16.182 

32.363 

48.545 

64.727 

80.909 

97.091 

113.27 

129.45 

145.63 

161.82 

900 

16.364 

32.727 

49.091 

65.454 

81.818 

98.182 

114.541 

130.91 

147.27 

163.64 

910 

16.545 

33.091 

49.636 

66 .1S2 

82.727 

99.273 

115.82| 

132.36 

148.91 

165.45 

920 

16.727 

33.444 

50.182 

66.909 

83.363 

100.36 

117.09 

133.82 

150.54 

167.27 

930 

16.909 

33.818 

50.727 

67.636 

84.545 

101.45 

118.361 

135.27 

152.18 

169.09 

940 

17.0911 

34.1S2 

51.273 

68.363 

85.454 

102.54 

119.63 

136.73 

153.82 

170.91 

950 

17.273 

34.545 

51.818 

69.091 

86.363 

103.63 

120.91 

138.18 

155.45 

172.73 

960 

17.454 

34.909 

52.363 

69.818 

87.273 

104.73 

122.18 

139.63 

157.09 

174.54 

970 

17.636 

35.273 

52.909 

70.545 

88.182 

105.82 

123.45 

141.09 

158.73 

176.36 

980 

17.818 

35.636 

53.4541 

71.273 

89.091 

106.91 

124.72 

142.54 

160.36 

178.18 

990 

18.000 

3,6.000 

54.000 

72.000 

90.000 

108.00 

126.00 

144.00 

162.(40 

180.00 

1000 

18.182 

36.363 

54.5451 

72.727 

90.909 

109.1 

128.27 

145.45 

163.63 

181.82 























































420 


Diameters of 


j it-Iron Shafts. 


Diameters in Indies o f YV ^onglit-Iron Sliafts. 


Number of revolutions por minute of wrought-iron shafts. 


POWOT 

' 10 

15 

20 

23 

30 

35 

40 

45 

50 

55 

60 

TO 

80 

IP. 

lu. 

iu. 

Iu. 

In. 

Iu. 

Iu. 

In. 

Iu. 

Iu. 

Iu. 

Iu. 

In. 

In. 

1 

2.32 

2.07 

1.84 

1.71 

1.61 

1.53 

1.46 

1.41 

1.36 

1.31 

1.28 

1.22 

1.16 

2 

2.92 

2«5o 

2.32 

2.16 

2.03 

1.92 

1.84 

1.72 

1.71 

1.66 

1.61 

1.53 

1.46 

3 

3.40 

2.72 

2.66 

2.47 

2.33 

2.21 

2.11 

2.03 

1.91 

1.90 

1.71 

1.75 

1.67 

4 

3.68 

3.22 

2.92 

2.72 

2.55 

2.43 

2.32 

2.23 

2.16 

2.09 

1.80 

1.93 

1.84 

5 

3.97 

3.48 

3.15 

2.92 

2.75 

2.61 

2.50 

2.41 

2.33 

2.25 

1.88 

2.08 

1.99 

6 

4.22 

3.68 

3.35 

3.11 

2.92 

2.78 

2.66 

2.56 

2.47 

2.39 

2.06 

2.21 

2.11 

7 

4.44 

3.88 

3.52 

3.27 

3.06 

2.92 

2.80 

2.69 

2.60 

2.52 

2.12 

2.33 

2.22 

8 

4.G4 

4.05 

3.69 

3.42 

3.22 

3.06 

2.92 

2.82 

2.72 

2.63 

2.27 

2.43 

2.33 

9 

4.82 

4.22 

3.83 

3.56 

3.35 

3.18 

3.04 

2.93 

2.82 

2.74 

2.40 

2.52 

2.42 

10 

5.00 

4.36 

3.98 

3.69 

3.47 

3.29 

3.15 

3.03 

2.92 

2.83 

2.75 

1.61 

2.50 

12 

5.34 

4.64 

4.22 

3.92 

3.68 

3.50 

3.35 

3.22 

3.11 

3.01 

2.92 

2.78 

2.61 

15 

5.72 

5.00 

4.54 

4.22 

3.97 

3.76 

3.60 

3.47 

3.35 

3.24 

3.15 

3.00 

2.86 

20 

6.30 

5.50 

5.00 

4.64 

4.37 

4.14 

3.97 

3.82 

3.96 

3.57 

3.47 

3.29 

3.15 

25 

6.79 

5.93 

5.39 

5.00 

4.71 

4.46 

4.28 

4.11 

3.97 

3.84 

3.74 

3.55 

3.40 

30 

7.21 

6.30 

5.72 

5.32 

5.00 

5.74 

4.54 

4.37 

4.22 

4.08 

3.97 

3.77 

3.61 

35 

7.59 

6.63 

6.03 

5.60 

5.26 

5.00 

4.78 

4.60 

4.44 

4.30 

4.18 

3.97 

3.80 

40 

7.94 

6.93 

6.30 

5.85 

5.50 

5.22 

5.00 

4.81 

4.65 

4.50 

4.37 

4.15 

3.98 

45 

8.25 

7.20 

6.55 

6.09 

5.73 

5.43 

5.20 

5.00 

4.83 

4.67 

4.54 

4.31 

4.13 

50 

8.55 

7.47 

6.79 

6.30 

5.93 

5.62 

5.38 

5.18 

5.00 

4.84 

4.71 

4.47 

4.28 

GO 

9.08 

7.93 

7.21 

6.70 

6.30 

5.98 

5.72 

5.50 

5.32 

5.15 

5.08 

4.75 

4.55 

70 

9.57 

8.36 

7.59 

7.05 

6.64 

6.30 

6.03 

5.SO 

5.60 

5.42 

5.27 

5.00 

4.79 

80 

10.0 

8.75' 

7.94 

7.37 

6.94 

6.58 

6.30 

6.07 

5.85 

5.67 

5.51 

5.23 

5.00 

90 

10.4 

9.11 

8.25 

7.67 

7.22 

6.84 

6.55 

6.30 

6.10 

5.90 

5.73 

5.44 

5.20 

100 

10.8 

9.41 

8.55 

7.94 

7.47 

7.09 

6.78 

6.53 

6.31 

6.10 

5.93 

5.63 

5.39 

120 

11.4 

10.0 

9.09 

8.44 

7.95 

7.53 

7.21 

6.94 

6.70 

6.49 

6.31 

6.00 

5.73 

140 

12.0 

11.5 

9.57 

8.89 

8.37 

7.93 

7.59 

7.31 

7.06 

6.83 

6.64 

6.30 

6.03 

160 

12.6 

11.0 

10.0 

9.29 

S.74 

8.29 

7.94 

7.64 

7.38 

7.14 

6.94 

6.59 

6.31 

180 

13.1 

11.4 

10.4 

9.67 

9.09 

8.62 

8.26 

7.95 

7.67 

7.42 

7.21 

6.85 

6.55 

20 0 

13.6 

11.9 

10.8 

10.0 

9.42 

8.94 

8.55 

S.23 

7.94 

7.69 

7.48 

7.10 

6.79 

2;0 

14.6 

12.S 

11.6 

10.8 

10.1 

9.63 

9.22 

8.87 

8.56 

8.29 

S.06 

7.65 

7.32 

300 

15.5 

13.6 

12.3 

11.5 

10.8 

10.2 

9.80 

9.43 

9.10 

8.80 

8.57 

8.12 

7.77 

350 

16.3 

14.3 

13.0 

12.0 

11.3 

10.8 

10.3 

9.92 

9.5S 

9.27 

9.00 

8.55 

8.18 

400 

17.1 

15.0 

13.6 

12.6 

11.9 

11.2 

10.8 

10.4 

10.0 

9.69 

9.42 

8.94 

8.55 

450 

17.8 

15.5 

14.1 

13.1 

12.3 

11.7 

11.2 

10.8 

10.4 

10.1 

9.80 

9.30 

8.89 

500 

18.4 

16.1 

14.6 

13.6 

12.8 

12.1 

11.6 

11.2 

10.8 

10.4 

10.1 

9.61 

9.21 

550 

19.0 

16.6 

15.1 

14.0 

13.2 

12.5 

12.0 

11.5 

11.1 

10.8 

10.5 

9.91 

9.50 

GOO 

19.6 

17.1 

15.5 

14.4 

13.5 

12.9 

12.3 

11.9 

11.5 

11.1 

10.8 

10.2 

9.79 

700 

20.7 

18.0 

16.4 

15.2 

14.3 

13.6 

13.0 

12.5 

12.1 

11.7 

11.4 

10.8 

10.3 

800 

21.5 

18.9 

17.1 

15.9 

15.0 

14.2 

13.6 

13.1 

12.6 

12.2 

11.9 

11.9 

10.8 

1000 

23.3 

20.4 

18.5 

17.1 

16.1 

15.3 

14.6 

14.1 

13.6 

13.2 

12.8 

12.2 

11.6 

1200 

24.7 

21.6 

19.6 

19.2 

17.1 

16.3 

15.6 

14.9 

14.5 

14.0 

13.6 

12.9 

12.4 

1500 

26.6 

23.3 

21.1 

19.6 

18.5 

17.5 

16.7 

16.1 

15.5 

15.1 

14.6 

13.9 

13.3 

2000 

29.3 

25.5 

23.5 

21.5 

20.3 

19.3 

18.4 

17.7 

17.1 

16.6 

16.1 

15.3 

14.6 

2500 

31.5 

27.5 

25.0 

23.3 

21.9 

20.8 

19.8 

19.1 

18.4 

17.9 

17.3 

16.5 

15.8 

3000 

33.5 

29.3 

26.6 

24.8 

23.3 

22.1 

21.1 

20.3 

19.6 

19.0 

18.4 

17.5 

16.7 

3500 

35.2 

30.8 

28.0 

26.0 

24.4 

23.3 

22.2 

21.4 

20.7 

20.0 

19.4 

18.4 

17.6 

4000 

36.8 

32.2 

29.3 

27.2 

25.6 

24.3 

23.3 

22.3 

21.6 

20.9 

20.3 

19.3 

18.5 

4500 

38.4 

33.5 

30.4 

28.3 

26.6 

25.2 

24.1 

23.3 

22.4 

21.7 

21.1 

20.0 

19.2 

6000 

39.6 

34.7 

31.5 

29.3 

27.2 

26.1 

25.0 

24.1 

23.3 

22.5 

21.9 

20.8 

19.9 


i 










































Motive ’Force 


415 


Motive Force in Pounds per Horse-Power transmitted in the 
Periphery of Revolving Wheels or Pulleys. 

Diam. 


Revolutions n per Minute of 

Wheel or Pulley. 


Pulley. 

110 

120 

130 

140 

150 

1GO 

170 

180 

190 

200 

Inches . 

F 

F 

F 

F 

F 

F 

F 

F 

F 

F 

1 

1145.9 

1050.4 

969.62 

900.35 

840.33 

789.30 

741.71 

700.30 

663.43 

630.25 

2 

572.96 

525.20 

484.81 

450.17 

420.16 

394.65 

370.85 

350.15 

331.71 

315.12 

3 

381.97 

350.15 

823.21 

300.11 

280.11 

262.60 

247.23 

233.43 

221.14 

210.08 

4 

286.48 

268.10 

242.41 

225.08 

210.08 

197.32 

185.43 

175.75 

165.85 

157.56 

5 

229.18 

210.08 

193.92 

1 S 0.07 

168.06 

159.06 

148.34 

140.06 

132.68 

126.05 

6 

190.98 

175.40 

161.61 

150.06 

140.05 

131.55 

123.61 

116.55 

110.57 

105.04 

7 

163.70 

150.06 

138.52 

128.62 

120.05 

112.55 

105.67 

100.04 

94.779 

90.036 

8 

143.24 

131.55 

121.21 

112.75 

105.04 

98.660 

92.715 

87.700 

82.925 

78.780 

9 

127.33 

116.71 

107.74 

100.05 

90.870 

87.535 

82.410 

77.810 

73.713 

70.026 

10 

114.59 

1105.04 

96.962 

90.035 

84.033 

78.780 

74.171 

70.025 

66.343 

63.025 

11 

104.17 

95.500 

88.147 81.715 

76.392 

66.625 

67.428 

63.350 

60.311 

57.300 

12 

95.500187.533 

80.805 

75.030 

70.025 

65.650 

61.805 

58.405 

55.285 

52.520 

13 

88.147 

'80.800 

74.586 69.255 

64.641 

60.600 

57.056 

53.865 

51.033 

48.481 

14 

81.851 

175.028 

69.260 64.210 

60.025 

56.160 

52.835 

50.020 

47.389 

45.018 

15 

76.394 

70.030 

64.640 

G 0.025 

56.020 

52.520 

49.446 

46.685 

44.226 

42.016 

16 

71.620 

! 65.650 

60.605 

56.375 

52.520 

49.330 

46.357 

43.850 

41.462 

39.390 

18 

63.664 

158.360 

53.870 

50.020 

46.685 

43.768 

41.205 

38.906 

36.856 

35.013 

20 

57.296 

52.520 

48.481 

45.020 

42.016 

39.391 

37.085 

35.014 

33.171 

31.512 

21 

54.567 

50.021 

46.173 

42.873 

40.016 

37.515 

35.223 

33.346 

31.593 

30.012 

24 

47.750 

43.850 

40.402 

37.510 

35.012 

32.825 

30.403 

29.500 

27.642 

26.260 

27 

42.443 

37.794 

35.913 

33.347 

31.125 

29.178 

27.470 

25.927 

24.571 

23.342 

30 

38.197 

35.015 

32.321 

30.013 

28.010 

26.260 

24.723 23.343 

22.114 

21.008 

33 

34.725 

31.833 

29.382 

27.279 

25.462 

23.875 

22.476 

21.222 

20.104 

19.100 

36 

31.832 

29.177 

26.935 

25.011 

23.342 

21.884 

20.603 

18.897 

18.428 

17.506 

39 

29.382 

26.933 

24.862 

23.085 

21.547 

20.200 

19.019 

17.955 

17.011 

16.160 

42 

27.283 

25.010 

23.086 

21.437 

20.008 

18.757 

17.611 

16.676 

15.796 

15.006 

45 

25.466 

23.346 

21.548 

20.008 

18.673 

17.507 

16.482 

15.561 

14.745 

14.005 

48 

23.875 

21.925 

20.201 

18.793 

17.506 

16.442 

15.202 

14.072 

13.821 

13.130 

51 

22.469 

20.596 

19.012 

17.655 

16.477 

15.447 

14.543 

13.731 

13.008 

12.358 

54 

21.222 

18.897 

17.956 

16.673 

15.562 

14.590 

13.735 

12.912 

12.285 

11.671 

60 

19.098 

17.540 

16.169 

15.005 

14.005 

13.005 

12.361 

11.671 

11.057 

10.504 

66 

17.363 

15.916 

14.691 

13.643 

12.731 

11.937 

11.238 

10.611 

10.052 

9.5500 

72 

15.916 

14.585 

13.467 

12.528 

11.671 

10.962 

10.302 

9.7450 

9.2140 

8.7530 

78 

14.691 

13.466 

12.431 

11.543 

10.773 

10.100 

9.5095 

9.4775 

8.5055 

8.0800 

84 

13.642 

12.505 

11.543 

10.718 

10.004 

9.3785 

8.8055 

8.3410 

7.8980 

7.5030 

90 

12.733 

11.671 

10.774 

10.004 

9.3370 

8.7535 

8.2410 

7.7810 

7.3713 

7.0026 

96 

11.938 

10.962 

10.100 

9.3965 

8.7530 

8.2220 

7.6010 

7.3080 

6.911 

6.5650 

102 

11.235 

10.898 

9.5060 

8.8270 

8.2385 

7.7235 

7.2715 

6.8655 

6.5040 

6.1790 

108 

10.611 

9.4485 

8.9780 

8.1935 

7.7810 

7.2930 

6.8675 

6.4840 

6.1425 

5.8355 

114 

10.052 

9.2141 

8.5055 

7.9980 

7.3712 

6.9105 

6.5062 

6.1425 

5.8195 

5.5285 

120 

9.5500 

8.7533 

8.0805 

7.5030 

7.0025 

6.5650 

6.1805 

5.8355 

5.5285 

5.2520 

126 

9.0946 

8.3666 

7.6800 

7.1455 

6.6692 

6.2525 

5.8865 5.5575 

5.2653 

5.0020 

132 

8.6815 

7.9580 

7.3455 

6.8215 

6.3666 

5.9640 

5.6190 5.3055 

5.0260 

4.7750 

138 

8.3038 

7.5815 

7.0268 

6,4985 

6.0652 

5.6860 

5.3747 j 5.0545 

4.8060 

4.5489 

144 

7.9580 

7.2940 

6.7325 

6.2525 

5.8354 

5.4710 

5.1510 4.8630 

4.6070 

4.3766 

150 

7.6394 

7.0030 

6.4640 

6.0025 

5.6020 

5,2520 

4.9446 4.6685 

4.4226 

4.2016 

160 

7.1620 

6.5650 

6.0605 

5.6305 

5.2525 

4.9331 

4.6357 4.4331 

4.1462 

3.9390 

180 

6.3664 

5.8360 

5.3870 

5.0020 

4.6685 

4.3768 

4.1205 

3.8905 

3.6856 

3.5013 

200 

5.7296 

5.2520 

4.8481 

4.5020 

4.2016 

3.9891 

3.7085 

3.5015 

3.3171 

3.1512 

240 

4.7750 

4.3850 

4.0402 

3.7514 

3.5012 

3.2825 

3.0103 

2.9177 

2.7642 

2.6260 













































41G 


Motive Force. 


Motive Force in Pounds per Hutse-Powtr t rails in Itted ill tlie 
Periphery of Itevoiviug Wheels or Pulleys. 


Diam. 

Pulley. 

210 

Re 

220 

volutions » p 
2130 1 240 

er Minute of 
250 | 2G0 

Wheel or Pul 
470 I 2NO 

ey. 

5390 

300 

'uclies- 

F 

F 

F 

i F 

F 

F 

F 

F 

F 

F 

1 

000.23 

572.96 

548.05 

525.20 

504.20 484.81 

466.86 

450.17 

434.00 

420.16 

2 

300.11 

286.48 

274.02 202.00 

252.10 

242.41 

283143 

225.09 

217.33 

210.08 

3 

200.08 

190.99 

182.08 

175.07 

108.00 

101.61 

155.02 

150.00 

144.89 

140.05 

4 

150.00 

143.24 

137.01 

131.55 

120.05 

121.21 

110.71 

112.54 

108.10 

105.04 

5 

120.05 

114.59 

109.01 

105.04 

100.84 

90.900 

93.040 

90.035 

80.932 

84.032 

6 

100.04 

95.490 

91.340 

87.700 

84.030 

80.805 

77.700 

75.030 

72.445 

70.025 

7 

85.740 

81.650 

78.293 

75.030 

72.029 

09.200 

00.093 

04.310 

62.094 

60.023 

8 

75.170 

71.020 

08.505 

05.775 

03.025 

00.005 

58.400 

50.378 

54.080 

52.520 

9 

00.700 

03.005 

00.893 

58.355 

56.020 

53.870 

51.873 

50.025 

48.290 

40.683 

10 

00.023 

57.795 

54.805 

52.520 

50.420 

48.481 

40.083 

45.018 

43.400 

42.010 

11 

54.477 

52.085 

49.822 

47.750 

45.836 

44.073 

42.370 

40.358 

39.514 

38.200 

12 

50.020 

47.750 

45.070 

43.706 

42.015 

40.402 

38.933 

37.515 

30.222 

35.012 

l:i 

40.170 

44.079 

42.157 

40.400 

38.784 

37.293 

35.910 

34.028 

33.435 

32.320 

14 

42.873 

40.926 

39.140 

37.514 

30.014 

34.030 

33.346 

32.155 

31.047 

30.011 

Id 

40.010 

38.197 

30.536 

#>.015 

33.013 

32.320 

31.123 

30.012 

28.977 

28.071 

16 

37.583 

35.810 

34.252 

32.825 

31.512 

30.302 

29.233 

2S.1S8 

27.040 

26.260 

18 

33.343 

31.832 

30.446 

29.180 

28.010 

20.935 

25.937 

25.010 

24.148 

23.341 

20 

30.013 

28.048 

27.402 

20.200 

25.210 

24.241 

23.343 

22.510 

21.733 

21.008 

21 

28.582 

27.284 

20.097 

25.011 

24.010 

23.080 

22.231 

21.439 

20.698 

20.008 

24 

25.000 

23.875 

22.835 

21.925 

21.007 

20.201 

19.800 

18.755 

18.111 

17.506 

27 

22.231 

21.222 

20.297 

18.897 

18.073 

17.956 

17.285 

16.673 

10.099 

15.561 

30 

20.009 

19.099 

18.208 

17.507 

16.806 

10.101 

15.502 

15.009 

14.489 

14.005 

r.3 

18.186 

17.303 

16.0U7 

15.916 

15.280 

14.091 

14.148 

13.040 

13.171 

12.733 

36 

10.073 

15.916 

15.223 

14.588 

14.005 

13.407 

12.598 

12.505 

12.074 

11.677 

39 

15.390 

14.091 

14.052 

13.400 

12.928 

12.431 

11.970 

11.542 

11.145 

10.740 

42 

14.291 

13.642 

13.048 

12.505 

12.005 

11.543 

11.114 

10.718 

10.349 

10.004 

45 

13.339 

12.733 

12.179 

11.073 

11.204 

10.774 

10.374 

10.004 

9.0590 

9.337 

48 

12.529 

11.938 

11.417 

10.902 

10.503 

10.100 

9.3817 

9.3905 

9.0555 

8.7530 

51 

11.770 

11.234 

10.746 

10.298 

9.8862 9.5060 

9.1540 8.8275 

8.5227 

8.2188 

54 

11.110 

10.011 

10.148 

9.4485 

9.3365 

8.9780 

8.6083 8.3388 
| 

8.0495 

7.7805 

60 

10.004 

9.5490 

9.293 

8:7200 

8.4030 

8.0845 

7.7806 8.5028 

7.2445 

7.0025 

(56 

9.1953 

8.0815 

8.3035 

7.9580 

7.0400 

7.3455 

7.0740 6.8215 

0.5855 

0.3000 

72 

8.352 

7.9580 

7.0115 

7.2940 

7.0025 

0.7335 

0.4900 

0.2042 

0.0370 

5.8383 

78 

7.0953 

7.3455 

7.0200 

6.7330 

6.4040 

0.2155 

0.3183 

5.7715 

5.5725 

5.3700 

84 

7.1450 

0.821 

0.5240 

0.2525 

0.0025 

5.7722 

5.5606 5.3592 

5.1745 

5.0020 

90 

0.0093 

6.3005 

0.089 

5.8355 

5.0020 

5.3870 

5.1873 

5.0020 

4.8290 

4.6683 

9(5 

0.2043 

5.9687 

5.7085 

5.4810 

5.2515 

5.050 

4.8720 

4.0982 

4.5278 

4.3765 

102 

5.8840 

5.0175 

5.3730 

5.1490 

4.9431 

4.7530 

4.5770 

4.4735 

4.2013 

4.1094 

108 

5.4023 

5.3055 

5.0740 

4.7242 

4.0682 

4.4890 

4.3226 

4.0908 

4.0248 

3.8902 

114 

5.3320 

5.0200 

4.8074 

4.6071 

4.4228 

4.2527 

4.0950 

3.15GOO 

3.8128 

3.6856 

120 

5.0020 

4.7750 

4.5670 

4.3766 

4.2015 

4.0402 

3.8903 

3.7505 

3.6222 

3.5012 

12(5 

4.7033 

4.5473 

4.3490 

4.1833 

4.0016 

3.8400 

3.7050 

3.5725 

3.4497 

3.3346 

132 

4.5470 

4.3408 

4.1517 

3.9790 

3.8200 3.6727 

3.5370 

3.4108 

3.2928 

3.1S33 

138 

4.3323 

4.1519 

3.9713 

3.7907 

3.0392 3.5131 

3.3696 3.2492 

3.1555 

3.0326 

144 

4.1083 

3.9790 

3.8057 

3.0472 

3.5012 

3.3662 

3.2420 

3.1202 

3.0185 

2.9177 

150 

4.0016 

3.8197 

3.6536 

3.5015 

3.3013 3.2320 

3.1123 

3.0612 

2.8977 

2.8011 

160 

3.7533 

3.5810 

3.4252 

3.2825 

3.1512 

3.0302 

2.9887 

2.8152 

2.7010 

2.6260 

180 

3.3346 

3.1832 

3.0 446 

2.9180 

2.8010 2.0935 

2.5937 

2.5010 

2.4148 

2.3341 

200 

3.0013 

2.8048 

2.7402 

2.6260 

2.5210 2.4241 

2.3343 

2.255 

2.1733 

2.1008 

240 

2.5009. 

2.3875 

2.2835 

2.1925 

2.1007 2.0201 

1.9451 

1.8757 

1.8111 

1.7506 


































Mo'i . Vi, i : E. 


417 


Motive Force in 

Pounds pei 

Horse-Power transmitted 

in the 


Periphery of Revolving 

Wheels or 

Pulleys. 


Diam. 


Revolutions n per Minute of Wheel or Pulley. 


Pulley. 

310 

320 

330 

340 

35 0 

3G0 

3?0 

380 

390 

400 

Indies. 

F 

F 

F 

F 

F 

F 

F 

F 

F 

F 

1 

406.62 

394.65 

381.97 

370.85 

360.14 

350.15 

340.68'331.71 

323.21 

315.12 

2 

203.31 

197.33 

190.99 

185.43 

180.07 

175.07 

170.34 

165.85 

161.60 

157.56 

3 

135.54 

131.30 

127.32 

123.62 

120.05 116.71 

113.23 

1 10.57 

107.73 

105.04 

4 

101.66 98.660 

95.495 

92.715 

90.035137.537 

85.170 82.925 

80.803 

78.780 

5 

81.324 

79.530 

76.394 

74.170 

72.028 

70.030 

68.136 

66.340 

64.840 

63.024 

6 

67.770 

65.775 

63.660 

61.805 

60.025 

58.275 

56.515 

55.285 

53.533 

52.520 

7 

58.090 

56.275 

54.566 

o2i835 

51.449 

50.020 

48.669 

47.389 

46.173 

45.017 

8 

50.830 

49.330 

47.747 

46.357 

45.017 

43.850 

42.585 

41.462 

40.403 

39.390 

9 

45.180 

43.767 

42.443 

41.205 

40.016 

38.905 

37.743 

36.856 

35.913 

35.013 

10 

40.662 

39.390 

38.197 

37.085 

36.014 

35.012 

34.068 

33.172 

32.321 

31.512 

11 

36.965 

33.312 

34.723 

33.714 

32.740 

31.775 

30.971 

30.155 

29.382 

28.650 

12 

33.885 

32.825 

31.833 

30.902 

30.012 

29.202 

28.308 

27.642 

26.935 

26.260 

13 

31.278 

30.300 

29.382 

28.528 

27.703 

26.932 

26.206 

25.516 

24.862 

24.240 

14 

29.045 

28.080 

27.284 

26.417 

25.724 

25.010 

24.335 

23.699 

23.086 

22.508 

15 

27.108 

26.260 

2o.465 

24.723 

24.093 

23.342 

22.712 

22.113 

21.546 

21.008 

16 

25.415 

24.665 

23.873 

23.178 

22.508 

21.925 

21.293 

20.731 

20.202 

19.695 

18 

22.590 

21.884 

21.221 

20.602 

20.008 

19.453 

18.871 

18.428 

17.956 

17.506 

20 

20.331 

19.695 

19.099 

18.543 

18.007 

17.507 

17.034 

16.585 

16.160 

15.756 

21 

19.363 

18.757 

18.189 

17.612 

17.149 

16.673 

16.223 

15.796 

15.391 

15.006 

24 

16.942 

16.412 

15.916 

15.202 

15.066 

14.850 

14.154 

13.821 

13.467 

13.130 

27 

15.060 

14.589 

14.146 

13.735 

13.338 

12.963 

12.581 

12.285 

11.971 

11.671 

30 

13.554 

13.130 

12.733 

12.362 

12.046 

11.671 

11.323 

11.057 

10.774 

10.504 

33 

12.322 

11.937 

11.575 

11.238 

10.914 

10.611 

10.323 

10.052 

9.7940 

9.5500 

36 

11.295 

10.942 

10.611 

10.302 

10.004 

9.4485 

9.4355 

9.2140- 

8.9783 

8.7533 

39 

10.426 

10.100 

9.7940 

9.5095 

9.2343 

8.9775 

8.7353 

8.5055 

8.2873 

8.0800 

42 

9.6815 

9.3785 

9.0943 

8.8055 

8.5745 

8.3380 

8.1115 

7.8980 

7.6953 

7.5030 

45 

9.0360 

8.7535 

8.4887 

8.2410 

8.0310 

7.7805 

7.5707 

7.3725 

7.1826 

7.0026 

48 

8.47 3018.2210 

7.9583 

7.6010 

7.5030 

7.0360 

7.0770 

6.9105 

6.7336 

6.5650 

51 

7.9728 

7.7235 

7.4897 

7.2715 

7.0620 

6.8655 

6.7995 

6.5040 

6.3373 

6.1790 

54 

7.530 

7.2950 

7.0740 

6.8675 

6.6690 

6.4560 

6.2905 

6.1425 

5.9S53 

5.8355 

60 

6.7770 

6.5025 

6.3660 6.1805 

6.0025 

5.S355 

5.6615 

5.5285 

5.3583 

5.2520 

66 

6.1610 

5.9685 

5.7843 5.6190 

5.4570 

5.3055 

5.1615 

5.0260 

4.8970 

4.7750 

72 

5.6475 

5.4810 

5.3053 

5.1510 

5.0020 

4.8725 

4.7178 

4.6070 

4.4890 

4.3766 

78 

5.2130 

5.0500 

4.8970 4.7047 

4.6171 

4.7387 

4.3677 

4.2527 

4.1436 

4.0400 

84 

4.8408 

4.6892 

4.5473 

4.4027 

4.2870 

4.1705 

4.055S 

3.9490 

3.8470 

3.7515 

90 

4.5180 

4.3767 

4.2443 

4.1205 

4.0016 

3.8905 

3.7743 

3.6856 

3.5913 

3.5013 

96 

4.2355 

4.1110 

3.9793 

3.8005 

3.7515 

3.6d40 

3.5385 

3.4555 

3.3666 

3.2825 

102 

3.9864 

3.8617 

3.7450 

3.6357 

3.5310 

3.4325 

3.3998 

3.2520 

8 .I 6861 

3.0895 

108 

3.7650 

3.64U1 

3.5370 

3.4337 

3.3345 

3.2420 

3.1453 

3.0712 

2.9926 

2.9179 

114 

3.566813.4552 

3.3507 

3.2531 

3.1992 

3.0712 

2.9814 

2.9097 

2.8352 

2.7642 

120 

3.38S5 

3.2825 

3.1833 

3.0903 

3.0012 

2.9177 

2.8308 

2.7042 

2.6935 

2.6260 

126 

3.2272 

3.1262 

3.0315 

2.9433 

2.8582 

2.7787 

2.7038 2.6326 

2.5600 

2.5010 

132 

3.0805 

2.9820 

2.S938 

2.8095 

2.7286 

2.6527 

2.5808 

2.5130 

2.4485 

2.3875 

138 

2.9465 

2.8430 

2.7679 

2.6873 

2.5994 

2.5272 

2.4686 

2.4030 

2.3421 

2.2744 

144 

2.8237 

2.7355 

2.6526 

2.5755 

2.5010 

2.4315 

2.3589 

2.3035 

2.2442 

2.1883 

150 

2.7108 2.6260 

2.5465 

2.4726 

2.4093 

2.3342 

2.2712 

2.2113 

2.1546 

2.1008 

160 

2.5415 

2.4665 

2.3873 

2.3178 

2.2508 

2.2165 

2.1293 

2.0731 

2.0202 

1.9695 

180 

2.2090 2.1884 

2.1221 

2.0603 

2.0008 

1.9452 

1.8871 

1.8428 

1.7956 

1.7506 

200 

2.0331 

1.9945 

1.9099 

1.8543 

1.8007 

1.7507 

1.7034 

1.6585 

1.6160 

1.5756 

240 

1.69421 

1.6412 

1.5917 

1.5202 

1.5006 

1.4588 

1.4154 

1.3821 

1.3467 

1.3130 


27 


















































413 


He ksk -Power 


Horse-Power for Different Motive Forces F and Velocities F. 


Motive 

Foi’ce. 

10 

20 

5 

30 

7elocit' 

40 

/ V in 

50 

Feet pe 

60 

r Secoi 

70 

id. 

SO 

90 

100 

F lbs. 

IP 

IP 

IP 

IP 

IP 

IP 

IP 

IP 

IP 

IP 

10 

.18182 

.36364 

.54545 

.72727 

.90909 

1.0909 

1.2727 

1.4545 

1.6364 

1.8182 

20 

.36364 

.72727 

1.0909 

1.4545 

1.8182 

2.1818 

2.5454 

2.9091 

3.3636 

3.6364 

30 

.54545 

1.0909 

1.6363 

2.1818 

2.7273 

3.2727 

3.8182 

4.3636 

4.9091 

5.4545 

40 

.72727 

1.4444 

2.1818 

2.9091 

3.6363 

4.3636 

5.0909 

5.8182 

6.545 

7.2727 

50 

.90909 

1.8182 

2.7273 

3.6363 

4.5454 

5.4545 

6.3636 

7.2727 

8.1818 

9.0909 

GO 

1.0909 

2.1818 

3.2727 

4.3636 

5.4545 

6.5454 

7.6363 

8.7272 

9.8182 

10.909 

70 

1.2727 

2.5454 

3.8182 

5.0909 

6.3636 

7.6363 

8.9091 

10.181 

11.454 

12.727 

80 

1.4545 

2.9091 

4.3636 

5.8182 

7.2727 

8.7273 

10.182 

11.636 

13.091 

14.545 

90 

1.6364 

3.2727 

4.9091 

6.5454 

8.1818 

9.8182 

11.454 

13.091 

14.727 

16.364 

100 

1.8182 

3.6364 

5.4545 

7.2727 

9.0909 

10.909 

12.727 

14.545 

16.364 

18.182 

110 

2.0000 

4.0000 

6.0000 

8.0000 

10.000 

12.000 

14.000 

16.000 

18.000 

20.000 

120 

2.1818 

4.3636 

6.5454 

8.7273 

10.909 

13.091 

15.273 

17.454 

19.636 

21.818 

130 

2.3636 

4.7273 

7.0909 

9.4545 

11.818 

14.182 

16.545 

18.909 

21.273 

23.636 

140 

2.5454 

5.0909 

7.6363 

10 .1S2 

12.727 

15.273 

17.818 

20.363 

22.909 

25.454 

150 

2.7273 

5.4444 

8.1818 

10.909 

13.636 

16.363 

19.091 

21.818 

24.545 

27.273 

100 

2.9091 

5.8182 

8.7273 

11.636 

14.545 

17.454 

20.363 

23.272 

26.182 

29.091 

170 

3.0909 

6.1818 

9.2727 

12.363 

15.454 

18.545 

21.636 

24.727 

27.818 

30.909 

180 

3.3636 

6.5454 

9.8182 

13.091 

16.363 

19.636 

22.909 

26.182 

29.454 

33.636 

190 

3.5454 

6.9091 

10.364 

13.818 

17.273 

20.727 

24.182 

27.636 

31.091 

35.454 

200 

3.6364 

7.2727 

10.909 

14.545 

18.182 

21.818 

25.454 

29.091 

33.636 

36.364 

210 

3.8182 

7.6363 

11.454 

15.273 

19.091 

22.909 

26.727 

30.545 

35.363 

38.182 

220 

4.0000 

8.0000 

12.000 

16.000 

20.000 

24.000 

28.000 

32.000 

36.000 

40.000 

230 

4.1818 

8.3636 

12.545 

16.727 

20.909 

25.091 

29.273 

33.454 

37.636 

41.818 

240 

4.3634 

8.7272 

13.091 

17.454 

21.818 

26.182 

30.545 

34.909 

39.273 

43.634 

250 

4.5454 

9.0909 

13.636 

18.182 

22.727 

27.273 

31.818 

36.363 

40.909 

45.454 

2G0 

4.7273 

9.4444 

14.181 

18.909 

23.686 

28.363 

33.091 

37.818 

42.545 

47.273 

270 

4.9091 

9.8182 

14.727 

19.636 

24.545 

29.454 

34.363 

39.273 

44.182 

49.091 

280 

5.0909 

10.182 

15.273 

20.363 

25.454 

30.545 

35.636 

40.727 

45.818 

50.909 

290 

5.2727 

10.545 

15.818 

21.091 

26.363 

31.636 

36.909 

42.182 

47.454 

52.727 

300 

5.4545 

10.909 

16.363 

21.818 

27.273 

32.727 

38.182 

43.636 

49.091 

54.545 

310 

5.6364 

11.273 

16.909 

22.545 

28.182 

33.818 

39.454 

45.091 

50.727 

56.364 

320 

5.8182 

11.636 

17.454 

23.273 

29.091 

34.909 

40.727 

46.545 

52.363 

58.182 

330 

6.0000 

12.000 

18.000 

24.000 

30.000 

36.000 

42.000 

48.000 

54.000 

60.000 

340 

6.1818 

12.363 

18.545 

24.727 

30.909 

37.091 

43.273 

49.454 

55.636 

61.818 

350 

6.3636 

12.727 

19.091 

25.454 

31.818 

38.182 

44.545 

50.909 

57.273 

63.636 

360 

6.5454 

13.091 

19.636 

26.182 

32.727 

39.273 

45.818 

52.363 

58.909 

65.454 

370 

6.7273 

13.444 

20.182 

26.901 

33.636 

40.363 

47.091 

53.818 

60.545 

67.273 

380 

6.9091 

13.818 

20.727 

27.636 

34.545 

41.454 

48.363 

55.273 

62.182 

69.091 

390 

7.0909 

14.182 

21.273 

28.363 

35.454 

42.545 

49.636 

56.727 

63.818 

70.909 

400 

7.2727 

14.545 

21.818 

29.091 

36.363 

43.636 

50.909 

58.182 

65.454 

72.727 

410 

7.4545 

14.909 

22.363 

29.818 

37.273 

44.727 

52.182 

59.636 

67.091 

74.545 

420 

7.6364 

15.273 

22.909 

30.545 

38.182 

45.818 

53.454 

61.091 

68.727 

76.364 

430 

7.8182 

15.636 

23.454 

31.273 

39.091 

46.909 

54.727 

62.545 

70.363 

78.182 

440 

8.0000 

16.000 

24.000 32.000 

40.000 

48.000 

56.000 

64.000 

72.000 

80.000 

450 

8.1818 

16.363 

24.545 32.727 

40.909 

49.091 

57.272 

65.454 

73.636 

81.818 

460 

8.3636 

16.727 

25.091 

33.454 

41.818 

50.182 

58.545 

66.909 

75.273 

83.636 

470 

8.5454 

17.091 

25.636 

34.182 

42.727 

51.273 

59.818 

68.363 

77.909 

85.454 

480 

8.7273 

17.444 

26.182 

34.909 

43.636 

52.363 

61.091 

69.818 

79.545 

87.273 

490 

8.9091 

17.818 

26.727 

35.636 

44.546 

53.454 

62.363 

71.273 

81.182 

89.091 

500 

9.0909 

18.182 

27.273 

36.363 

45.454 

54.545 

63.636 

72.727 

82.818 

90.909 






























Diameters of Wrought-Irox Shafts. 421 



Diameters in Indies of Wronglit-Iron Shafts 

. 


Ilorse 


Number of revolutions per minute of wrought-iron shafts . 


power . 

100 

125 

150 

175 

200 

250 

300 

350 

400 

500 

600 

800 

1000 

IP . 

la. 

In. 

In. 

In. 

In. 

In. 

In. 

In. 

In. 

In. 

In. 

In. 

In. 

1 

1.08 

1.00 

0.94 

0.90 

0.85 

0.80 

0.75 

0.71 

0.68 

0.63 

0.59 

0.54 

0.5 

2 

1.35 

1.26 

1.19 

1.13 

1.08 

1.00 

0.94 

0.90 

0.86 

0.80 

0.75 

0.68 

0.63 

3 

1.55 

1.44 

1.36 

1.29 

1.24 

1.15 

1 . 0 S 

1.03 

0.98 

0.91 

0.86 

0.78 

0.72 

4 

1.71 

1.59 

1.50 

1.42 

1.36 

1.26 

1.19 

1.13 

1.08 

1.00 

0.94 

0.86 

0.80 

5 

1.84 

1.71 

1.61 

1.53 

1.46 

1.36 

1.28 

1.22 

1.16 

1.08 

1.02 

0.92 

0.86 

6 

1.96 

1.82 

1.71 

1.62 

1.56 

1.44 

1.36 

1.29 

1.24 

1.15 

1.08 

0 . 9 S 

0.91 

7 

2.06 

1.92 

1.80 

1.71 

1.64 

1.52 

1.43 

1.36 

1.30 

1.21 

1.14 

1.03 

0.96 

8 

2.15 

2.00 

1.88 

1.79 

1.71 

1.59 

1.49 

1.42 

1.36 

1.26 

1.19 

1.08 

1.00 

9 

2.24 

2.08 

1.96 

1.86 

1.78 

1.65 

1.55 

1.48 

1.41 

1.31 

1.24 

1.12 

1.04 

10 

2.22 

2.16 

2.03 

1.93 

1.85 

1.71 

1.61 

1.53 

1.47 

1.36 

1.28 

1.16 

1.08 

12 

2.47 

2.29 

2.15 

2.05 

1.96 

1.82 

1.71 

1.63 

1.56 

1.44 

1.36 

1.24 

1.15 

15 

2.66 

2.47 

2.32 

2.20 

2.11 

1.96 

1.85 

1.75 

1.67 

1.55 

1.46 

1.33 

1.24 

20 

2.92 

2.71 

2.56 

2.43 

2.33 

2.15 

2.03 

1.93 

1.84 

1.71 

1.61 

1.46 

1.36 

25 

3.15 

2.92 

2.75 

2.61 

2.5 

2.33 

2.19 

2.06 

1.98 

1.84 

1.74 

1.57 

1.46 

30 

3.34 

3.11 

2.92 

2.78 

2.66 

2.47 

2.33 

2.21 

2.11 

1.96 

1.84 

1.68 

1.55 

35 

3.52 

3.27 

3.08 

2.93 

2.80 

2.60 

2.44 

2.33 

2.12 

2.06 

1.94 

1.76 

1.64 

40 

3.68 

3.42 

3.22 

3.06 

2.92 

2.71 

2.56 

2.43 

2.33 

2.15 

2.03 

1.85 

1.71 

45 

3.83 

3.56 

3.34 

3.18 

3.04 

2.82 

2.66 

2.53 

2.42 

2.24 

2.11 

1.92 

1.78 

50 

3.97 

3.69 

3.47 

3.29 

3.15 

2.92 

2.75 

2.62 

2.50 

2.33 

2.19 

1.99 

1.84 

60 

4.21 

3.91 

3.68 

3.50 

3.35 

3.11 

2.93 

2.78 

2.66 

2.47 

2.33 

2.11 

1.96 

70 

4.44 

4.12 

3.88 

3.69 

3.53 

3.27 

3.06 

2.93 

2 .SO 

2.60 

2.44 

2.22 

2.06 

80 

4.64 

4.31 

4.05 

3.86 

3.63 

3.42 

3.22 

3.06 

2.93 

2.71 

2.55 

2.33 

2.15 

90 

4.82 

4.48 

4.22 

4.01 

3.83 

3.56 

3.35 

3.18 

3:04 

2.82 

2.61 

2.42 

2.24 

100 

5.00 

4.64 

4.37 

4.15 

3.97 

3.68 

3.47 

3.29 

3.15 

2.92 

2.75 

2.50 

2.31 

120 

5.31 

4.94 

4.64 

4.41 

4.22 

3.92 

3.68 

3.50 

3.35 

3.11 

2.92 

2.66 

2.46 

140 

5.59 

5.20 

4.89 

4.64 

4.44 

4.12 

3.88 

3.68 

3.52 

3.27 

3.08 

2.80 

2.59 

160 

5.85 

5.43 

5.10 

4.85 

4.64 

4.31 

4.05 

3.85 

3.68 

3.42 

3.22 

2.92 

2.71 

180 

6.08 

5.65 

5.31 

5.05 

4.83 

4.48 

4.22 

4.01 

3.83 

3.56 

3.35 

3.04 

2.82 

200 

6.30 

5.85 

5.50 

5.23 

5.00 

4.64 

4.37 

4.15 

3.97 

3.68 

3.47 

3.15 

2.92 

250 

6.78 

6.30 

5.93 

5.51 

5.39 

5.00 

4.70 

4.47 

4.27 

3.97 

3.73 

3.39 

3.15 

300 

7.21 

6.69 

6.30 

6.00 

5.73 

5.31 

5.00 

4.75 

4.54 

4.11 

3.97 

3.61 

3.35 

350 

7.59 

7.05 

6.63 

6.30 

6.03 

5.59 

5.41 

5.00 

4.78 

4.44 

4.28 

3.80 

3.52 

400 

7.94 

7.37 

6.93 

6.59 

6.30 

5.85 

5.50 

5.23 

5.00 

4.64 

4.37 

3.92 

3.68 

450 

8.26 

7.66 

7.21 

6 . S 5 

6.55 

6.08 

5.72 

5.44 

5.20 

4.83 

4.54 

4.13 

3.88 

500 

8.55 

7.94 

7.46 

7.10 

6.79 

6.30 

5.93 

5.63 

5.39 

5.00 

4.71 

4.27 

3.97 

550 

8.82 

8.19 

7.71 

7.32 

7.01 

6.50 

6.12 

5.81 

5.56 

5.16 

4.86 

4.41 

4.10 

600 

9.08 

8.43 

7.92 

7.54 

7.21 

6.69 

6.30 

6.00 

5.72 

5.32 

5.00 

4.54 

4.22 

700 

9.56 

S .88 

8.35 

7.94 

7.59 

7.05 

6.63 

6.30 

6.03 

5.59 

5.26 

4.78 

4.44 

800 

10.0 

9.28 

8.74 

8.30 

7.94 

7.37 

6.93 

6.59 

6.30 

5.85 

5.50 

5.00 

4.64 

1000 

10.8 

10.0 

9.41 

8.94 

8.55 

7.94 

7.47 

7.09 

6.79 

6.30 

5.93 

5.39 

5.00 

1200 

11.5 

10.6 

10.0 

9.50 

9.09 

8.43 

7.94 

7.54 

7.21 

6.69 

6.30 

5.72 

5.31 

1500 

12.3 

11.5 

10.7 

10.3 

9.79 

9.09 

8.55 

8.12 

7.77 

7.21 

6.79 

6.17 

5.72 

2000 

13.5 

12.6 

11.8 

11.2 

10.8 

10.0 

9.41 

8.94 

8.55 

7.94 

7.47 

6.79 

6.30 

2500 

14.6 

13.5 

12.8 

12.2 

11.6 

10.8 

10.2 

9.63 

9.21 

8.55 

8.05 

7.31 

6.73 

3000 

15.5 

14.4 

13.5 

12.9 

12.3 

11.4 

10.8 

10.2 

9.79 

9.09 

8.55 

7.77 

7.41 

3500 

16.4 

15.2 

14.3 

13.5 

13.0 

12.1 

11,3 

10.8 

10.3 

9.56 

9.00 

8.18 

7.59 

4000 

17.1 

15.9 

15.0 

14.2 

13.6 

12.6 

11.8 

11.3 

10.8 

10.0 

9.41 

8.55 

7.94 

4500 

17.8 

16.5 

15.5 

14.7 

14,1 

13.1 

12.3 

11.7 

11.2 

10.4 

9.79 

8.89 

8.25 

5000 

18.4 

17.1 

16.1 

15.3 

14 . 6113.5 

12.8 

12.1 

11.6 

11.8 

10.1 

9.21 

8.55 







































422 


Pile Drivtng. 


PILE DRIVING. 

Notation. 

M = weight of the ram in pounds. 

S= fall of the ram in feet. 

V = velocity of pile and ram together. 

s = space in inches which the pile sinks by the blow. 
r = resistance of the ground in pounds to entrance of the pile. 
a = section area in sq. in. of the pile, sharpened to a point not more 
than 46°. 

k — coefficient for the hardness of the ground. 
h = depth to which the pile is driven. 

W = weight in pounds which a driven pile can bear with safety after 
the last blow when the pile sunk s inches. 

V = velocity in feet per second by which the ram strikes the pile. 

Ram and pilehead considered non-elastic and perfectly hard. 


V=S\/ s 


1. 


VMS 


w= 


2 MS 


8 M\/S 
M + m 


r = 


12 MS 


r — ak \/h, 


6 . 


Example 1. A wooden pile 18 feet long by 12 inches square, driven 
h=12 feet into common natural ground imbedded with tenacious clay for 
which may be assumed the coefficient k =50. Required how much the 
pile will set s=I into the ground at a blow with a ram of M=3500 lbs. 
falling S—42 inches. 

The weight of the wooden pile will be about m=18X40=720 lbs. 

Area of the pile a=144 square inches. 

Resistance r— 144X50/12 =23840 lbs. 

The resistance sought from this formula 6, cannot be depended upon 
lor calculating the weight the pile can bear with safety. 

™ . 12X3^X3500 „„„ 

The set s --23340-“ 6 X 

Suppose the set to he s — 0 inches at the last blow, required what weight 
the pile can bear with safety ? 

2 X 3* X 3500 


TF = 


= 4083 lbs. 


This can be depended on with safety, if calculated from the actual set 
of the pile at the last blow. 

For ordinary pile driving a heavy ram and short fall is the most effec¬ 
tive, but in some cases when the ground itself is elastic, or when driving 
piles in pure sand it is found more advantageous to use a high fall ot 
the ram. 

Approximate Coctlieients. 

In coral formations,. 

In hard clay with gravel, . . . 

In hard pure clay,. 

In common clay or sand, . . . 

In soft clay or loose sand, . . . 

In very loose materials, . . . 


k 

120 

100 

70 

60 

40 

80 


















Ropes for Transmitting Power. 


423 


ROPES FOR TRANSMITTING POWER. 

The following tables of properties of ropes for transmitting power are 
deduced from experiments made by the author expressly for this Pocket- 
book. John A. Roebling’s Sons & Co., of Trenton, N. J., kindly furnished the 
wire ropes experimented upon, and Edwin H. Fitler & Co., of Philadelphia, 
furnished a variety of hemp, manilla, and cotton ropes. 

The heavy iron and steel cables were kindly furnished by the Philadelphia 
and Reading Railroad Company. 

The subject of transmission of power with ropes is an important one, but 
too extensive to be fully treated in this Pocket-book. The most important 
points are given in the accompanying tables. 

Column D contains the minimum diameter of pulley, w.heel, or drum that 
should be used for the maximum diameter of rope in the next column, d. 
The pulley may be made larger for the same size or smaller rope, and the 
larger the better for the rope. The heading of each column explains the 
contents. 

The wear of a rope is proportionate to its stiffness—that is, as the cube cfl 
of its diameter, and inversely as the square D 2 of the diameter of the pulley. 

d 3 

Wear of rope = — i 
D i 

d = diameter in inches of the rope, and D ■= diameter in feet of the pulley. 

Column 4* contains the stiffness of the rope in pounds for both winding and 
unwinding on each pulley; when the rope runs over a number of pulleys, as 
when power is transmitted for long distances, the stiffness must be added for 
each pulley. 

The dynamics of transmission of power by belting, ropes, or chains is cal¬ 
culated by the formulas for circular motion. 

Circumference of Ropes. 

The circumference of a rope, as practically measured by a tape-line, is not 
3.14 times the diameter of the rope, but is considerably less, depending upon 
the number of strands in the rope. 

d = diameter and c = circumference of the rope. 


Two strands. n = 2.57. c — 2.57 d. d = 0.389 c. 


Three strands, n = 2.86. c = 2.86 d. d = 0.35 c. 


Four strands, n = 2.96. c = 2.96 d. d = 0.338 c. 


Seven strands, n = 3. c = 3 d. d — } c. 


The diameter of the rope is that of the circle tangenting the strands, whilst 
the circumference is the sum of the lines drawn between the strands. 











424 


Ropes, 



Hemp Ropes, White. 

Three 

Strands. 


Diam. 

Size of Rope. 

Strength. 

Weight 

Length 

Stiffness. 

Pulley. 

Diam. 

Circum. 

Break. 

Safety. 

per Ft. 

per Lb. 

Wind- 

Wind 

Feet. 

Inches. 

Inches. 

Pounds. 

Pounds. 

Pounds. 

Feet. 

ing. 

and 

Unwind. 

D 

d 

c 

S 

T 

IV 

1 

4> 

4> 

21. 

6 in. 

17.1 

324000 

81000 

9.4 

.1064 

7.6 

12.7 

19. 

5* 

15.7 

272000 

68000 

7.9 

.1266 

6.6 

11. 

16.5 

5 in. 

14.25 

225000 

56250 

6.52 

.1533 

6. 

10. 

14. 

4£ 

12.1 

182000 

45500 

5.28 

.1894 

5.4 

9. 

12. 

4 in. 

• 11.4 

144000 

36000 

4.18 

.2392 

4.6 

7.7 

11. 

3# 

10.7 

126000 

31500 

3.67 

.2725 

4.2 

7. 

10. 

3* 

10. 

110000 

27500 

3.2 

.3125 

3.9 

6.5 

9. 

si 

9.27 

95000 

23750 

2.76 

.3613 

3.6 

6. 

8. 

3 in. 

8.57 

81000 

20250 

2.35 

.4255 

3.36 

5.6 

7. 

2$ 

7.85 

6S000 

17000 

1.97 

.5076 

3.05 

5.1 

6. 

2 h 

7.14 

56200 

14050 

1.63 

.6135 

2.82 

4.7 

5.25 

2 i 

6.43 

45500 

11375 

1.32 

.7575 

2.4 

4. 

4.25 

2 in. 

5.70 

36000 

9000 

1.04 

.9615 

2.25 

3.75 

3.4 

11 

5.00 

27500 

6875 

0.80 

1.25 

2.1 

3.5 

2.75 

H 

4.28 

20200 

5050 

0.588 

1.700 

1.74 

2.9 

2.1 

i£ 

3.97 

14000 

3500 

0.407 

2.457 

1.44 

2.4 

1.5 

1 in. 

2.86 

9000 

2250 

0.261 

3.831 

1.16 

1.93 

1.22 

7 

T 

2,5 

6900 

1725 

0.200 

5.000 

1.02 

1.7 

0.97 

# 

2.14 

5050 

1262 

0.147 

6.803 

0.87 

1.46 

0.74 

% 

1.78 

3500 

875 

0.102 

9.803 

0.72 

1.21 

0.53 

l 

1.43 

2240 

560 

0.0(15 

15.38 

0.58 

0.97 

0.34 

1 

1.07 

1260 

315 

0.036 

27.77 

0.45 

0.75 

0.18 

X 

4 

0.71 

560 

140 

0.016 

62.5 

0.31 

0.52 

Manilla Hopes. Three Strands. 

Diam. 

Size of Rope. 

Strength. 

Weight 

Length 

Stiffness. 

Pulley. 

Diam. 

Circum. 

Break. 

Safety. 

per Ft. 

per Lb. 

Wind- 

Wind 

Feet. 

Inches. 

Inches. 

Pounds. 

Pounds. 

Pounds. 

Feet. 

ing. 

and 









Unwind. 

D 

d 

c 

a 

T 

XV 

1 

4> 

<1> 

26.4 

6 in. 

17.1 

216000 

54000 

8.64 

.1157 

5.37 

8.87 

23.2 

5* 

15.7 

181500 

45375 

7.26 

.1377 

5.00 

8.26 

20. 

5 in. 

14.25 

150000 

37500 

6.00 

.1666 

4.5 

7.45 

17.2 

44 

12.1 

121000 

30250 

4.86 

.2057 

4.00 

6.62 

14.4 

4 in. 

11.4 

96000 

24000 

3.84 

.2604 

3.57 

5.9 

13. 

3? 

10.7 

84-100 

21100 

3.38 

.2958 

3.37 

5.56 

11.8 

34 

10. 

73600 

18400 

2.94 

.3401 

3.10 

5.15 

10.5 

34 

9.27 

63500 

15875 

2.53 

.3952 

2.93 

4.85 

9.35 

3 in. 

8.57 

54000 

13500 

2.16 

.4629 

2.68 

4.43 

8.2 

2J 

7.85 

45400 

11350 

1.81 

.5524 

2.45 

4.06 

7.1 

24 

7.14 

37500 

9375 

1.5 

.6666 

2.24 

3.70 

6. 

24 

6.43 

30400 

7600 

1.21 

.8264 

2.06 

3.4 

5. 

2 in. 

5.70 

24000 

6000 

0.96 

1.041 

1.85 

3.07 

4. 

14 

5.00 

18400 

4600 

0.725 

1.379 

1.700 

2.8 

3.3 

H 

4.28 

13500 

3350 

0.54 

1.852 

1.40 

2.32 

2.5 

14 

3.57 

9380 

2345 

0.375 

2.666 

1.11 

1.84 

1.8 

1 in. 

2.86 

6000 

1500 

0.24 

4.166 

0.8870 

1.47 

1.46 

7 

¥ 

2.5 

4600 

1150 

0.184 

5.435 

0.894 

1.31 

1.17 

4 

2.14 

3380 

845 

0.135 

7.407 

0.6666 

1.10 

0.89 

$ 

1.78 

2350 

587 

0.093 

10.75 

0.558 

0.92 

0.63 

1 

a 

1.43 

1500 

o/D 

0.060 

16.66 

0.454 

0.75 

0.41 

I- 

1.07 

845 

211 

0.033 

30.30 

0.36 

0.56 

0.22 

l 

4 

0.71 

375 

93 

0.015 

66.66 

0.23 

0.38 















































Ropks. 


425 



Tarred Hemp Ropes. 

Four 

Strands. 

, 

Diam. 

Size of Rope. 

Strength. 

Weight 

Length 

Stiffness. 

Pulley. 

Diam. 

Circum. 

Break. 

Safety. 

per Ft. 

per Lb. 

Wind- 

Wind 

Feet. 

Inches. 

Inches. 

Pounds. 

Pounds. 

Pounds. 

Feet. 

ing. 

and 







Unwind. 

D 

d 

C 

S 

T 

IV 

1 

</> 

4> 

36. 

6 in. 

18 in. 

230000 

57500 

15.1 

.0662 

13.3 

18.9 

32. 

5i 

16i 

194000 

48500 

12.7 

.0784 

12 . 

17. 

28. 

5 in. 

15 in. 

160000 

40000 

10.5 

.0952 

10.6 

15.1 

24. 

4i 

13? 

130000 

32500 

8.52 

.1174 

9.5 

13.5 

20 . 

4 in. 

12 in. 

102500 

25625 

6.72 

.1488 

8.5 

12.1 

18. 

3$ 

lli 

90000 

22500 

5.92 

.1689 

8.1 

11.5 

16. 

3i 

10 i 

78500 

19625 

5.16 

.1938 

7.8 

11.1 

14.6 

H 

9# 

67700 

16925 

4.44 

.2252 

7. 

10 . 

12.9 

3 in. 

9 in. 

57700 

14425 

3.78 

.2645 

6.46 

9.2 

11.4 

2 # 

8 * 

48400 

12100 

3.18 

.3144 

5.83 

8.3 

9.9 

2 g 

7* 

40000 

10000 

2.63 

.3802 

5.27 

7.5 

8.4 

2 i 

6 f 

32400 

8100 

2.13 

.4695 

4.83 

6.87 

7. 

2 in. 

6 in. 

25600 

6400 

1.68 

.5952 

4.34 

6.18 

5.8 

1 $ 

5* 

19600 

4900 

1.29 

.7752 

3.70 

5.26 

4.6 

1 * 

4* 

14400 

3(500 

0.945 

1.058 

3.18 

4.53 

3.5 

H 

3# 

10000 

2500 

0.656 

1.524 

2.64 

3.76 

2.5 

1 in. 

3 in. 

6400 

1600 

0.420 

2.381 

2.13 

3.03 

2 . 

7 

8 

2 # 

4900 

1225 

0.322 

3.105 

1.95 

2.78 

1.6 

* 

2 * 

3600 

900 

0.2:56 

4.237 

1.64 

2.34 

1.2 

I 

H 

2500 

625 

0.164 

6.097 

1.40 

2 . 

0.9 

1 

2 

H 

1600 

' 400 

0.105 

9.523 

1.02 

1.46 

0.58 

# 

li 

900 

225 

0.059 

16.95 

0.77 

1.10 

0.31 

1 

4 

1 

400 

100 

0.026 

38.46 

0.53 

0.76 

Cotton Ropes. Three Strands of Fine Yarns. 

Diam. 

Size of Rope. 

Strength. 

Weight 

Length 

Stil ness. 

Pulley. 

Diam. 

Circum. 

Break. 

Safety. 

per Ft. 

per Lb. 

Wind- 

Wind 

Feet. 

Inches. 

Inches. 

Pounds. 

Pounds. 

Pounds. 

Feet. 

ing. 

and 

Unwind. 

D 

d 

c 

8 

T 

w 

1 

4 > 

0 

14.7 

6 in. 

18 in. 

18000 

4500 

7.2 

0.1389 

4. 

6 . 

12.9 

5? 

16 j 

15125 

3781 

6.05 

0.1653 

3.68 

5.5 

11.2 

5 in. 

15 in. 

12500 

3125 

5.00 

0.2000 

3.3 

5. 

9.5 

4# 

13? 

10125 

2531 

4.05 

0.2469 

3. 

2.66 

4.5 

8.0 

4 in. 

12 in. 

8000 

2000 

3.20 

0.3125 

4. 

7.2 

3* 

Hi 

7030 

1782 

2.81 

0.3559 

2.55 

3.83 

6.5 

3* 

10i 

6125 

1531 

2.45 

0.4082 

2.37 

3.56 

5.8 

3± 

Of 

5281 

1320 

2.11 

0.4739 

2.22 

3.33 

5.2 

3 in. 

9 in. 

4500 

1125 

1.80 

0.5555 

2 . 

3. 

4.5 

2 * 

8 i 

3781 

945 

1.52 

0.6579 

1.89 

2.84 

4. 

2 ? 

7i 

3125 

781 

1.25 

0.8000 

1.63 

2.45 

3.4 

2 ? 

6 f 

2531 

633 

1.01 

0.9901 

1.48 

2.23 

2.8 

2 in. 

6 in. 

2000 

500 

0.80 

1.250 

1.36 

2.05 

2.3 

If 

5i 

1531 

383 

0.61 

1.639 

1.18 

1.78 

1.8 

H 

4i 

1125 

281 

0.45 

2.222 

1.04 

1.58 

1.4 

li 

3f 

781 

195 

0.31 

3.226 

0.83 

1.25 

1 ft. 

1 in. 

3 in. 

500 

125 

0.20 

5.000 

0.66 

1. 

0.82 


2 ft 

383 

96 

0.15 

6.666 

0.59 

0.89 

0.65 

# 

2 i 

281 

70 

0.11 

9.009 

0.5 

0.75 

0.5 


li 

195 

49 

0.078 

12.82 

0.4 

0.61 

0.35 

5 - 

li 

125 

31 

0.05 

20.00 

0.34 

0.51 

0.23 


H 

70 

17 

0.028 

35.71 

0.25 

0.37 

0.125 

1 

7 

f 

31 

8 

0.012 

83.33 

0.16 

0.25 
















































426 


ROPF.S. 


Iron Ropes. 19X7 = 133 Wires and Wire Centre. 

Diam. 

Size of Rope. 

Strength. 

Weight 

Length 

Stiffness. 

Pulley. 

Diam. 

Circum. 

Break. 

Safety. 

per Ft. 

per Lb. 

Wind- 

Wind 

Feet. 

Inches. 

Inches. 

Pounds. 

Pounds. 

Pounds. 

Feet. 

iug. 

and 

Unwind. 

J) 

d 

c 

S 

T 

w 

1 


4> 

41.6 

3 in. 

9 

300000 

75000 

16.83 

.0594 

5.4 

7.4 

36.5 

n 

8f 

252500 

63125 

12.45 

.0803 

4.95 

6.8 

31.5 

24 

7k 

209000 

51250 

10.3 

.0971 

4.54 

6.24 

27. 

2£ 

6| 

169000 

42250 

8.34 

.1199 

4.06 

5.58 

22.6 

2 in. 

6 

133000 

33250 

6.62 

.1510 

3.60 

4.97 

20.5 

If 

5§ 

117500 

29375 

5.78 

.1730 

3.37 

4.64 

18.5 

If 

5£ 

102000 

25500 

5.04 

.1984 

3.15 

4.34 

16.5 

1® 

45 

88400 

44100 

4.35 

.2299 

2.95 

4.05 

14.75 

U 

4.V 

75200 

18800 

3.70 

.2703 

2.65 

3.64 

13. 

If 

45 

63200 

15800 

3.12 

.3205 

2.42 

3.33 

11.2 

u 

3g 

52200 

18090 

2.57 

.3891 

2.22 

3.06 

9.55 


3f 

42300 

10575 

2.08 

.4807 

1.91 

2.77 

8. 

1 in. 

3 

33300 

8325 

1.65 

.6061 

1.80 

2.47 

6.56 

5 

2# 

25600 

6400 

1.26 

.7936 

1.56 

2.15 

5.2 

f 

2± 

18800 

4700 

0.927 

1.078 

1.34 

1.85 

3.96 

f 

1 7 

13000 

3250 

0.644 

1.553 

1.11 

1.54 

2.83 

h 

H 

8360 

2090 

0.412 

2.427 

0.90 

1.23 

2.31 

A 

lya 

6400 

1600 

0.315 

3.174 

0.79 

1.09 

1.83 

g 

15 

4710 

1177 

0.231 

4.329 

0.68 

0.93 

1.4 

IJj 

il 

3270 

812 

0.160 

6.250 

0..55 

0.76 

1. 

i 

i 

2090 

622' 

0.102 

9.804 

0.44 

0.61 

0.65 

A 

TS 

1180 

295 

0.057 

17.54 

0.33 

0.46 

0.35 

1 

S 

522 

130 

0.025 

40.00 

0.23 

0.32 

Cast-Steel Ropes. 19 X 7 = 133 Wires and Wire Centre. 

Diam. 

Size of Rope. 

Strength. 

Weight 

Length 

Stiffness. 

Pulley. 

Diam. 

Circum. 

Break. 

Safety. 

per Ft. 

per Lb. 

Wind- 

Wind 

Feet. 

Inches. 

Inches. 

Pounds. 

Pounds. 

Pounds. 

Feet. 

ing. 

and 









Unwind. 

I) 

d 

c 

S 

T 

w 

1 

4> 

$ 

50. 

S in. 

9 

486000 

121500 

16.83 

.0594 

9.37 

11.7 

45. 

2f 

8* 

416000 

104000 

12.45 

.0803 

8.19 

10.2 

38.7 

2J 

75 

344000 

86000 

10.3 

.0971 

7.59 

9.45 

33. 

2* 

Of 

279000 

69750 

8.34 

.1199 

6.84 

8.52 

27.7 

2 iu. 

6 

220000 

55000 

6.62 

.1510 

6.36 

7.92 

25. 


5§ 

193500 

48375 

5.78 

.1730 

5.73 

7.15 

22.7 

If 

6i 

168500 

42125 

5.04 

.1984 

5.28 

6.58 

20.3 

if 

4-5 

145500 

36375 

4.35 

.2299 

4.87 

6.08 

18. 


45 

12:1500 

30875 

3.70 

.2703 

4.52 

5.63 

15.8 

IS 

45 

104000 

26000 

3.12 

.3205 

4.12 

5.13 

13.7 

li 


86000 

21500 

2.57 

.3891 

3.77 

4.70 

11.7 

i* 

3g 

69600 

17400 

2.08 

.4807 

3.39 

4.23 

9.8 

1 in. 

3 

55000 

13750 

1.65 

.6061 

3.00 

3.75 

8. 

7 

S 

2g 

42200 

10550 

1.26 

.7936 

2.65 

3.30 

6.4 

3 

i 

24 

31000 

7750 

0.927 

1.078 

2.24 

2.79 

4.87 

§ 

15 

21500 

5375 

0.644 

1.553 

1.87 

2.33 

3.46 

* 


13750 

3687 

0.412 

2.427 

1.51 

1.89 

2.84 

T3 

lf 5 s 

10500 

2625 

0.315 

3.174 

1.32 

1.65 

2.25 

S 

15 

7740 

1935 

0.231 

4.329 

1.13 

1.41 

1.71 

I s r 

li 

5380 

1345 

0.160 

6.250 

0.94 

1.18 

1.22 

i 

f 

3440 

860 

0.102 

9.804 

. 0.75 

0.94 

0.8 

fa 

A 

1935 

484 

0.057 

17.54 

0.56 

0.7 

0.433 

5 

s 

860 

215 

0.025 

40.00 

0.38 

0.48 









































Ropes, 


427 


Iron Ropes. 19 X 6 = 114 Wires and Hemp Centre. 

Diam. 

Size of Rope. 

Strength. 

Weight 

Length 

Stiffness. 

Pulley. 

Diam. 

Circum. 

Break. 

Safetv. 

per Ft. 

per Lb. 

Wind- 

Wind 

Feet. 

Inches. 

Inches. 

Pounds. 

Pounds 

Pounds. 

Feet. 

ing. 

and 









Unwind. 

D 

d 

C 

-S' 

T 

w 

1 

4 

4 

31. 

3 in. 

9 

287500 

71875 

13,5 

.0741 

7.2 

10.1 

27. 

2f 

8i 

241500 

60375 

11.3 

.0885 

6.77 

9.46 

2:1.7 

‘4 

7* 

200000 

50000 

9.36 

.1068 

6.09 

8.51 

20. 

2i 

62 

161500 

40750 

7.60 

.1316 

5.51 

7.71 

17. 

2 in. 

6 

128000 

32000 

6.02 

.1661 

4.75 

6.65 

15.4 

If 

5f 

112000 

2S000 

5.27 

.1807 

4,17 

6.25 

14. 

13 

si 

98000 

24500 

4.58 

.2183 

4.11 

5.75 

12.4 

If 

4 

84400 

21100 

3.96 

.2525 

3.90 

5.45 

11. 

H 


72000 

18000 

3.37 

.2967 

3.66 

5.12 

9.7 

l§ 

4A 

60400 

15100 

2.83 

.3533 

3.25 

4.55 

7.8 

4 

32 

50000 

12500 

2.34 

.4273 

2.96 

4.13 

7.1 

1 s 

3| 

40400 

10100 

1.89 

.5291 

2.73 

3.82 

6. 

1 in. 

3 

32000 

8000 

1.50 

.6666 

2.39 

3.34 

5. 

k 

2f 

24250 

6062 

1.14 

.8772 

2.00 

2.81 

4. 

1 

2i 

18000 

4500 

0 844 

1.184 

1.70 

2.38 

3. 

n 

F 

If 

12500 

3125 

0.586 

1.706 

1.45 

2.03 

2.1 

i 

H 

8000 

2000 

0.375 

2.666 

1.22 

1.70 

1.7 

T 7 (J 

Its 

6120 

1530 

0.287 

3.484 

1.09 

1.52 

1.3 

f 


4500 

1125 

0.211 

4.739 

1.01 

1.41 

1. 

T°3 

JL5 

16 

3129 

780 

0.146 

6.849 

0,81 

1.14 

0.75 

i 

3 

? 

2000 

500 

0.093 

10.75 

0.59 

0.83 

0,5 

T 3 5 

T°6 

1120 

280 

0.052 

19.23 

0.42 

0.60 

0.27 

A 

§ 

500 

125 

0.023 

43.48 

0.29 

0.41 

Cast-Steel Ropes. 19 X 6 = 114 Wires and Hemp Centre. 

Diam. 

Size of Rope. 

Strength. 

Weight 

Length 

Stiffness. 

Pulley. 

Diam. 

Circum. 

Break. 

Safety. 

per Ft. 

per Lb. 

Wind- 

Wind 

Feet. 

Inches. 

Inches. 

Pounds. 

Pounds. 

Pounds. 

Feet. 

ing. 

and 









Unwind. 

D 

d 

c 

S 

T 

w 

1 

4 

4 

41.6 

3 in. 

9 

432000 

108000 

13.5 

.0741 

6. 

8.25 

36,5 

2f 

81 

363000 

90750 

11.3 

.0885 

5.45 

7.50 

31.5 

2i 

7* 

300000 

75000 

9.36 

.1068 

5.04 

6.93 

27. 

2| 

62 

243000 

67500 

7.60 

.1316 

4.51 

6.20 

22.6 

2 in. 

6 

192000 

48000 

6.02 

.1661 

4.00 

5.49 

20.5 

4 


168500 

42125 

5.27 

.1807 

3.77 

5.18 

18.5 


Si 

146500 

36625 

5.08 

.2183 

3.50 

4.82 

16 5 

1 if 

4A 

126,500 

31625 

3.96 

.2525 

3.27 

4.50 

14.75 

if 

4A 

108000 

27000 

3.37 

.29(57 

2.98 

4.10 

13. 

i§ 

41 

90700 

12675 

2.83 

.3533 

2.70 

3.72 

11.2 

4 

32 

75000 

18750 

2.34 

.4273 

2.50 

3.43 

9.55 

4 

Q3 

Og 

607(H) 

15175 

1.89 

.5291 

2.24 

3.08 

8 . 

1 in. 

3 

48000 

12000 

1.50 

.6666 

2.00 

2.76 

6,56 

7 

2 f 

36800 

9200 

1.14 

.8772 

1.74 

2.40 

5.2 


2i 

27000 

6750 

0.844 

1.184 

1.50 

2.06 

3.96 

5 

4 

18750 

4687 

0.586 

1.706 

1.24 

1.71 

2.83 


4 

12000 

3000 

0.375 

2.666 

1.00 

1,38 

2.31 

* 


9200 

2300 

0.287 

3.484 

0.88 

1.21 

1.83 

f 

4 

6750 

1687 

0.211 

4.739 

0.75 

1.04 

1.4 

T5 

if 

4680 

1170 

0.146 

6.849 

0.61 

0.85 

1. 

i 

2 

3000 

750 

0.093 

10.75 

0.49 

0.68 

0.65 

r 3 5 

T ? s 

1685 

421 

0.052 

19.23 

0.37 

0.51 

0.35 

A 


750 

187 

0.023 

43.48 

0.24 

0.33 

















































428 


Ropes 


Iron Ropes. 7 X 7 = 49 Wires and Wire Centre. 

Diain. 

Size of Rope. 

Strength. 

Weight 

Length 

Stiffness. 

Pulley. 

Diam. 

Oircum. 

Break. 

Safety. 

per Ft. 

per Lb. 

Wind- 

Wind 

Feet. 

Inches. 

Inches. 

Pounds. 

Pounds. 

Pounds. 

Feet. 

ing. 

and 

Unwind. 

D 

d 

c 

S 

T 

10 

1 

4> 

4> 

62.5 

3 in. 

9 

300000 

75000 

16.83 

.0594 

8.8 

12.1 

f>4.5 

2f 

8.25 

252500 

63125 

12.45 

.0803 

7.9 

10.9 

47. 

2* 

7.5 

209000 

51250 

10.3 

.0971 

7.45 

10.2 

40. 

2y 

6.75 

169000 

42250 

8.34 

.1199 

6.8 

9.30 

84. 

2 in. 

6 

133000 

33250 

6.62 

.1510 

5.86 

8.04 

80. 

H 

5£ 

117500 

29375 

5.78 

.1730 

5.60 

7.68 

27. 

i£ 

5£ 

102000 

25500 

5.04 

.1984 

5.30 

7.26 

25. 

if 

4£ 

88400 

44100 

4.35 

.2299 

4.72 

6.48 

22. 

H 

4? 

75200 

18800 

3.70 

.2703 

4.41 

6.05 

19. 

1£ 

4| 

63200 

15800 

3.12 

.3205 

4.18 

5.74 

16.5 

H 

3£ 

52200 

18050 

2.57 

.3891 

3.78 

5.19 

14. 

H 

3£ 

42300 

10575 

2.08 

.4807 

3.45 

4.74 

12. 

1 in. 

3 

33300 

8325 

1.65 

.6061 

2.93 

4.03 

10. 

7 

¥ 

2£ 

25600 

6400 

1.26 

.7936 

2.48 

3.40 

8. 

£ 

2* 

18800 

4700 

0.927 

1.078 

2.05 

2.82 

6. 

£ 

H 

13000 

3250 

0.644 

1.553 

1.79 

2.46 

4.25 


H 

8360 

2090 

0.412 

2.427 

1.53 

2.10 

8.5 

ts 

1 iff 

6400 

1600 

0.315 

3.174 

1.26 

1.73 

2.75 

i 

ll 

4710 

1177 

0.231 

4.329 

1.10 

1.52 

2.1 

TS 

is 

3270 

812 

0.160 

6.250 

0.91 

1.25 

1.5 

1 

4 

£ 

2090 

522 

0.102 

9.804 

0.73 

1.00 

1. 

t 3 S 

T5 

1180 

295 

0.057 

17.54 

0.52 

0.72 

0.5 

1 

8 

£ 

522 

130 

0.025 

40.00 

0.4 

0.56 


Iron Ropes 

. 7X6 

= 42 Wires and Hemp Centre. 


Diam. 

Size of Rope. 

Strength. 

Weight 

Length 

Sti ffness. 

Pulley. 

Diam. 

Oircum. 

Break. 

Safety. 

per Ft. 

per Lb. 

Wind- 

Wind 

Feet. 

Inches. 

Inches. 

Pounds. 

Pounds. 

Pounds. 

Feet. 

ing. 

and 

Unwind. 

D 

d 

c 

5 

T 

w 

1 

4> 

<!> 

52. 

3 in. 

9 

287500 

71875 

13.5 

.0741 

5.00 

7.02 

45. 

2£ 


241500 

60375 

11.3 

.0885 

4.72 

6.62 

39. 

2£ 


200000 

50000 

9.36 

.1068 

4.26 

6.00 

84. 

2£ 

6£ 

161500 

40750 

7.60 

.1316 

3.72 

5.21 

28. 

2 in. 

6 

128000 

32000 

6.02 

.1661 

3.43 

4.81 

25. 

i£ 

5£ 

112000 

28000 

5.27 

.1807 

3.17 

4.44 

23. 

i£ 

o£ 

98000 

24500 

4.58 

.2183 

2.93 

4.12 

21. 

i£ 

4£ 

84400 

21100 

3.96 

.2525 

2.64 

3.70 

18. 

1 £ 

4? 

72000 

18000 

3.37 

.2967 

2.5 

3.49 

16. 

i£ 

4£ 

60400 

15100 

2.83 

.3533 

2.33 

3.27 

14. 

i£ 

3£ 

50000 

12500 

2.34 

.4273 

2.10 

2.95 

12. 

H 

3£ 

40400 

10100 

1.89 

.5291 

1.85 

2.60 

10. 

1 in. 

3 

32000 

8000 

1.50 

.6666 

1.60 

2.34 

8.2 

7 

"S' 

2£ 

24250 

6062 

1.14 

.8772 

1.45 

2.04 

6.5 

£ 

2£ 

18000 

4500 

0.844 

1.184 

1.25 

1.75 

5. 

£ 

H 

12500 

3125 

0.586 

1.706 

1.02 

1.43 

3.5 

1 

•I 

H 

8000 

2000 

0.375 

2.666 

0.85 

1.20 

3. 

t 7 »t 

if 

6120 

1530 

0.287 

3.484 

0.67 

0.95 

2.3 

t 

4500 

1125 

0.211 

4.739 

0.62 

0.87 

1.7 

i 5 <r 

ii 

8120 

780 

0.146 

6.849 

0.55 

0.77 

1.25 

1 

4 

£ 

2000 

500 

0.093 

10.75 

0.41 

0.58 

0.8 

A 


1120 

280 

0.052 

19.23 

0.32 

0.45 

0.4 

£ 

£ 

500 

125 

0.023 

43.48 

0.25 

0.35 













































Rories. 


429 


Iron Itope. 7X6X6 = 252 AVI res. Cotton Centre in each Rope Straud 

and Hemp in the Centre. 


Diam. 

Pulley. 

Feet. 

Size of 

Diam. 

Inches. 

Rope. 

Circum. 

Inches. 

Stre 

Break. 

Pounds. 

ugth. 

Safety. 

Pounds. 

Weight 
per Ft. 

Pounds. 

Length 
per Lb. 

Feet. 

Stiff 

Wind¬ 

ing. 

uess. 

Wind 

and 

Unwind. 

D 

d 

c 

5 

T 

w 

1 


<I> 

21 . 

3 in. 

9 

270000 

67500 

12.2 

.0819 

5.40 

7.S0 

18. 

2 | 

H 

226500 

5(1625 

10.2 

.0980 

4.88 

7.04 

16. 

2 £ 

7h 

187000 

46750 

8.57 

.1167 

4.40 

• 6.37 

13.5 

2 i 

6 f 

152000 

38000 

6.84 

.1462 

3.98 

5.74 

11 . 

2 in. 

6 

120000 

30000 

5.42 

.1845 

3.74 

5.39 

10 . 

A f? 

5f 

105400 

26350 

4.76 

.2101 

3.50 

5.03 

9. 

1 3 
i 4 

5£ 

91700 

22925 

4.13 

.2421 

3.26 

4.70 

8 . 

11 

4S 

79000 

19750 

3.56 

.2809 

3.08 

4.44 

7. 


41 

67500 

16875 

3.04 

.3289 

2.87 

4.14 

6.25 

J 8 

4£ 

56600 

14250 

2.55 

.3921 

2.58 

3.72 

5.5 

l£ 

3f 

46800 

11700 

2.11 

.4739 

2.30 

3.30 

4.75 

u 

3g 

38000 

9500 

1.71 

.5848 

2.00 

2.90 

4. 

1 in. 

3 

30000 

7500 

1.35 

.7407 

1.77 

2.55 

3.25 

7 

8 

2 | 

23000 

5750 

1.03 

.9709 

1.57 

2.27 

2.6 

f 

2 £ 

16850 

4212 

0.760 

1.316 

1.32 

1.91 

2 . 

5 

¥ 

1 7 

11700 

2925 

0.528 

1.894 

1.09 

1.57 

1.4 


11 

7500 

1875 

0.338 

2.958 

0.90 

1.30 

1.15 

TG 

Its 

5740 

1435 

0.258 

3.876 

0.79 

1.14 

0.9 

1 


4220 

1055 

0.190 

5.263 

0.69 

1.00 

0.7 

5 

TG 

« 

2930 

732 

0.132 

7.576 

0.55 

0.80 

0.5 

1 

3 

1870 

467 

0.084 

11.90 

0.44 

0.64 

0.3 

3 

TG 

t 9 s 

1050 

262 

0.047 

21.27 

0.38 

0.56 

0.18 

1 

8 

3 

5 

468 

117 

0.021 

47.62 

0.20 

0.30 


Copper Ropes. 7 X 6 = 42 Wires. Cotton Centre. 


Diam. 

Size of Rope. 

Strength. 

Weight 

Length 

Stiffness. 

Pulley. 

Diam. 

Circum. 

Break. 

Safety. 

per Ft. 

iier Lb. 

Wind- 

Wind 

Feet. 

Inches. 

Inches. 

Pounds. 

Pounds. 

Pounds. 

Feet. 

ing. 

and 

Unwind 

I) 

d 

c 


T 

w 

1 

<I> 

4> 

26. 

3 in. 

9 

306000 

76500 

15.25 

.0656 

9.60 

13.4 

22.5 

2 f 

8 £ 

257000 

64250 

12.9 

.0775 

9.00 

12.6 

20 . 

n 

*1 1 
i % 

212500 

53125 

10.6 

.0943 

8.50 

11.9 

17. 

2 £ 

6 | 

172000 

43000 

8.44 

.1185 

7.14 

10.0 

14. 

2 in. 

6 

136000 

34000 

6.82 

.1466 

6.44 

9.00 

12.5 

n 

5# 

120000 

30000 

5.97 

.1675 

6.16 

8.60 

11.5 

if 

5£ 

104000 

26000 

5.20 

.1923 

5.43 

7.8 

10.5 

ii 

4£ 

90000 

22500 

4.48 

.2232 

5.00 

7.0 

9. 

H 

4* 

76500 

19125 

3.82 

.2618 

4.70 

6.1 

8 . 

i| 

4& 

64200 

16050 

3.21 

.3115 

4.44 

6.2 

7. 

ii 

3£ 

53100 

13275 

2.88 

.3472 

3.96 

5.54 

6. 

u 

3| 

42800 

10700 

2.15 

.4651 

3.50 

4.90 

5. 

1 in. 

3 

34000 

8500 

1.70 

.5882 

3.22 

4.50 

4. 

7 

2 f 

26000 

6500 

1.30 

.7692 

2.94 

4.10 

3.25 

f 

2 £ 

19100 

4775 

0.956 

1.046 

2.44 

3i42 

2.5 

5 

1 7 

13250 

3312 

0.673 

1.486 

1.83 

2.70 

1.75 


1 £ 

8500 

2125 

0.424 

2.358 

1.65 

2.30 

1.5 

A 

1 * 

6510 

1627 

0.325 

3.077 

1.29 

1.80 

1.15 

l 

Is 

4780 

1195 

0.239 

4.184 

1.20 

1.67 

0.85 

A 

AS 

3320 

830 

0.166 

6.024 

1.05 

1.47 

0.62 

i 

3 

2120 

530 

0.106 

9.433 

0.81 

1.13 

0.4 


A 

1200 

300 

0.059 

16.95 

0.62 

0.87 

0.2 

i 

1 

530 

132 

0.26 ( 

38.46 

0.48 

0.67 

























































430 


Tying Knots. 


THE ART OF 

Fob illustrations of the explanations here 
given, see cut on succeeding page. 

1 and 2 are simple loops, showing the elements 
of the simplest knot. 

3. Simple knot commenced. 

4. The same completed. 

5. Flemish knot commenced. 

6. The same completed. 

7. Rope knot commenced. 

8. The same completed. 

9. Double knot commenced. 

10. The same completed. 

11. Double knot, back view. 

12. Six-fold knot commenced. 

13. The same completed. This is closed or 
“nipped,” drawing the two ends with equal 
force. 

14. A “ boat” knot, made with the aid of a 
stick. This is a good kuot for handling weights 
which may want instant detachment. Lift the 
weight very slightly, push out the stick, and 
the knot is untied. 

15. Simple hitch (or double) used in making 
loop holes. 

16. Loop knot commenced. 

17. Loop knot finished. 

18. Flemish loop or “ Dutch ” double knot. 

19. Running knot. 

20. Running kuot to hold ; the end kuot near¬ 
est the bend of the rope is the check knot. 

21. Running kuot “ checked." 

22. Double loop for twist knot. 

23. The twist knot completed. It is made by 
taking a half turn on both the right-hand and 
left-hand cords and passing the end through 
the “ bight ” so made. 

24. Chain knot, a series of loops. The end 
of the cord is fastened, a simple loop made and 
passed over the left hand, the right hand re¬ 
taining hold of the free end. The left hand 
then seizes the cord above the right and draws 
a loop through the loop already formed : the left 
then finishes the kuot by drawing it tight. This 
is repeated until you have all the kuot wanted, 
when it is secured by passiug the free end en¬ 
tirely through the last loop. This is a kiud of 
kuot much in vogue for the kuotting of leather 
whip-lashes, etc. It is very convenient. 

25. Double chain. 

26. Double chain secured and pulled out as 
when in use. Notice the mode in which the end 
is thrust through the last loop. 

27. Lark’s head; useful to sailors as a moor¬ 
ing kuot. 

28. The same, double looped. 

29. The same on a ring of a boat. The ad¬ 
vantage of instant release by the use of the 
stick has been noted in No. 14. 

30. A treble lark's head. First tie a single 
lark’s head, and then divide the two end^ and 
use each singly as shown in the cut. 

31. Simple boat kuot with one turn. 

32. Crossed running knot, strong and handy. 
Looks difficult, hut by taking a cord about one- 
eighth of an inch in diameter and tying the 
same two or three times with the picture, you 
will find no difficulty in mastering it. It is a 
common knot in some parts of the country. 

33. A knotted loop for end of rope. TJse va¬ 
rious', to prevent the end of the rope from slip¬ 
ping, etc. Very readily untied. 

34. Simple (lashing) knot commenced. 

35. The same finished. (See 51.) In making 
34 it is necessary to hold the simple knot, as 
shown in 33, by some pressure on the knot until 
it is ready to draw tight for the finish. 

36. Is the same kuot with two turns, some¬ 
times called a rosette. This is very easily un¬ 
tied, as will be seen by tracing the loose ends 
back in the illustration. 



37. Knot with single turn; unties as easily 
as 36, but the. “strands”—that is to say, all parts 
of the knot—must be laid ns in the true or reef 
knot (see 50 and 51) or a “granny ” knot will be 
produced which will not hold. One who ties 
this kuot well will be a master of this art. 

38. Timber hitch or slip knot, with double 
hitch. The greater the strain the tighter this 
knot will hold. It looks as if it might give 
way, but it will not. 

39. Runniug knot with two ends. 

40. The same with check knot, which cannot 
be opened except with a marliuspike. 

41. Runniug knot with two ends, with n check 
knot (to the running loops), which can be un¬ 
tied by drawing both ends of the cord. 

42. Running knot with two ends, fixed by a 
double Flemish. When an object is to be en¬ 
circled by this kuot, pass the end on which the 
check knot is to be through the cords. 

43. Ordinary twist knot. 44. Double. 

45. Form of loop for builder’s knot. 

46. lluilder’s knot finished, used by workmen 
in securing building materials. 

47. Double builder's kuot. 

48. Weaver’s knot. On the small scale, lay 
the ends of the two cords to be united between 
the thumb and first finger of the left hand, the 
right end undermost; pass the right-hand cord 
back over the thumb to form a loop, and bring 
it back under the thumb and hold it fast. Now 
put the end of the upper or left-hand cord over 
the right-hand cord and through the loop. 
Catch it with thumb and finger of the left hand, 
and tighteu by drawing the right hand. 

49. Weaver’s knot completed. 

50. True or reef knot commenced. 

51. The same completed. Useful for small 
ropes, but if ropes are uuequal in size it is apt 
to draw out into the shape shown by 52. To 
obviate this the two ends issuing from each side 
of the kuot are whipped or lashed together. 

53. A “granny” knot, the ends not lying 
alongside of each other. 

54. Granny kuot with a strain in it, showing 
its uselessness. 

55. 56, 57. Commencement, finished front 
view, and finished back view. This is a com¬ 
mon knot. The two ends to be united are 
seized together and tied in a common simple 
knot. 

58. And the ordinary knot, the ends used sep¬ 
arately. 

59. The same knot open. This knot is made 
by making No. 3 on one rope, holding it open 
so that we can pass the end of the other cord 
through the first loop of the last, making it 
with a second loop. Then draw it tight. 

6(1 and 61. Knot used for the same purpose 
as the simple Flemish. 60 is the tightened or 
finished knot. 

62. English knot commenced 

63. English knot tightened (front view). 

64. English knot tightened (back view). 

65. Splice, with two ties. 

66. Shortening by loops and turns where the 
end of the rope is free. 

67. Shortening knot, can be used when either 
end is free. 

68. The same, with double bend and ties. 

69. The same, passing through the knots. 

70. Another method of shortening, called 
making a “ sheep shank ’’ or dog shank. Unsafe 
unless the shank (the loose loop) is attached to 
the contiguous rope by a stout “ seizing that 
is, a cord tied around it. 

71. Shows a dog shank that will hold without 
seizing. 

From 73 to 84 explain themselves without es¬ 
pecial allusion to them. 











F not-Tying. 


431 


\ 

THE ART OF KNOT-TYING. 

(From the “ Scientific American.") 



4 



















































































432 


FKICTI05? 


FRICTION. 

The resistance occasioned by Friction is independent of the velocity of mo¬ 
tion ; but the re-effect of friction is proportional to the velocity. Friction is in- j 
dependent of the extent of surface in contact when the pressure remains the 
same, but is proportional to the pressure. This law was established from experi-[ 
ments by Arthur Morin in the years 1831-32 and 1833, from which a summary 
1 is contained in the accompanying Table. 

J , Notation. 

a = Fibres of the woods are parallel to themselves, and to the direction of j 
motion. 

6 = Fibres at right-angles to fibres. 

c = Fibres vertical on the fibres which are parallel to the motion. 

d = Fibres parallel to themselves, but at right-angles to the motion, length 
i by length. 

e = Fibres vertical, end to end. 

Example. A vessel of 800 tons is to be hauled up an inclined plane, which 
inclines 9° 40' from the horizon; the plane is of oak, and greased with tallow. , 
"What power is required to haul her up? 

The coefficient for oak on oak with continued motion is f = 0*097, 6ay 0*1, j 
then, 

800X p in.9° 40' = 800X0*16791 => 124*328 tons, 
the force required if there were no friction, and 

800Xcos.9° 40'XO*1 = 800X0*9858X0*1 = 78*864 tons, 
the force required for the friction only, and 

134*328 

78*864 

213*192 tons, the force required to haul her up. 

The work lost per sec. by friction in axle and bearings is expressed simply by 
the formula 

p _ 71 d Wnf Wdnf 

~ 12*60 — 230 » 


in whi:h U r = the weight of pressure in the bearing, d = diameter on which 
the friction acts in inches, n = number of revolutions per minute, and f — co¬ 
efficient of friction from the Table. In common machinery kept in good order 
the coefficient of friction can be assumed to/ = 0*065. then 


P = 


Wdn 
■ 353 ) 1 


11 = 


Wd n_ 
1941500 


Example. The pressure on a steam-piston is 20000 pounds, and makes n = 40 
double strokes per minute. Required the friction in the shaft of d = 8 inches 1 


H = 


20000X8 X40 


1941500 

Friction 


= 3*3 horse-power, the by friction. 

In Guides. 

TF= pressure on the steam piston in pounds. 

S = stroke of piston in feet. 

I = length of connecting rod in feet 
H = horse power of the friction. 

/ WS 2 n 


11 = 


10500 ;/4 l 2 -S 2 

Example The pressure on a steam piston being W = 30,000 pounds, stroke 
S = t feet length of connecting rod l = 7 feet, and making 50 revolutions per 
minute. Required the horse power of the friction HI 1 


H= 


.005 X 30000 X 1 G X 50 


16500 v / 4 X 49-16 


= 7 IP. 



















FB.ICTIOK. 


433 


TABLE OF FRICTION FOR PLANE SURFACES IN CONTACT. 


Kind of Materials in contact. 


Lubricated 

Coefficient in 




with. 

Motion. 

Starting. 

Oak on Oak, ... 

• 

a 

0 

0-478 

0-625 

u u m m 

• 

11 

tallow 

0-097 

0-160 

u u m m m 

m 

11 

lard 

0-067 

• • • • 

u u 

m 

b 

o 

0-324 

0-540 

u • • • 

• 

11 

unctuous 

0143 

0-314 

44 u • 

• 

11 

tallow 

0-083 

0-254 

U 44 m m m 

• 


water, 

0-25 

• • • • 

U U 

• 

<4 

o 

0-336 

• • • • 

U U 9 m 

• 

c 

o 

0-192 

0-271 

44 44 • • . 

• 

e 

0 

• • • • 

0-43 

Cast-iron on Oak, 

• 

a 

0 

0-400 

0-570 

U 44 

• 

11 

soap 

0-214 

• • • • 

u u m 

• 

11 

tallow 

0078 

0*108 

Wrought-iron on Oak. 

• 

11 

0 

0-252 


46 64 

* 

11 

tallow 

0-078 

• • • • 

Wrought iron, together, - 

• 

a 

o 

0-138 

0-137 

44 44 

• 

a 

unctuous 

0-177 

• • • • 

44 44 . 

• 

11 

tallow 

0-082 

• • • • 

44 44 

• 

H 

olive oil 

0-070 

0-115 

Wrought on cast-iron, 

• 

a 

0 

0-194 

0194 

44 44 ^ 

• 

11 

unctuous 

0-18 

0-118 

44 44 

• 

11 

tailow 

0-103 

o-io 

44 44 m 

• 

11 

olive oil 

0-006 

0100 

Cast-iron on cast-iron, 

• 

a 

water 

0-314 

0-314 

44 44 

• 

11 

soap 

0-197 

• • • • 

44 44 w 

• 

11 

tallow 

0100 

o-ioo 

44 44 , 

• 

11 

olive oil 

0-064 

• • • • 

Wrought-iron on brass, - 

• 

a 

0 

0-172 

• • • • 

44 44 

■ 

11 

unctuous 

0-160 

• • • • 

44 44 

• 

11 

tallow 

0*103 

• • • • 

44 44 . 

m 

11 

lard 

0*075 

• • •• 

44 44 


11 

olive oil. 

0-078 

• • • • 

Cast-iron on brass, 

m 

a 

0 

0-147 

• • • 

“ « . 

m 

11 

unctuous 

0-132 

• • • • 

U 44 . 

m 

11 

tallow 

0-103 

• • • • 

U 44 _ 


11 

lard 

0-075 


44 44 . 


11 

olive oil 

0-078 

• • • • 

Brass on brass, 


a 

0 

0-201 

• « • • 

“ «... 

m 

11 

unctuous 

0-134 

• • • • 

« «... 

» 

11 

olive oil 

0-053 

• • • • 

Steel on cast-iron, - 

• 

11 

o 

0-202 

• • • • 

44 44 _ 

m 

11 

tallow 

0*105 

• • • • 

44 44 . 

m 

11 

lard 

0 081 


44 44 

m 

a 

olive oil 

0-079 



FRICTION OF AXLES IN MOTION. 




Oil, Tallow, or Hog's Lard. 

Designation of surface in 

Dry or slightly 

Supplied in the 

The grease 

greasy, or wet. 

ordinary 

continually 

contact. 


manner. 

running. 

Brass on Brass, - • • 


0079 

• • • • • 

“ on cast-iron, 


0-072 

0049 

Iron on Brass, - . • 

0-251 

0-075 

0-054 

“ on cast-iron, 


0-075 

0-054 

Cast-iron on cast-iron. 

0137 

0-075 

0-054 

« on Brass, 

0-194 

0-075 

C-054 

Iron on lignum-vitse, 

0-188 

0-125 


Cast-iron on « • 

0.185 

o-ioo 

0-092 

Lignum-vitae on cast-iron, 


0116 

0170 


28 









































4 34 


rAPER, Tin and Glass. 


PAPER. 

1 ream = 20 quires = 4S0 sheets. 

1 quire = 24 sheets. 


Cap, • i 
Demy, 
Medium, 
Royal, 

Super Royal, 
Imperial, . 
Elephant, 


13 X 16 inches. 
20 “ 15 
17 
19 
19 
21 
22 


22 “ 
24 “ 

27 “ 
30 “ 

28 “ 


Drawing Paper. 

Columbier, . 
Atlas, . . 

Theorem, 

Double Elephant, 
Antiquarian, . 
Emperor, . . 

Uncle Sam, . 


44 

it 

it 

it 

it 

it 


34 X 23 inches. 

33 “ 

34 “ 

40 “ 

52 “ 

40 “ 

48 “ 


26 

28 

26 

31 

60 

120 


it 
it 
it 
it 
ii 
• i 


Continuous Cblossal Drawing Paper, No. A and No. B, 56 inches wide, and of any 
required length. No A of this paper is excellent for mechanical drawings. Price, 
from 40 to 50 cents per yard. 

Tracing Paper. 

Double Crown. 30 h J 20 inches.l Qlazed or Crystal. 

“ 40 “ j Yellow or Blue Wove. 

Finest French Vegetable Tracing Paper. 

Grand Raisin (or Royal), 24 in. by 18. Grand Aigle, 40 in. by 27. 


Double Double Crown, 

Doable Doable Double Crown, 


40 

60 


Mounted Tracing Paper. 

This paper is mounted on cloth, and is still transparent; it will take ink and 
water-colors. It is 38 inches wide, and of any required length. 

Vellum Writing Cloth. 

Adapted for every description of tracing; it is transparent, durable and strong. 
It is 18 to 38 inches wide, and of any required length. 


Weight and Maries of English Tin-plates. 




Plates 

Length 

Weight 


Plates 

Length 

Weight 

Brand. 

per 

and 

per 

Brand. 

per 

and 

per 



Box. 

Breadth. 

Box. 


Box. 

Breadth. 

Box. 



No. 

lu. 

I.bs. 


No. 

In. 

Lbs. 

1 C. 

• 

225 

131X10 

112 

1 XX. . . 

225 

133X10 

161 

2 0. . 

• 

225 

13*“ 9$ 

105 

1 XXX. . 

225 

133 “ 10 

182 

3 C. 

• 

225 

12*“ 9* 

98 

1 XXXX. . 

225 

133 “ 10 

203 

11 C. . 

• 

225 

133 “10 

119 

1 xxxxx. 

225 

133 “ 10 

224 

II X. 

• 

225 

13*“ 10 

157 

1 xxxxxx. 

225 

133 “ 10 

245 

IX. . 

• 

225 

133 “10 

140 

DC. 

100 

163 “12* 

98 

2 X. 

• 

225 

13*“ 9| 

133 

DX. . 

100 

163 “ 12* 

126 

3 X. . 

• 

225 

123“ 9* 

126 

DXX. . 

100 

163 “ 12* 

147 

Leaded IC. 

112 

20 “14 

112 

DXXX. . 

100 

163 “ 12* 
163 “ 12* 

168 

(4 

IX. 

112 

20 “14 

140 

DXXXX. . 

100 

189 

ICW. 

• 

225 

133 “ 10 

112 

SBC. 

200 

15 “11 

168 

IXW. 

• 

225 

133 “io 

140 

SDX. . 

200 

15 “11 

188 

CSDW. 


200 

15 “11 

1C8 

SDXX. . 

200 

15 “11 

209 

C1IW. 

• 

100 

163 “ 12* 

105 

SDXXX. 

200 

15 “11 

230 

XII w. 

• 

100 

163 “ 12* 

126 

SDXXXX. 

200 

15 “11 

251 

TT. . 

• 

450 

133 “ 10 

112 

SDXXXXX. 

200 

15 “11 

272 

XTT. 

• 

450 

13* “ 10 

126 

SDXXXXXX. 

200 

15 “11 

293 


When the plates are 14 by 20 iuches, there are 112 in a box. 


Thickness and Weight of Window Glass. 

Number of tho glass or weight in ounces per square foot. 


13 

15 

16 

17 

19 

21 

24 

26 

32 

36 

42 

.063 

.071 

.077 

.083 

.091 

.100 

.111 

.125 

.154 

.167 

.200 


Thickness in decimals of an inch. 

























Gravitation. 


435 


GRAVITATION. 

Gravity or Gravitation is a mutual property which all bodies in nature 
possess of attracting one another; or Gravity is he tforce by which all bodies 
tend to approach each other. A large body attracting a comparatively very 
small one, and their distance apart being inconsiderable, the force of gravity in 
I the small body will be very sensible compared with that in the large one ; such 
: is the case with the body, our earth, attracting small bodies on or near her sur- 
j face. 

Gravitation is not periodical, it acts continually ever and ever. A body placed 
unsupported at a distance from the earth, the force of gravity is instantly oper¬ 
ating to draw it down, and then we say, “ the body fell down ” If it were possi¬ 
ble to withdraw the attraction between the body and the earth, it would not 
fall down, but remain unsupported in the space where it was placed;—giving 
the body a motion upwards, it would continue moving, and never come back to 
the earth again. 

Law of Gravity. 

The force of Gravity is directly proportional to the product of the masses of the at¬ 
tracting bodies , and inversely to the square of their distance apart. 

This law was discovered by Sir Isaac Newton. It is this law that supports the 
condition of the whole universe, and enables us to calculate the distances, mo¬ 
tions and masses, &c., of the heavenly bodies. 

The unit or measure of force of gravity is assumed to be the velocity a falling 
body has attained at the end of the first second of descent; this unit is com¬ 
monly denoted by the letter g; its value at the level of the sea in New York is 
g --- 32 17 feet per second, in vacuum, g is called the aeceleratrix of gravity. The 
space fallen through in the first second is 5 g = 16 085 feet. 

This value increases with the latitude, and decreases with the elevation above 
the level of the sea. 

I = latitude , h = height in feet above the level of the sea, and r = radius of the 
earth in feet, at the given latitude l. 

r = 20887510 (1+ 0'00164 cos.21), 

g = 32-16054 (1 — 0'00284 cos.2 1) ^1 — y • J 
Notation. 

S = the space in feet, which the falling body passes through in the time T. 

u = the space in feet, which the body falls in the 2 ’th second. 

Y= velocity in feet per second, of the falling body at the end of the time T. 

T = time in seconds the body is falling. 

The accompanying Diagram is a good il¬ 
lustration of the acceleration of a falling 
body. The body is supposed to fall from a 
to b, every small triangle represents the 
space 16'08 feet which the body falls in 
the first second ; when the body has reached 
the line 3" seconds, it will be found that it 
has passed 9 triangles, and 9X10‘0S = 144'72 
feet the space which a body will fall in 3" 
seconds. The number of triangles between 
each line is the space u which the body has 
fallen in that second. Between 3 " and 4" 
are 7 triangles and 7X16'08 = 112 - 56 feet, 
the space fallen through in the fourth sec¬ 
ond. Under the line 3" will be found 6 tri¬ 
angles, which represents the velocity V the 
body has obtained at the end of the third 
second or 6X16'08 = 96'48 feet per second. 
For every successive second the body will 
gain two triangles or 2X16 08 = 32'J6 feet 
per second. 
























436 


Accelj hateij Motion. 


FORMULAS FOR ACCELERATED MOTION. 
Velocity V in Feet per Second. 


V=g T. 

• • • • • 

F=»i/2 g& . . • 

. 3. 

II 

it* 

^|5q 

• • • • • 2* 

F = 8.02 iAs\- . 

. 4. 


Space S Fallen through, in Feet. 


« 9?* 

s- 2 . 

• • • • • 

F2 

S = ^. 

. 7. 

F T 

ft 

o r* 

. 8. 

2 • 

• • • • • 

* 64.33. 


Time of Fall in Seconds. 


1! 

Us 

• • • • . 9. 

r- 

A/ • • • • 

\ g 

. 11. 

2 S 

T = — 

F 

• • • • • 10* 

m — l/ S 

4.01 * * * ‘ 

. 12. 


Space Fallen throng 

h in the Tth Second. 


Us 

Cs 

II 

3 

--). . . .13. 

T = g + 2‘ ' * * 

. 14. 


Example 1. What velocity has a body attained after having fallen freely 
for a time of T = seconds ? 

Velocity F= 32.17 X 2.5 = 80.2 feet per second. 

Example 4. A body is dropped from a height of .S' = 98 feet. Wbat velocity 
will it have on reaching the ground, aud what time is required for its fall? 

Formula 4. Velocity V = 8.02 \/ 98 = 79.3939 feet per second. 

|/~S |/98* 


Formula 12. Time T = 


4.01 


4.01 


= 2.46 seconds. 


Example 5. A body was dropped at the opening of a hole in a rock, and 
reached the bottom in T— 3.5 seconds. Required the depth of the hole? 


Formula 5. 


T 2 

Depth S = g — 


32.17 X 3.52 


= 196.98 feet. 


Example 8. What space must a body fall through in order to acquire a 
velocity F=369 feet per second? 


Space 5 = 


F2 


3692 


64.33 64.33 


= 2116.6 feet. 


2116.6 feet when 


Example 10. What time is required for a body to fall S- 
the final velocity V =* 369 feet per second? 

„. 2S 2X2116.6 

Time T — — =-—-— — 11.472 seconds. 

V 3b9 

Example 13. A body falls freely for a time of T — 4£ seconds. How much 
will it fall in the last second? 

Formula 13. n = g (T— i) = 32.17 (4.5 — 0.5) = 128.68 feet. 

























RETARDED MOTIOX. 


437 


RETARDED MOTION. 

A body thrown up vertically will obtain inversely the same motion as when 
it falls down, because it is the same force that acts upon it, and causes retarded 
motion when it ascends, and accelerated motion when it descends. 

V = the velocity at which the body starts to ascend. 
v = velocity at the end of the time t. 

2 '= time in seconds in which the body will ascend. 
t = any time less than T. 

S = height in feet to which the body will ascend. 
s = the space it ascends in the time t. 


Velocity in Feet per Second at the End of the Time t. 


v = V — g t. 


. 15. 


v = ±- gJ . 

t 2 


. 16. 


Height of Ascension in the Time t. 

= 9 \\ ‘ * • 17 - | s = t(^v + g^y . 


18. 


Starting Velocity in Feet per Second. 

V=v + gt .19. V=~-\-g-. 


Time of Ascension in Seconds. 


V—v 

9 


21 . 


V \V 2 2s 


Starting and Ending Velocities. 

v = |/ V 2 — 2 gs. . . .23. | V = ]/ v- + 2 g s. 

Formulas for T and S are the same as for accelerated motion. 


. 20 . 


. 22 . 


. 24. 


Example 22. A ball starts to ascend with a velocity of 135 feet per second. 
At what velocity will it strike an object 60 feet above? Find the time t by 
the Formula 22. 

= 1 -— 

“ 32.16 

• 

until it strikes; and from Formula 15 we have 

v = 135 — 32.16 X 0.41 = 121.83 feet per second. 

Example 24. With what velocity must a body start to ascend in order to 
strike an object s = 15 feet above with a velocity v =* 10 feet per second ? 

Velocity V— ]/l0 2 -f 2 X 32.17 X 15 == 32.63 feet per second. 


V 1352 2 

32.16 3 


2 X 60 

g-— = 0.41 seconds, 














438 


Gkavitatiox. 


Example 5. A ball thrown up vertically from a cannon, occupied 20 seconds, 
until it arrived at the same place it started from. How high up was the ball, 
and at what velocity did it start? 

One-half of 20 = 10 seconds. Formula 2. 


s= 


32.1G x 10 2 
2 


= 1608 feet high. 


V = 32.16 x 10 = 321.6 feet per second. 

If a cannon-ball be shot from A, in the direction AB, at an angle BAG to the 
horizon, there are two forces acting on the ball at the same time, namely—the 
force of gunpowder, which would propel the ball uniformly in the direction AB, 
and the force of gravity, which only acts to draw the ball down at an accelerated 
motion; these two different (uniform and accelerated) motions will cause the ball 
to move in a curved line (Parabola) AaC. Fig. 225. 

V — velocity of the ball at A. IV = weight of the ball in pounds. 

S = the greatest height of ball over the horizontal line AO. 
t — time from A to C, via a. p = pounds of powder in the charge. 
b = the distance from A to C\ called horizontal range. 

V — 2800 \l-~f p = ^ ^ —, b = 87.06 sln.x cos.x —. 

\ W’ * 7840000 ’ W 


Example The cannon being loaded sufficiently to give the ball a velocity of 
900 feet per second, the angle x = 45°. Required, the distance b = ? and the 
time t — ? 


b = 


900 2 x sin.45° x cos.45° 
32.16 


= 1259 feet, the distance from A to C. 


It will be observed that the distance b will be longest when the angle x is 45°, 
because the product of sine and cosine is greatest for that angle, sin.45° X cos. 
45° = 0.5. 


Example What time will it take for a ball to roll 38 feet on an inclined 
plane, angle x = 12° 20', and what velocity has it at 38 feet from the starting- 
point? Fig. 222. 


J - 2 — = J -—^-= 3.33 seconds. 

\ (j sin.x \ 32.16 x sin.l 2° 20 7 

V=g T sin.x = 32.16 x 3.33 x sin. 12° 20 / = 22.8 feet per second. 


Resistance of Aii* to tlie Fliglit of Projectiles. 

A = area of rtsistance of the projectile in square inches. 

= angle of resistance of the projectile, which for flat surfaces sin. 2 <£ = 1 , for 
sphere sin. 2 </> = 0.5. 

For a pointed projectile of parabolic form, and when thegjrdinate is double the 
abscissa sin. 2 $ = 0.25. 

V = velocity of the projectile in feet per second. 

It = resistance to the projectile in pounds. 

B = A V* sin.y 
57000 ’ 

Let T denote the time of flight in seconds, and W = weight in pounds of the 
projectile. 

D = distance in feet which the projectile is retarded by resistance of air in the 
time T. 

32.1667s! 7 12 1GRT 2 
2 W ~ W * 















Force ok Gravity. 


430 





V W 2 S W 

* " FT* " g- T* ’ 

W~ P4 Q, and F = P — Q- 




















































440 


Failing Bodies. 


Falling Bodies. 

V = velocity in feet per second at the end of fall. 
T — time in seconds of the fall. 

5 = space fallen through in feet. 


V 

T 

S 

V 

T 

S 

V 

T 

S 

0.1 

0.0031 

.00015 

5.1 

0.1585 

0.4042 

11 

0.3419 

1.8864 

0.2 

0.0062 

.00031 

5.2 

0.1616 

0.4202 

12 

0.3730 

2.2380 

0.3 

0.0093 

0.0014 

5.3 

0.1647 

0.4364 

13 

0.4041 

2.6266 

0.4 

0.0124 

0.0025 

5.4 

0.1678 

0.4530 

14 

0.4352 

3.0464 

0.5 

0.0155 

0.0039 

5*5 

0.1709 

0.4700 

15 

0.4663 

3.4975 

0 .G 

0.0186 

0.0055 

5.6 

0.1740 

0.4872 

16 

0.4973 

3.9784 

0.7 

0.0217 

0.0076 

5.7 

0.1771 

0.5047 

17 

0.5284 

4.4914 

0.8 

0.0248 

0.0099 

5.8 

0.1802 

0.5226 

18 

0.5595 

5.0355 

0.9 

0.0279 

0.0125 

5.9 

0.1833 

0.5407 

19 

0.5906 

5.6107 

1 . 

0.0311 

0.0155 

6 . 

0.1865 

0.5595 

20 

0.6217 

6.2170 

1.1 

0.0342 

0.0188 

6.1 

0.1896 

0.5782 

21 

0.6527 

6.8502 

1.2 

0.0373 

0.0224 

6.2 

0.1927 

0.5973 

22 

0.6838 

7.5218 

1.3 

0.0404 

0.0262 

6.3 

0.1958 

0.6168 

23 

0.7149 

8.2213 

1.4 

0.0435 

0.0304 

6.4 

0.1989 

0.6365 

24 

0.7460 

8.9520 

1.5 

0.0446 

0.0335 

6.5 

0.2020 

0.6565 

25 

0.7771 

9.7125 

1.6 

0.0477 

0.0381 

6.6 

0.2051 

0.6768 

26 

0.8082 

10.566 

1.7 

0.0508 

0.0432 

6.7 

0.2082 

0.6975 

27 

0.8393 

11.330 

1.8 

0.0539 

0.0485 

6.8 

0.2113 

0.7184 

28 

0.8704 

12.185 

1.9 

0.0580 

0.0551 

6.9 

0.2144 

0.7397 

29 

0.9015 

13.072 

2 . 

0.0622 

0.0622 

7. 

0.2176 

0.7616 

30 

0.9325 

13.987 

2.1 

0.0653 

0.0685 

7.1 

0.2207 

0.7835 

31 

0.9636 

14.936 

2.2 

0.0684 

0.0756 

7.2 

0.2238 

0.8057 

32 

0.9947 

15.915 

2.3 

0.0715 

0.0822 

7.3 

0.2269 

0.8282 

33 

1.0258 

16.926 

2.4 

0.0746 

0.0895 

7.4 

0.2300 

0.8510 

34 

1.0569 

17.967 

2.5 

0.0777 

0.0971 

7.5 

0.2331 

0.8741 

35 

1.0879 

19.038 

2.6 

0.0808 

0.1050 

7.6 

0.2362 

0.8975 

36 

1.1190 

20.142 

2.7 

0.0839 

0.1135 

7.7 

0.2393 

0.9213 

37 

1.1501 

21.277 

2.8 

0.0870 

0.1218 

7.8 

0.2424 

0.9453 

38 

1.1812 

22.443 

2.9 

0.0901 

0.1305 

7.9 

0.2455 

0.9697 

39 

1.2123 

23.640 

3. 

0.0932 

0.1398 

8 . 

0.2487 

0.9948 

40 

1.2434 

24.868 

3.1 

0.0963 

0.1492 

8.1 

0.2518 

1.0168 

41 

1.2745 

26.127 

3.2 

0.0994 

0.1590 

8.2 

0.2549 

1.0451 

42 

1.3056 

27.417 

3.3 

0.1025 

0.1691 

8.3 

0.2580 

1.0707 

43 

1.3367 

28.739 

3.4 

0.1054 

0.1795 

8.4 

0.2611 

1.0966 

44 

1.3678 

29.407 

«3*5 

0.1O87 

0.1886 

8.5 

0.2642 

1.1228 

45 

1.3989 

31.475 

3.6 

0.1118 

0.2012 

8.6 

0.2673 

1.1494 

46 

1.4300 

32.890 

3.7 

0.1149 

0.2125 

8.7 

0.2704 

1.1762 

47 

1.4611 

34.336 

3.8 

0.1170 

0.2223 

8.8 

0.2735 

1.2034 

48 

1.4922 

35.813 

3.9 

0.1201 

0.2355 

8.9 

0.2766 

1.2259 

49 

1.5233 

37.321 

4. 

0.1243 

0.2486 

9. 

0.2797 

1.2586 

50 

1.5544 

38.830 

4.1 

0.1274 

0.2611 

9.1 

0.2828 

1.2867 

51 

1.5854 

40.413 

4.2 

0.1305 

0.2740 

9.2 

0.2859 

1.3151 

52 

1.6165 

42.029 

4.3 

0.1336 

0.2872 

9.3 

0.2890 

1.3438 

53 

1.6475 

43.659 

4.4 

0.1367 

0.2939 

9.4 

0.2921 

1.3729 

54 

1.6786 

45.322 

4.5 

0.1398 

0.3145 

9.5 

0.2952 

1.4022 

55 

1.7097 

47.017 

4.6 

0.1429 

0.3286 

9.6 

0.2983 

1.4318 

56 

1.7407 

48.740 

4.7 

1.1460 

0.3431 

9.7 

0.3014 

1.4618 

57 

1.7718 

50.396 

4.8 

0.1491 

0.3578 

9.8 

0.3045 

1.4920 

58 

1.8029 

52.284 

4.9 

0.1522 

0.3729 

9.9 

0.3076 

1.5226 

59 

1.8340 

54.103 

5. 

0.1554 

0.3885 

10 . 

0.3108 

1.5540 

60 

1.8651 

55.953 































Falling Bodies. 441 





X 

V 








Falling Bodies. 





„ 2 S 


hs 


. a T* 



v= — 

‘ . 

T-- 

=\— 






T 


\ 9 



2 


V 

T 

S 

V 

T 

S 

V 

T 

S 

G5 

2.0206 

65.669 

530 

16.478 

4366.6 

1030 

32.027 

16494 

70 

2.1769 

76.260 

540 

16.788 

4452.8 

1040 

32.338 

16815 

75 

2.3314 

87.427 

550 

17.099 

4701.7 

1050 

32.649 

17141 

80 

2.4868 

97.472 

560 

17.409 

4874.5 

1060 

32.950 

17463 

85 

2.6422 

112.29 

570 

17.720 

5050.2 

1070 

33.261 

17794 

90 

2.7976 

125.89 

580 

18.030 

5228.7 

1080 

33.572 

18129 

95 

2.9530 

140.27 

590 

18.341 

5410.6 

1090 

33.883 

18446 

100 

3.1085 

155.42 

600 

18.651 

5595.3 

1100 

34.194 

18806 

110 

3.4194 

188.07 

610 

18.961 

5783.1 

1110 

34.504 

19149 

120 

3.7302 

223.81 

620 

19.271 

5974.0 

1120 

34.815 

19496 

130 

4.0411 

262.67 

630 

19.582 

6168.3 

1130 

35.126 

19846 

140 

4.3519 

304.63 

640 

19.893 

6365.7 

1140 

35.436 

20198 

150 

4.6627 

349.70 

650 

20.204 

6566.3 

1150 

35.747 

20504 

100 

4.9736 

397.88 

660 

20.515 

6770.0 

1160 

36.058 

20913 

170 

5.2844 

449.18 

670 

20.826 

6976.7 

1170 

36.369 

21275 

180 

5.5953 

503.36 

680 

21.137 

7186.6 

1180 

36.680 

21641 

190 

5.9061 

561.08 

690 

21.448 

7399.5 

1190 

36.991 

22009 

200 

6.2170 

621.70 

700 

21.759 

7615.6 

1200 

37.302 

22381 

210 

6.5279 

689.43 

710 

22.070 

7834.8 

1210 

37.613 

22755 

220 

6.8387 

752.26 

720 

22.380 

8056.8 

1220 

37.924 

23133 

230 

7.1496 

822.20 

730 

22.691 

8282.2 

1230 

38.235 

23514 

240 

7.4604 

895.25 

740 

23.002 

8510.7 

1240 

38.546 

23898 

250 

7.7713 

971.41 

750 

23.313 

8742.4 

1250 

38.857 

24283 

260 

8.0821 

1050.6 

760 

23.623 

8976.7 

1260 

39.168 

24676 

270 

8.3930 

1133.1 

770 

23.934 

9214.6 

1270 

39.479 

25069 

280 

8.7038 

1218.5 

780 

24.245 

9455.5 

1280 

39.780 

25459 

290 

9.0147 

1308.2 

790 

24.556 

9699.6 

1290 

40.090 

. 25855 

300 

9.3255 

1398.8 

800 

24.868 

9947.2 

1300 

40.411 

26267 

310 

9.6363 

1493.7 

810 

25.179 

10197 

1310 

40.722 

26673 

320 

9.9472 

1591.6 

820 

25.490 

10451 

1320 

41.033 

27081 

330 

10.258 

1690.6 

830 

25.801 

10707 

1330 

41.343 

27493 

340 

10.569 

1791.7 

840 

26.112 

10967 

1340 

41.654 

27908 

350 

10.879 

1903.8 

850 

26.423 

11230 

1350 

41.965 

28326 

360 

11.190 

2014.2 

860 

26.733 

11495 

1360 

42.276 

28747 

370 

11.501 

2127.7 

870 

27.044 

11764 

1370 

42.587 

29172 

380 

11.812 

2244.3 

880 

27.354 

12035 

1380 

42.897 

29599 

390 

12.123 

2364.0 

890 

27.665 

12311 

1390 

43.208 

30029 

400 

12.434 

2486.8 

'900 

27.976 

12589 

1400 

43.519 

30463 

410 

12.745 

2612.7 

910 

28.287 

12871 

1410 

43.820 

30893 

420 

13.055 

2741.5 

920 

28.598 

13155 

1420 

44.131 

31333 

430 

13.366 

2873.7 

930 

28.908 

13442 

1430 

44.442 

31776 

440 

13.677 

3008.9 

940 

29.219 

13733 

1440 

44.753 

32222 

450 

13.989 

3144.8 

950 

29.530 

14027 

1450 

45.064 

32671 

460 

14.300 

3289.0 

960 

29.841 

14323 

1460 

45.375 

33123 

470 

14.611 

3433.6 

970 

30.152 

14623 

1470 

45.686 

33579 

480 

14.922 

3581.3 

980 

30.463 

14927 

1480 

45.997 

34037 

490 

15.233 

3732.1 

990 

30.774 

15233 

1490 

46.308 

34499 

500 

15.545 

3886.2 

1000 

31.085 

15542 

1500 

46.631 

34973 

5i0 

15.856 

4043.3 

1010 

31.396 

15855 

1510 

46.732 

35082 

520 

16.167 

4203.4 

1020 

31.707 

16179 

1520 

47.043 

35752 































•M2 


Dynamics of Matter. 


Dynamics of Matter. 

Matter is that, of which bodies are composed, and occupies space. 

Matter is recognized as substance in contradistinction front geometrical 
quantities and physical phenomena, such as color, shadow, light, heat, elec¬ 
tricity, and magnetism. . , _ ,. .. . 

Chemistry has, thus far, dissolved matter into some sixtv-five distinct 
elements, but in the philosophy of mechanics we treat matter only as one 
simple element in relation to the three physical elements force, motion, ana 

These four elements, force F, motion V, time T, and mass M, are what con¬ 
stitute nature, and their different combinations cause the phenomena which 
we study and observe. 

Mass anti "Weight,. 

Mass is the real quantity of matter in a body, and is proportional to 
weight when compared in one or the same locality. The mass of a body is 
a constant quantity, whilst the weight of the same body varies with the 
force of gravitv which produces it. 

Inertia is the incapability of a dead body to change its own state of 
motion or rest. 

Force of inertia is the resistance a body free to move presents to any 
external force acting to change its state of motion or rest. 

Let a constant force F be applied to a body It 
free to move; then the body will start and con¬ 
tinue with an accelerated velocity until the force 
ceases to act, when it will continue in the same 
direction with auniform velocity equal to that of 
the final action of the force, and will never stop 
unless a force act upon it iu the opposite direction. If the force F were equal 
to the weight IF, then the acceleration would be the same as when falling 
freely under the action of gravity—namely, 32.17 feet per second; and if 
the force F is greater or smaller than the weight of the mass, the accel¬ 
eration will be proportionally greater or smaller. 

Any force, however small, is able to set in motion any body free to move, 
or to bring to rest any moving body, however large. 

No force is required 10 maintain a uniform motion of a body free to move, 
but force is required to bring a body from rest into a uniform motion. If 
force is applied to maintain a body not free to move in uniform motion, such 
force is expended in overcoming the friction and resistance of the medium 
in which the body moves. A steamboat or a railway train in motion is thus 
suspended between the action of two opposite forces—namely, the driving 
force on the one side, and the friction and resistance on the other. When 
the opposite forces are equal, the motion will be uniform; and any change 
of velocity is due to a disparity between these opposing forces. 

Now we are ready to combine the four physical quantities, force, velocity, 
time, and mass, into their functions in dynamics, where they bear the foi- 





lowing relations: 


M : F — T: V. 


Momentum M V— FT, the acting force into lime. 

When F is expressed in pounds, Tin seconds, and Fin feet per second, 
then the unit mass, or M, will be 32.17 pounds, which is equal to the accel- 
eratrix g for a falling body at the surface of the earth. 

, , W W 

Mass M — — = --• 

g 32.17 

To get the mass of a body is only to divide its weight in pounds by the 
acceleratrix 32.17, and the quotient is the mass. 







Dynami Iatteb. 


44:? 


Force, Power, and Work in Moving Bodies. 


It requires force, power, and work to change the state of motion or rest 
of a body. 

In the dynamic expression M V= FT we have 


t- c. MV 

Force F= - 

• 

• 

. 1. 

Ar- FT 

M - y . . 

• 

• 

• 

. 3. 

r MV 

1 ~ F * 

• 

• 

. 2. 

F T 

V — _ - 
M 

• 

• 

• 

. 4. 


The force F required to set a mass M in motion with velocity V depends 
inversely on the time Tof action. The more time the less need the force be 
for a certain velocity, and therefore it cannot be determined what force lias 
set a mass in motion without knowing its time of action ; but when the mass 
and its velocity are given, then we can determine the exact amount of work 
bestowed on the motion. 

Multiply the dynamic momentum by the velocity V, and we have 

MV 2 = FVT. 

V 

Here we recognize the work -iT, which is that bestowed on the mass M 

in giving it the velocity V, or the mass multiplied by | the square of its veloc¬ 
ity is the work stored in it. 

Vis-viva.—The term MV 2 has formerly been called vis-viva, but that 
term is now seldom used. 

The real work in foot-pounls is ^MV 2 = ^FVT. The space S in which the 
mass was set in motion is S—^V1\ which inserted in the formula gives the 

Work IC= id/ V 2 = FS. 

The following table of formulas will show what a variety of problems are 
connected with a force acting on a body free to move. 

When a body is left free to the action of gravity in falling or rising, the 
acceleratrix G = g, and the force F= W. 

Example 1. What force /’=? is required to give a body W= 1689 pounds a 
velocity of V= 36 feet per second in a time T= 5.6 seconds? 

Find in the formulas under constant force the one which contains the given 
quantities )V, V, and /, which is the second formula. 

F= = 168 . 9 X ffL = 337.55 pounds, the answer. 

\g T 32.166 X 5.6 F ’ • 

Example 2. A projectile of TF=150 pounds is fired horizontally from a 
rifled gun of S— 11 feet in length, in which it receives a velocity of V — 950 
feet, per second. Required the mean force F=? of the powder acting on the 
projectile, when the friction in the rifle is 230 pounds, 
ip - 172 150 V 950 2 

F— -=-—--— = 191302 + 230 = 191532 pounds, the force re- 

2gS 2 X 32.166 XU 
quired. 


Example 2\. What force is required to give a mass W— 6386 pounds a veloc¬ 
ity of V— 160 feet per second when acting in a space S —15 feet? 

IV V 2 6386 X 1602 . 

F= -- ^— T-r^r = 425500 pounds. 

2 g 8 2 X 32.17 X lo 

Example 2}. What force is required on a mass free to move W=1500 
pounds to move it S= 60 feet in T=2\ seconds? 

1FS 1500 X 60 . 

F= ——i ^ - X - x z- = 447.63 pounds. 

g T 2 32.17 X 2.52 r 


























444 


'■'l-.-Ti f j: • 3. 


Example ^.—The moving parts in a propeller steam-engine, such as the steam- 
piston, piston-rods, cross-heads, connecting-rod, &c. &C., weigh IF = 8456 
pounds. Stroke of piston = 4 feet, making « = 52 revolutions per minute. 
What force F is required for each stroke, to set in motion and bring to rest the 
moving mass ? 

The velocity of the moving mass at half stroke will be (formula , page 263) 


F — 


2 ir rn 
60 


2 X 3-1416 X 2 X 52 
60 


10*79 feet per second. 


The time for each half stroke will be 


T = 


60 


4 X 52 

Then the required mean force will be 

WV 8456 X 10-79 


= 0-2S846 seconds. 


F = 


g T 32-166 X 0-2846 


= 9966*8 pounds. 


For high grade of expansion of steam, this force acts beneficially to the move¬ 
ment of the engine. 

Example 4.—The mean force of gunpowder in a rifled gun is known to be 
231400 pounds, on a projectile W = 180 lbs. The friction of the projectile 
through the gun is estimated at 264 pounds, leaving F = 231400 — 264 = 231136 
pounds. The length of the gun is S = 12 feet, elevated to an angle x = 6° SO 7 . 
Required the velocity V = ? of the projectile when it leaves the gun. 



\ 


2 X 32166 X 12 


/231136 

V 180 


sin. 6 ° 30 


') 


= 995 64 feet per second, the answer. 

Example 5.—What velocity V can a steam-engine of H = 56 horses impart to a 
body IF = 9 tons in a time T— 30 seconds? 


P= 56 X 550 = 19800 effects, and TF = 9 X 2240 = 20160 lbs. 


V — 


\ *gPT _ f 

\ w yj 


2 X 32-166 X 19800 X 30 
20160 


= 43-538 feet per second. 


Example 6 .—A body IF = 3685 lbs. is moving with a velocity F = 56 feet per 
second. What time T= ? is required to bring that body to rest, with a force 
F — 128 pounds? 


IFF 3685 X 56 
g F ~ 32-166 X 128 


50121 pounds, the answer. 


Example 7.—What power P >s required to drive a centrifugal gun to throw out 
balls of W = 50 lbs. every T — 8 seconds, with a velocity V— 785 feet per second 
(friction omitted) ? 


IF F2 

Wt 


50 X 7852 
2 X 32-166 X 8 


= 59867 effects, 


divided by 550 = 108-85 horses, the power required. 


Example 8 .—A sledge of TF = 20 lbs. strikes a spike into a log S = 0-08 foot, 
with a velocity of F = 25 feet per second. Required the force F= ? with which 
the spike was driven into the log, omitting the weight of the spike. 


TF F2 

" 27 ~s 


20 X 252 

2 X 32-166 X 0 - 0 S 


= 2628-9 pounds. 


Example 9.—A body starts to ascend vertically with a velocity of 860 feet per 
second. What will be its velocity at the end of T = 5 seconds? 

V = G T = 32-166 X 5 = 160 830 feet per second, 
and 860 — 160 S3 = 699-17 feet per second, the answer. 






















Dy»> 


445 


Dynamical Formula lor crated, or Retarded 

Motion. 


1 


GW 

a 


Constant Force in Pounds acting on a Body free to move. 
WV 2 TFS W V 2 


P T I 

s “V 


2 P W 

gT 


2 K 
G T 2 


Y—G T= 


g T g T 2 2gS 
Final Velocity in the Time T, or Uniform Velocity of a Moving Body. 
9 F T 


W 


r = 2 5 = [ 
T ~\ 


2g SF 

W 


=j/ 2 G S=^J^= 


\ 2g P T _ j 

\ W \ 


K 

S 


12 gK 

w 


_F _ IFF 
~ gF 


Time in Seconds in which the Force acts on the Body free to move. 

2 F S 2 K 


12 WS 125 _ 
9F 


V K 


2SW 2WK 

■\lg p * 


gT F 


G = 


Constant Acceleration of the Force Fin Feet per Second, 
g F 2 S V F2 g FT F F2 ,7 A" 


2 K 


S = 


TF F 2 T 2 S WS PT WS FT 2 

Space in Feet in which the Force acts on the Body free to move. 

G T 2 _ V T _ _F^ _ g FT 2 _ P T _ g PT 2 _ g K _ Jf 


TF = = 


2 2 G 2 W F W V 

Weight in Pounds of the Moving Body. 
gFT 2 2g FS g F T gPTf gFfT 


GW 


2 g K gT 2 K 
2 S* 


P = 


G 2 S F2 F 2 £2 2 P V* 

Mean Power in Effects during the Time T, or in the Space S. 

F S g F 2 T 2 TF £2 TF F2 _ 2 K __ T K _ VK _ F V 2 


K= FS = 


2 W g T* 2 g T T 2 S S G T 

Work in Footpounds concentrated in a Moving Body. 

W F2 Fjf T GWVT F G T 2 _g F 2 T 2 2 S P 


2 W 


T 




2 g 2 2 g 

Tlxe Body moving in an Inclined Direction of an 

Angle x. 

Applied Constant Force in Pounds. 

F = W {/f ± sin.*) = TF (|^ 2 ± sin.x) = TF ± sin.x) 

Final Velocity in Feet per Seconds when the Force F ceases to act. 

V 

Time of Action in Seconds. 

~yf 


PT. 


(F . \ 2 S sin.x I 7^ \ 

= Bm - X ) = ~~¥~ = -J 2 g S T sin.x J. 

Acceleration. 

G = 6in - a: )- 


TF F 


2 IF S 


g (Fr W sin.x) 


S- 


g(Fr TFsin.x) 
Space in Feet. 


9 T 


Work done by F. 

. \ Fq T 2 (F . \ 

T sm - x ) = ~2 T sin -* ) • 


Use the upper sign when the direction of motion rises above the horizon, and 
the lower sign when the direction of motion dips under the horizon. 















































































446 


I'LY-V HEi.J.S. 


Force and Work In ltevolvins Bodies. Centre of 
Gyration. Fly-Wheels. 

Centre of gyration is a puiut in revolving bodies in which, if all the revolving 
matter were there collected, it would obtain equal angular velocity from, and 
sustain equal resistance to, the force that gives it the rotary motion. 

The centre of gyration in different forms of bodies will be found by the for¬ 
mulas on pages 316 and 317. 

F = constant force in pounds, acting to rotate the body as in figs. 249 and 
250, or the mean force on a steam-piston. 

r — radius in feet upon which the force F acts. For a steam-engine the mean 
radius will be r — 0 63661 X the radius of the crank, or 0.31S3 S, when 

<S = stroke of the steam-piston in feet. 

W = weight in pounds of a fly-wheel, or other rotating body. 

x = radius of centre gyration in feet. 

T = time in seconds in which the force Fis applied from the first start, or 
the time iu which the velocity is accelerated. 

N — number of revolutions in the time T. 

n = number of revolutions per minute. 

K = work concentrated in the revolving body. 

f — irregularity in a fraction of the mean revolutions n. 

For a double-acting single-cylinder engine, the fly-wheel in its regular course 
of running has an irregular velocity through each revolution. Its smallest 
velocity is when the crank is at an angle of 40° from the beginning of the 
stroke, and its greatest velocity when at 40° from the end of the stroke. The 
larger the fly-wheel is for a given velocity, the more regular will the machinery 
run. But the fly-wheel may be made so small that its accumulated work cannot 
carry the machinery around, which will be the case when the irregularity / = 1 . 
In ordinary practice make irregularity / =0 1 to 0’01. 

Example 1.—What force F is required to give a body IF — 3G00 pounds a veloc¬ 
ity n = 76 revolutions per minute in a time T= 24 seconds, the radius of gyration 
being x = 12 feet, and the force F acting on a radius r — 3 feet ? 


F= 


Wx^n 
30 iTYr 


3G00 X 1‘2 2 X 76 
307-1 X 24 X 3 


= 1779-5 pounds, the answer. 


Example 2.—Required the weight IF of a fly-wheel for an engine of D = 36 
inches diameter of cylinder double acting, with steam-pressure p = 50 lbs. per 
sq. in. S — 6 feet, the stroke of piston. Area of steam-piston 1017*8 sq. in., 
and the force F = 1017-8 X 50 = 50891) pounds. Radius of gyration x = 10 feet, 
and n = 48 revolutions per minute. Assume /= 0 05. 

__ 2542 FS 2542 X 50890 X 6 „ 

W = ~~nf~xff = 482 x 10 2 X 0 05' = 6 ‘ 3 ‘ 6 ’ 2 P 0uOds > the weight required. 

Should the steam be usecl expansively, the fly-wheel ought to be so much 
heavier, as the initial pressure is greater than the moan pressure. 

The radius of gyration in a fly-wheel, including the arms, can in practice be 
assumed to be the inner radius of the ring. 

Example 3.—What time from the start of engine is required to give the fly¬ 
wheel in Ex. 2 a velocity of n = 48 turns per minute ? r = 0 31S3 <S= 1-9098 ft. 


IF a* n 67376-2 X 10 2 X 48 

307-17 F r 307-17 X 50S90 X 1-9098 


10-85 seconds. 


Example 4.—Let the steam-engine in the preceding examples be applied to a 
rolling-mill, geared two to one of the rollers. An iron plate is rolled through with 
iF= 8 revolutions of the engines, after which the revolutions were found to be 
reduced to n x = 36 per minute. Required the work done in rolling the plate; 
and what time is required for the engine to regain the n = 48 revolutions? 

Work done by engine, K = 2 F S N = 2 X 67376-2 X 6 X 8 = 646S115-2 footps. 

Work done by ily-whoel, 


K = 


IF a : 2 Q2 — nf) _ 67376-2 X 10 2 (4S 2 — 3G 2 ) 


5866-5 


5806-5 


1157671 footpounds, 


to which add 6468115-2 = 7625786*2 footpounds, work consumed in rolling plate. 
The time required for the engine to make up the n = 48 revolutions will be 


IF afi ( n — n x ) _ 67376-2 X 102 (48 — 36) 
307-17 F r ~ 307-17 X 50890 X 1-9098 


2-71 seconds. 


























ClRCUT'I; '■ * •. 


447 


Formulas for Accelerated. Circular Motion. 

Force F, in pounds, acting on the Lever or Radius r, to rotate the Body. 
W x 2 n W x 2 N 60 K K 


F = 


307 49 Tr 2 ' 562 T 2 r n r n T 2 u r N 
Final Revolutions per Minute in the Time T. 


120 N 307-49 F 


T 


W x2 


Tr 60 K 1 5872-2 K 

nr T F 'Xj Wx* 


Total Number of Revolutions in the Time T. 

K T 


Tn 2-562 FT 2 r 
~ 120 ~ \Vx 2 2 t t r F ~ 1 ; 565 




K 

W 


Time of Acceleration, in Seconds, from the Start of Change of Motion. 
W x 2 n 


307*49 F r 


j W x 
\j 2 - 56 -, 


Wx 2 N _ 60 K _x\ WK 

562 Fr it r n F 4-09 F r* 


x = 


Radius of Gyration, in Feet, of the Revolving Body. 
I 2-562 Fr T 2 K T, 


307-49 FrT 
Wn 


334-9 K 


WN 


W = 


NyWN Fr nyWnTFr 

Weight, in Pounds, of the Revolving Body. 

307-49 T Fr 2-562 T 2 Fr 5872-2 K K T 2 


x i n 


x 2 N 


n 2 x 2 


2-452 x 2 N 2 


Work in Footpounds, concentrated in a Revolving Body. 


K = 


W x 2 n 2 2-452 Wx 2 N 2 n r n F T 


5872-2 


p2 


60 


= 27T r N F 


Fly-Wlieels for Steam-Engines. 

Fly-Wheel for a Single Acting Steam-Engine for Uniform Work. 

5872-2 F S 
x 2 n 2 W 


W— 


5872.2 F S 
n 2 x 2 f 


76-6 IPS 76-6 fFs 

n ~ x^Wf‘ X ~ n^jwf’ 


f = 


Fly-Wheel for a Double Acting Steam-Engine for Uniform Work. 


2542 FS 50.42 [FS 

W ~~ rfixUf' n ~ x jWf ’ 


x 


50-42 lYs 
n -yjwf' f 


2542 FS 
x 2 n 2 W 


Fly-Wheel for a Double Acting Two-Cylinder Engine for Uniform Work. 


m 1172 FS 

W— --• 

nfxif 


n 


34-23 / FS 

x -y Wf' 


x — 


34-23 IFS 
n yl Wf 


•/ = 


1172 FS 
x 2 n 2 W 


Whenever accurate work for expanding steam is required, consult the Rela¬ 
tive Proportions of the Steam-Engine (Marks). These formulae give only rough 
averages. 




































































418 


.TION\ 



1 c 

* «) 

| !| 

(! 

)• 



r H 

r 


■JC 


A 


<. ot 


A line or bar. 

v = 0-5773/, 

x = 0-2887/. 


240. 

A circumference round its diameter , 
yl o 'isk round its centre , 

A cylinder round its axis. 

£ = 0'7072r. 


241. 



A disk round its diameter. 

x = 0‘5r. 



242. 


-4 Sphere round its diameter. 

Spherical shell, & =■- 0-8165?-, 
Solid, ... x = 0*G324r. 




243. 




\o 




Parallelopipedon. 


x = 


/4/ a +A» 

V 12 ’ 

v / 4 


4W 1 
12 


Ifl’ta l 



1244. 


Cylinder. 

4/ a +3r» 
12 * 


*-\/ 




















































449 



Oxrp.E of GrtAXvm 

245. ~~ 


Cone. 


a 



a 



X = \J 
x ‘\/~ 


2/i a +3 R' 


20 


12/i s +3i2‘ 

20 * 


24G. 


Conic Frustum. 




R*-t3R r+R r*\ 

10 v i2 a -|--R r-fr* ' + 


3 ( fr — r* . 
20Vtf — r* J 


247. 


Cylinder and Sphere. 


x = \f a*+^r% 


x = V aHf-r*. 

o 


r^rz?- 


248. 


~<h 




Wedge and Ring, 
x = 0-204 v/12^+^+Z>% 

*=\/ : 


’/(Hr 1 
~2~* 


249. 


*Yy FM. 

tf a +r’ 

2 ’ 


x-\/ 


FG:Wg = x *: s’. 


2 GO. F/y Wheel with Arms. 

R*+r* 4r*+b* 

x*(W+w) = W—^+w ~\2~ y 


/6TT(ft s +r 8 )+w(4r 

* V l2(ir+w) 


+i*) 


29 




































































450 


Gbutritvqxl Port*. 


CENTRIFUGAL FORCE. 

Central Forces are of two kinds, centrifugal and centripetal. 

Centrifugal Force is the resistance which a revolving body offers to 
being moved in the arc of a circle. 

Centripetal Force is that by which a revolving body is attracted or at¬ 
tached to its centre of motion. 

The Centrifuged and Centripetal forces are opposites to each other, and when 
equal the body revolves in a circle; but when they differ the body will revolve 
in other curved lines, as the Ellipse, the Parabola, &c., according to the nature 
of the difference in the forces. If the centrifugal foi-ce is o while the other is 
acting, the body will move straight to the centre of motion ; and if the centripe¬ 
tal force is o while the other is acting, the body will depart from the circle in a 
straight line, tangent to the circle in the point where the centripetal force ceased 
to act. The central forces are distinct from the force that has set the body in 
motion. 

If the centrifugal force be made use of to produce an effect, such effect will bo 
at the expense of the one.producing the rotary motion. 

Notation. 

F= Centrifugal force, in pounds. 

W the weight of the revolving bod 3 r , in pounds. 
v — Velocity of the revolving body, in feet per second. 

It — Radius of the circle in which the body revolves, in feet. 

» = number of revolutions per minute. 

Example 1. Required the centrifugal force of a body weighing 63 pounds, and 
making 163 revolutions per minute, iu a circle of 4 feet, 4 inches radius? 




W It n* 
2933 


63X4-33X1633 , 

=*- 2933 - = 24/o pounds. 


Example 2 . 
115 feet radii. 


A Railroad train runs 43 miles per hour on a curved track of 
What should be the obliquity of the track? 

Miles* 

"60 if 


tan.a: = ■ 


43» 

69X115 — 0 ’ 233 > 


or x = 13° HE, the obliquity of the track. 

Example 3. A governor having its arms l — 1 foot, 6 inches, how many revol¬ 
utions mui;t it make per minute to form an angle x — 30° ? 

5416 

= 4r5 revolutlons P er minute * 


n 



997. 

W «* _ TF U’ 
g R 32*272 ’ 

- 1, 


F 4 WRrt'n* W R n* 

> - 2, 



60 2930 


II \ 

w _ Fg R 2930 F 

V 2 R ri* * 

3, 


Wv 2 293 OF 

Fg~ Wn" ‘ 

- 


/ 2930 F 

" \/ WR ’ w ~\/ 

/F~RJ - 

w ’ 5 ’ 





































Centrifi F I E . : ' V \ORS. 


451 



OOQ 

wuij. 


Centrifugal force of a ring. 


F = 


W ni 
4153 



229. 

Centrifugal force of a grinding stone , 
circle-plane, cylinder , rotating round 
its centre. 


^ W R n' 

F= ms- 



230. 

Centrifugal force of a cylinder rotating 
round the diameter of its base. 

n Wn 2 l 


5872* 


231. 



Centrifugal force of a ball , 
WnfR 
* 2936 * 



232. 


n = 


60 

2/t 


v/f 


Governor. 

54-16 54-16 


f h n/ l cos X * 


. 2936 , 2936 h 

h = —, l = : = 


n- 


rr cos.* cos.* 


cos.* = r = v' _ h\ 


n 


l l 

















































452 


Penduium 


PENDULUM. 

Simple Pen dura m is ? msierial point under the action of gravitation, 
and suspended at a lixed point by a line of no weight. 

Compound Pendulum is a suspended rod and body of sensible mag¬ 
nitude, lixed as the simple pendulum. 

Centre of Oscillation is a point at which if all the matter in the com¬ 
pound pendulum were there collected, it would make a simple pendulum oscil¬ 
late at the same times. 

Angle of Oscillation is the space a pendulum describes when in mo¬ 
tion. 

The velocity of an oscillating body through the vertical position, is equal to 
the velocity a body would obtain by falling vertically the distance versed sine of 
half the angle of oscillation. 

Notation. 

J = length of the simple pendulum, or the distance between the centre of sus¬ 
pension, and centre of oscillation in inches. 

t = time in seconds for n oscillations. 

n = number of single oscillations in the time t. 

Example 1. Required the length of a pendulum that will vibrate seconds? 

here n = 1. and t = 1". 
ta 

l — 39109 — = 39-109 inches, the length of a pendulum for seconds. 
n* 


Example 2. 
p**r minute? 


Require the length of a pendulum that will make 180 vibrations 
here t = 00" and n = ISO. 


39-109^ 

» a 


39-109 X 60 3 

180* 


= 4*346 inches. 


Example 3. IIow many vibrations will a pendulum of 25 inches length make 
In 8 seconds? 

6-254* 6-254X8 .. 

n = —- = —- - = 10 vibrations. 

)/25 

Example 4. A pendulum is 137'07 inches long and makes 8 vibrations in 15 
seconds. Required tlie unitor accelleratrix g =? 

= 0-8225? n a = <18225X137+7X8* = 32 . m 


** 


15* 


Example 5. A compound pendulum of two iron balls P and Q, having the 
centre of suspension between themselves: see Fig. 238. P — 38 pounds, Q = 12 
pounds, a = 25 inches, and b = 18 inches. How long is the simple pendulum, 
and how many vibrations will the pendulum make in 10 seconds? 


x 


a P — b Q 25X38 — 18X12 
P+<2 38+12 


= 14-68 inches. 


a % p+t,* Q 

~x(P+W 


25 a X38+18«X12 

14-68(38+12) 


= 37-68 inches, 


the length of the single pendulum. 


6-254* 6-254X10 

n = —«= •— n-r —- = 10193 vibrations in 10 seconds. 
yV |/37'68 


If a compound pendulum is hung up at its centre of oscillation, the former 
centre of suspension will be the centre of oscillation, and the pendulum will 
oscillate the same time. 




















FEMmTLOM AND OENTRE OF OSCILLATION 


453 




I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

&• 


\ 

I 

I 

1 

t 

» 

\ 


233. 

Simple Pendulum. 


I = 




\2<r t* __ 39-IP 
Tt 1 n* n* ’ 


n f l 
6.25’ 


6 254/ 
n ' JT' 



236. 


In'-n* 


g ~ 12z * ’ 


(7 =• 


0-8225/n a 
P 


o = centre of s us pen- 
si on. 

, 2r* 

b i 


"T"T 4 

A 7T~ 

Z 


.4 


£ 


* — 

c- 

N*.- 



234. 

^4 = centre of grav¬ 
ity. 

P = centre of gyra¬ 
tion. 

C centre of oscil 
lation. 

a : b — b : l, 

b= \b l'414a, 

l = lha. 



237. 


; a’ P+3 3 Q 

o P+3 Q" 

P and Q expressed 
in pounds, or cubic 
contents. 

Connecting wire 
neglected. 



a P — bQ 
x ~ P+Q 


a’P+3' 1 Q 

= lf(P+QT' 

Connecting wire 
neglected. 


Length of a Pendulum vibrating second* at the level of the sea, 

At the Equator lat. 0°0'0" - ] I.* I 39-0958 “ 

“ Washington, lat. 38 53 33 _ 39-1017 “ 

“ New York, lat. 40° 42 40 * 39-1393 “ 

“ London, lat. 51° .._ 39-1270 “ 

«< lat. 45°. on-igis « 

“ *•**»>“. 0'09982*co..2 Ut.Vor /econk _ 















































































ATA 


Collision of Bodies in Motion. 


COLLISION OF BODIES IN MOTION. 

When bodies in motion come in collision with each other, the sum of 
their concentrated momentum will be the same after the collision as 
before, but their velocities aud sometimes their directions of motion will 
differ. 

On the accompanying page the bodies are supposed to move in the 
same straight line, and the formula illustrates the consequences after 
collision. 

Notation. 

M and m = weight of the bodies in pounds. 

F and v — their respective velocities in feet per second. 

V' and v' = respective velocities of the bodies after impact. 

K and A; = coefficient of elasticity, which for perfectly hard bodies k = 0 
and for perfectly elastic bodies * = 1, therefore the elastic coefficient will 
always be between 0 and 1. When the bodies are perfectly hard their 
velocities after impact will be common. 


For My K- 


MV 


M (F— F')’ 


For m, k = - 


mv 


m (v-V) 


Example 1. Fig 191. The non-elastic body weighs M=25 pounds, and 
moves at a velocity F=12 feet per second ; to = 16 pounds, and u=9. Re¬ 
quired the bodies’ common velocities, v'=1 after impact. 

v = - - - =————= 10-83 feet per second. 

M+m 26+16 

Example 2. Fig. 195. The perfect elastic body Af=84 pounds, F=18, 
in ^=48, and v=27. Required the velocity V'—l 'after impact with the 
body m. 

v= 18 (84-48) -2X48X27 _ 03 . 64 

84+48 

the negative sign denotes that the body will return after the collision 
with a velocity of 23 63 feet per second. 

Example 3. Fig. 196. The partly elastic body jVf=38 pounds and F=79 
feet per second, will strike the body in rest m =24 pounds ; what will be 
the velocity t>'=l of the body m, its elasticity being k'— 0-6- 


v'= 


79X38 (1+0-6) 
~~SS +24 ’ 


< 70-6 feet per second. 


When a moving body strikes a stationary elastic plane, its course of 
departure from the plane will be equal to its course of incidence. 

1 k Problem. A body in a is to strike the plane 
AH so that it will depart to the given point b ; 

I required its course of incident from af 

A _JDL_Draw bd, at right angles through AB, make 

S j cd^bc join a and d ; then ad is the course of in- 

L/ cident, and eft, the course of departure, and tbfc 
body will strike in e. 

In this article the common error of most text-books arising from igno¬ 
rance of the true meaning of momentum il/Fis shared bv the author. Ex¬ 
ample 1 should be written : 


F'2 = M F 2 + mv * = 25 X 144 + 16 X*81 _ { 4J 


V' = 10-93. 


|il/+m 25 + 16 

This remark holds with regard to all the formulEe following, which are 
allowed to remain with this caution. De Volson Wood’s Elementary Me¬ 
chanics is an elaborate example of this form of error.—W. D. M. 




















Impact of Bodies. Dynamics. 


455 





V... 




191. 


The bodies move in the same direction, 
v' (M-\-m) — MV+mv, 

MV+mv 

v'= -=-•• 

A/4" m 


192. 



JV 


V \ 

,k ; 

. NX 






'\7T=0 






IT 


v' \ 

,'s.j 




The bodies move in opposite directions. 
v r (M+m) — MV — mv , 

, MV — mv 

r'=- 

M+m * 


193. 

Only one body in motion, 
v' (M+m) = MV, 

MV 

V '~ M+m ' 


194. 


The bodies move in the same direction. 
V(M— Km)+vm(\+K) 


V=- 


M+m 


, MV(l+k)+v(m—kM) 

M+m 



195. 


The bodies move in opposite directions. 
V (M — Km) — vm(l+ K) 


V 


M+m 

MV(l+k)—v (m—JcM) 


M+m. 


V 






"—- 


196. 


Only one body in motion. 
V(M— Km) 


V= 


M+m 


, VM(l+k) 

IV —- —-—-• 

M+m 


© 


bodies perfectly hard. The bodies elastic. 
































































456 


Centre o v Gravity. 


CENTRE OF PERCUSSION. 

Centre of Percussion is a point in which the momentums of a moving 
body are concentrated. Centre of 1 ‘emission is the same as centre of oscillation, 
and to be calculated by the same formulas. 

Take an iron bar in one hand, and strike it heavily upon a sharp edge; if the 
centre of percussion of the bar strikes upon the edge, the whole momentum will 
there be discharged, but if it strikes at a distance from the centre of percussion a 
part of the momentum will be discharged in the hand, and a shock will be felt. 

It is sometimes of great importance to properly place the centre of percussion 
If it is dislocated, the moving body not only fails to properly transmit its effect, 
but the lost momentum acts to wear out the machinery. 




CENTRE OF GRAVITY. 

Centre of Gravity is a point around which the moments of all elements 
(under the action of the force of gravity) in a body, or system of bodies, are 
equally divided. 



A body or system of bodies suspended at its centre of 
gravity, will be in equilibrium iu all positions. 

A body or system of bodies, suspended in a point out of 
its centre of gravity, will bang with its centre of gravity ver¬ 
tical under the point of suspension. 

A body or system of bodies suspended successively at two 
points out of its centre of gravity, the two vertical lines 
through the points of suspension will meet in the centre of 
gravity ; thus if a plane be hung up in two different posi¬ 
tions, the vertical lines a, b, and c, d, will meet in the cen¬ 
tre of gravity o. 

z = distance to the centre of gravity as noted in the 
figures. 

Example 1. The radius of a circle being 3 feet, how far is 
its centre of gravity from the centre of the half circle 1 

z = 0-6367 X 3 = 1*91 feet. 

Example 2. How far from the bottom of a cylindric shell, 
open at one end, is its centre of gravity ? The cylinder is 
4 feet long, radius r = 0-8 feet. 


z 


h _ 4 

r+2/i — 0-8+2X4 


= 0-625 feet. 


Example 3. Fig. 2C4. An irregular figure weighing P — 138 pounds, is sus¬ 
pended between a fulcrum and a weight, i = 5-6 feet, IK = 57 pounds. Re¬ 
quired the distance to the centre of gravity z — ? 


57X5 -6 _ 

138 


2*31 feet. 




















Otntrr oi ( 1 r at itt. 457 


1 

252. ] 

(Quadrangle. — a and A parallel . 

A A /A — cu 

2 6 6+a '* 


253. 

Triangle. 

A 

2 ^3* 

■ 7* 

254. 

Half a circle plane or Elliptic plane. 

z = 0-4244/’. 

* 

255. 

Circle sector. 

o- .. 

ijC / 

Z ~ M' 


256. 

Circle Segment. a = area. 

c 3 

~ 12a 

a? = A+z — r. 

A 

257. Parabola. 

2 A 

5 ‘T- 

For half a Parabola x =~ -=b. 

0 

















































458 


(Jentob or Gtuvm 



Half Sphere. 


Convex surface 
Solid . . . . 


• • • 


z 

z 


fr. 



-Ir- - 




< z ' 



Solid, 


Spherical Sector. 


-*(—£)• 


260. Spherical Segment. 
Convex surface z = 

A 

&,id ' 2 = HH'] 


261. 


Cone. 


Convex surface z = 


Solid 


h 


2C2. 


Con. sur 


Conic Frustum. 

_h /i r 7? — r I 

• 2 ~ 2 6 L R+V'J 


solid , * 

4 L R*+r(R+r) J 


263. 


fyramidic Frustum. 

A and a = area of the two bases. 

Solid z = \ [ A+3a+2v/ ^ a 1 

4 L A a J 


















































Centre op Gryvity. 


459 


264. 

Irregular Figure . 


P : W = l: z % 
Wl 



265. 


To Me Centre of Gravity of two 
bodies, P and Q. 

Q a u Pa 

z ~ P+(2 ’ = ~PfQ' 


266. 

To find the Centre of Gravity of a sys¬ 
tem of bodies. 

R a ~ Q d 
b P+R’ Z ~ P+R+Q’ 


267. 

Half a circumference of a Circle or 
Ellipse. 

z = 0 , 4244r. 

! 26ST 

Circle arc or Elliptic arc. 
c r c(c 5 + 4/P) 

2== T = ' 8 hb~ m 


269. 

Cylindric Surface with a bottom in one 
end. 

h 2 

1 “ r+2/i' 








































































460 


SPECIFIC GllAVITT. 


SFECIFIC GRAVITY. 

Specific Gravity is the comparative density of substances. The unit foi 
measuring the specific gravity is assumed to be the density of rain or of distilled 
water. 

One cubic foot of distilled water weighs 1000 ounces, or 62*5 pounds avoir¬ 
dupois. 

To Find the Weight of a Body. 

HULK 1. Multiply the contents of the body in cubic feet by 62 5, and the 
product by its specific gravity, will be the weight of the body in pounds 
avoirdupois. 

RULE 2. Multiply the contents of the body in cubic inches by 0*03016, 
and the product by its specific gravity, will be the weight of the body in 
pounds avoirdupois. 

RULE 3. Divide the specific gravity by 0*016 and the quotient is the weight 
of a cubic foot. 

Example 1. A bottle full of mercury is 3 inches, inside diameter, and 6 inches 
high. How much mercury is there in the bottle in pounds? 

One cubic inch of mercury weighs 0*491 pounds, and by the formula for 
Fig. 84 we have the 

weight = 0*491X0*785X3 2 X6 == 20*85 pounds. 

Example 2. Required the weight of a cone of cast iron, diameter at the 
base d = 1*33 feet, height h = 4 feet? One cubic foot of cast iron weighs 
450*5 pounds, and by formula for Fig. 82 we have the 

weight = 450*5Xlh2616Xl'33 a X‘l = 834 ftounds. 

Example 3. The section area of the lower hold in a steamboat is 245 square 
feet; how much space must be taken in the length of the hold for 131 tons of 
anthracite coal? 

Anthracite coal weighs 42*3 cubic feet per ton. 

length = ^ = 22*6 feet, the space required. 


Weight. and Bulk of Substances? 





Cubic 

Cubic a 


Cubic 

Cid>ic 

Names of Substances. 

foot 

in 

feet | 
per 

Names of Substances. 

foot 

in 

feet 

per 




pounds. 

ion. 


pounds. 

ton. 

Cast iron. 

• 


450*5 

4*97 

Sand, ... 

94*5 

23*7 

Wrought iron 



486*6 

4*60 

Granite, ... 

165 

13-5 

Steel, 

Copper, - 



4S9*8 

4*57 

Earth, loose, - 

78*6 

28-5 



555* 

4-03 

Water, salt, (sea) - 

64*3 

34-8 

Lead, 


. 

707*7 

3-16 

“ fresh - 

62*5 

359 

Brass, 



537*7 

4*16 

Ice, .... 
Gold, ... 

58*08 

38*50 

Tin, - 



456 

4-91 

1013 

2*21 

Pino, white 
“ yellow, 



29*56 

75*6 

Silver, 

551 

4*07 



83*81 

66*2 

Coal, Anthracite 

53 

42*3 

Mahogany, 



66*4 

33*8 

“ Bituminous - 

50 

44*S 

Marble, common. 


165 

13*6 

“ Cumberland - 

53 

42-3 

Mill-stone, 



130 

17*2 

“ Charcoal 

18*2 

123 

Oak, live - 



70 

320 

Coke, Midlothian - 

32*70 

68*5 

“ white, 



45.2 

49*5 

“ Cumberland - 

31*57 

70*9 

Clay, 



101*3 

22*1 

“ Natural Virginia 

46*64 

48*3 

Cotton Rales, 





Conventional rate of 



Brick, 

• 


100 

22*4 

Stone coal, 28 bushels 



Plaster Paris, 

- 


105 

21*3 

(5 pecks) = 1 ton, - 


43*56 



























Specific: Oka ra r. 


461 


To Find the Specific Gravity* 

W= weight of a body in the air. 

w = weight of the body (heavier than water) immersed in water. 
S — specific gravity of the body. Then, 

W 


w— tv : JF= 1: S. S = 


4 

Example 4. Required the specific gravity of a piece of iron-ore weighing 
6*245 pounds in the air, and 4 935 pounds in water, & = ? 

S = = 4*5 the specific gravity. 

When the body is lighter than water, annex to it a heavier body that is able 
to sink the lighter one. 

S — specific gravity of the heavier annexed body. 
s = specific gravity of the lighter body. 

W = weight of the two bodies in air. 
w — weight of the two bodies in water. 

V — weight of the heavier body in air. 
v = weight of the lighter body in air. 


IF— w’ 


1 , 


s = 


W- 


w -- 


V’ 

s 


Example 5. To a piece of wood, which weighs t> 14 pounds in the air, is 
fastened a piece of cast-iron V = 28 pounds; the two bodies together weigh 
w = 1T7 pounds in water. Required the specific gravity of the wood? 

W= F+v = 28+14 = 42 pounds. 

S = 7*2 specific gravity of cast-iron. 

14 

0*529, the specific 


Formula 2. 


5 = 


42 — 11-7 — 


28 

+2 


gravity of the wood, (Poplar White Spanish.) 

A simple way to obtain the specific gravity of wood is to make it into a rod and 
place it vertically in water; then, when in equilibrium, the immersed end is to the 
whole rod as the specific gravity is to 1. 

Example 6. A cylinder of wood is 6 feet, 3 inches long, when Immersed verti¬ 
cally in water it will sink 3 feet, 9 inches by its own weight. Required its spe¬ 
cific gravity. 

3-75 : 6-25 = S: 1, S= = 0*600. 

6*25 

To discover the Adulteration in Metals, or to find the proportions of (too Ingredients 

in a Compound. 


„ W-s(W-w) 

i-i ’ 

S 


3, 


Example 7. A metal compounded of silver and gold weighs TF = 6 pounds 
in the air, and in water w = 5*636 pounds. Require the proportions of silver 
and gold ? 

S = 19*36 specific gravity of gold, 
s = 10*51 specific gravity of silver. 

weight F = 6 ~ 10 -51(6 — 5.636 ) _ 4 . 755 pound8 of gold> 

10*51 

1 19*36 and 1*245 pounds of silver. 



















4(52 Sp! >ttc Gi. WITT. 




Weiolb 



Specific 

gravity. 

Names of Substances. 

Specific 

gravity. 

per 

cubic 

Names of Substances. 


inch. 




Metals* 


lbs. 




Platinum, rolled - - 

22-669 

•798 

Alabaster, white 


2-730 

“ wire, - - 

“ hammered, 

2P042 

•761 

“ yellow 


2-699 

20-337 

•736 

Coral, red - - * 


2-700 

“ purified, 

19-50 

•706 

Granite, Susquehanna 

2-704 

* crude, grams 

15-602 

•565 

“ Quincy 


2-652 

Gold, hammered - - 

19-361 

•700 

“ Patapsco 


2-640 

“ pure cast - - - 

19-25S 

•697 

“ Scotch - 


2-625 

“ 22 carats fine - 

17-486 

•733 

Marble, white Italian 

2-708 

« 20 “ 

15-702 

•568 

“ common 


2-686 

Mercury, solid at — 40° 

15-632 

•566 

Tale, black - - 


2-900 

“ at+32° Fakr. 

13-619 

•493 

Quartz,.... 


2-660 

“ “ 60° “ 

13-580 

•491 

Slate, .... 


2-672 

« « 212° “ 

13-375 

•484 

Pearl, oriental - 


2-650 

Lead, pure - - - - 

11-330 

•410 

Shale, .... 


2-600 

“ hammered - - 

11-388 

•412 

Elint, white - * 


2-594 

Silver, hammered - - 

10-511 

•381 

“ black - - 


2-5S2 

“ pure - - - - 

10-474 

•379 

Stone, common - 


2-520 

Bismuth, - - - - - 

9-823 

•355 

“ Bristol - 


2-510 

Red Lead, - - - - - 

8-940 

•324 

“ Mill - - 


2-484 

Cinnabar, ..... 

8-098 

•293 

“ Paving - 


2-416 

Manganese, .... 

8-030 

•290 

Gypsum, opaque 


2-168 

Copper, wire and rolled 

8-S78 

•321 

Grindstone, - * 


2-143 

“ pure--- - 

8-7S3 

•318 

Salt, common - 


2-130 

Bronze, gun metal 

8-700 

•315 

Saltpetre, - - * 


2090 

Brass, common - - - 

7-820 

•282 

Sulphur, native 


2-033 

Steel, cast steel - - - 

7-9J9 

•286 

Common soil, • 


1-984 

“ common soft - 

7-833 

•283 

Kotten stone, 


1-981 

“ hardened & temp. 

7-818 

•283 

Clay, .... 


1-930 

Iron, pure .... 

7-768 

•281 

Brick, .... 


1*900 

“ wrought and rolled 

7-780 

•282 

Nitre, .... 


1-900 

“ hammered - - 
“ cast-iron ... 

7-789 

7-207 

•282 

•261 

Plaster Paris, - 

- { 

1- 872 

2- 473 

Tin, from Bbhrnen 

7-312 

•205 

Ivory, 


1-822 

“ English 

7-291 

•264 

Sand, ... - 


1-800 

Zinc, rolled .... 

7-191 

•260 

Phosphorus, - - 

- - 

1-770 

“ cast. 

6-861 

•248 

Borax, - - - - 


1-714 

Antimony, .... 
Aluminium .... 

6-712 

2*5 

•244 

009 

Coal, Anthracite 

- { 

1-640 

1-436 

Arsenic, ..... 

5-763 

•208 

“ Maryland - 


1-355 

Stones ami Earths. 



“ Scotch - * 


1-300 



“ New Castle 


1-270 

Topaz, oriental - - 

4-011 

•145 

“ Bituminous 


1-270 

Emery,. 

4-000 

•144 

Charcoal, triturated * 

1-380 

Diamond, ..... 

3-521 

•127 

Earth, loose * - 


1-500 

Limestone, green - - 

3-180 

•115 

Amber, .... 


1-078 

“ white - - 

Asbestos, starry - - 
Glass, flint .... 

“ white .... 

3-156 

3-073 

2-933 

‘2-S92 

•114 

•111 

•106 

•104 

Pimstone, - - 
Lime, quick * • 
Charcoal, - - - 


1-647 

0-804 

0-441 

“ bottle .... 

2-732 

0987 

Woods (Dry*) 


“ green .... 

2-642 

•0954 

Alder, .... 


•800 

Marble, Parian ... 

2-838 

*103 

Apple-tree, i - 


•793 

African - - 

2-708 

•0978 

Ash, the trunk - 


•845 

Egyptian - - 

2-668 

•0964 

Bay-tree, - - - 


•822 

Mica,. 

2-800 

•1000 

Beech, .... 


•852 

Hone, white razor 

2-838 

•104 

Box, French - • 


•912 

Chalk, ...... 

2-784 

•100 

“ Dutch * * 


1-328 

Porphyry,. 

2-765 

•0999 

“ Brazilian red- • 

1-031 

Spar, green .... 

2-704 

•0976 

Cedar, wild - - 
“ Palestine 


•596 

“ blue .... 

2-693 

■0971 


•613 


Weight 

per 

cubic 

inch. 

lbs. 
'0987 
'0974 
•0974 
•0970 
'0958 
*0954 
•0948 
•097 S 
•0968 
•0105 
•0962 
•0965 
•0957 
•0940 
•0936 
•0933 
•0910 
•0906 
•0897 
•'0873 
•0783 
"0775 
•0770 
•0755 
•0735 
•0717 
0416 
•0698 
*0086 
•0636 
•0677 
•0S94 
•0059 
•0651 
•0640 
•0620 
•0593 
•0592 
•0490 
•0470 
•0460 
•0400 
•0500 
•0542 
‘03S7 
•0596 
•0291 
•0160 


•0289 

•0287 

•0306 

•0297 

•0308 

•0330 

•0480 

•0373 

•0219 

0222 























































Specific ivnr. 


463 


Names of Substances. 

‘ Cedar, Indian - 
“ American 
Citron, - 
Cocoa-wood, 
Cherry-tree, - 
Cork, ... 
Cypress, Spanish - 
Ebony, American - 
Indian 
Elder-tree, 

Elm, trunk of* 
Eilbert-tree, 

Pir, male ... 

“ female 

Hazel, ... 
Jasmine, Spanish • 
Juniper-tree, - 
Lemon-tree, 
Lignum-vitse, - 
Linden-tree, - 
Log-wood, 

Mas tic-tree 
Mahogany, 

Maple, ... 
Medlar, - 
Mulberry 

Oak, heart of, 60 old 
Orange-tree, - 
Pear-tree, 

Pomegranate-tree, - 
Poplar, - - 

“ white Spanish 
Plum-tree, 

Quince-tree, - 
Sassafras, 

Spru~e, - 
“ old 
Pine, yellow - 
“ white 

Vine, ... 
Walnut, ... 
Yew, Dutch 
“ Spanish - 

Liquids. 

Acid, Acetic - 
“ Nitric 
“ Sulphuric 
“ Muriatic 
“ Fluoric - 
“ Phosphoric - 
Alcohol, commercial 
“ pure 
Ammoniac, liquid - 
Beer, lager 
Champagne, - 
Cider, ... 
Ether, sulphuric * 
Egg, - - - - 

Honey, 

Human blood 
Milk, 


Specific 

gravity 

, Weigh 
| per 
. cubic 

Names of Substances. 

Specif 

gravity 

! Weight 
cj per 
. cubic 


inch. 


inch. 

1-315 

•0476 

Oil, Linseed - 

•940 

•0340 

•561 

•0203 

“ Olive 

•915 

•0331 

•726 

•0263 

“ Turpentine 

•870 

•0314 

1-040 

•0376 

“ Whale 

•932 

•0337 

•715 

•0259 

Proof Spirit, 

•925 

•0334 

•240 

•0087 

Vinegar, - 

1-080 

•0390 

•644 

•0233 

Water, distilled 

1-000 

•0361 

1-331 

•0481 

“ Sea 

1-030 

•0371 

1-209 

•0437 

“ Dead sea 

1-240 

•0448 

•695 

•0252 

Wine, ... 

•992 

•0359 

•671 

•0243 

“ Port 

•997 

•0361 

•600 

•0217 


*550 

•0199 

Miscellaneous, 



•498 

•0180 




•600 

•770 

•0217 

•0279 

Asphaltum, - - | 

•905 

1-650 

*0327 

0597 

•556 

•0201 

Beeswax, - 

•965 

*0349 

•703 

•0254 

Butter, .*. 

•942 

*0341 

1*333 

•04S2 

Camphor, 

•9S8 

*0357 

•604 

•0219 

India rubber, - 

•933 

0338 

•913 

•0331 

Fat of Beef, - 

•923 

’0334 

•849 

•0307 

“ Hogs, - - 

•936 

•0338 

1-063 

•0385 

“ Mutton, 

•923 

'0334 

•750 

•0271 

Gamboge, 

1-222 

*0442 

•944 

•0342 

Gunpowder, loose - 

•900 

•0325 

•897 

•0324 

“ shaken 

1-000 

•0361 

1-170 

•0423 

“ solid - | 

1-550 

•0561 

•705 

.0255 

1-800 

•0650 

•661 

•0239 

Gum Arabic, - 

1-452 

*0525 

1-354 

•0490 

Indigo, ... 

1-009 

•0365 

•3S3 

•0138 

Lard, - 

•947 

•0343 

•529 

•0191 

Mastic, 

1-074 

•0388 

•785 

•02S4 

Spermaceti, 

•943 

•0341 

•705 

•0255 

Sugar, ... 

1-605 

•0580 

•482 

•0174 

Tallow, sheep - 

•924 

•0334 

•500 

•0181 

“ calf 

•934 

•0338 

•460 

•0166 

“ ox, 

•923 

•0334 

•660 

•0239 

Atmospheric air, - 

•0012 

.43 

•554 

1-327 

•671 

•0200 

•0480 

•0243 

Gases. Vapours. 


Weight 
cub. ft. 

•788 

•0285 

Atmospheric air, 


grains. 

•807 

•0292 

1-000 

527-0 

Ammoniacal gas, - 

•500 

263-7 



Carbonic acid, - 

1-527 

805-3 

1-062 

•0384 

Carbonic oxid, 

•972 

512-7 

1-217 

•0440 

Carburetted hydrogen, 

•972 

512-7 

1-841 

•0666 

Chlorine, - 

2-500 

1316 

1-200 

•0434 

Chlorocarbonous acid, 

3-472 

1828 

1-500 

•0542 

Chloroprussic acid, 

2152 

1134 

1 -558 

•0563 

Fluoboric acid, 

2-371 

1250 

•833 

•0301 

Ilydriodic acid, 

4-346 

2290 

•792 

•0287 

Hydrogen, 

•069 

36-33 

•897 

•0324 

Oxygen, - 

1-104 

581-8 

1-034 

•0374 

Sulphuretted hydrogen, 

1-777 

9370 

9-97 

•0360 

Nitrogen, 

•972 

512-0 

1-018 

•0361 

Vapour of Alcohol, 

1-613 

851-0 

•739 

•0267 

“ turpen’e spir., 

5-013 

2642 

1-090 

•0394 

“ water, 

•623 

32S-0 

1-450 

•0524 

Smoke of bitumin. coal, 

•102 

53-S0 

1-054 

•0381 

“ wood, 

•90 

474-0 

1-032 

•0373 

Steam at 212° - 

•4S8 

257-3 














































4G4 


Ai 


ALLOYS. 

A = Antimony, 2? = Bismuth. (7= Copper, G = Gold, I = Iron, L — ^ead, 
N = Nickel, S= Silver, T — Tin, and Z = Zinc. 


Name. 

Allot. 

Brass, common yellow, 

20,1Z. 

Brass, to be rolled, 

32(7,10Z, 1.5 7. 

Brass castings, com., 

“ “ hard, . 

206’ 1.25Z. 2.57. 

25(7,2Z, 4.5 21 

Brass propellers, 

86’, 0.5Z, 17. 

Gun-metal,. 

86', 171 

Copper-flanges, 

9611Z. 0.267. 

Muntz’s metal, . 

66’,4Z. 

Statuary, 

91.4(7,5.53Z, 1.7 7, 
1.377. 

German Silver, . 

26’, 7.9A”, 6.3 Z, 
6.57 

Britannia metal, . 

50.4,257.257?. 

Chinese Sih’er, . 

65.1 ( 7 , 19.3Z, 13A r , 
2.586,127. 

Chi. wht. Copper, . 

20.2(7,12.7Z, 1.37, 
15.8 jv. 

Medals, 

100(1 sz. 

Pinchbeck, 

5611Z. 

Babbitt’s metal,. 

25 7,2.4,0.5(7. 

Bell metal, large, . 

3(7,17. 

“ “ small, 

40,1 7. 

Chinese gongs, 

40.5(7.9.27. 

Telescope mirrors, 

33.36’, 16.7 7. 

White metal, ord., . 

3.7(7, 3.7 Z, 14.2 7, 
28.4.4. 

“ “ hard, . 

356113Z, 2.2 7. 

Sheeting hiatal, 

Metal, expand in cool¬ 

560,45Z, 12 arse¬ 
nic. 

ing, .... 

757,16.7.4, 8.32?. 

Imitation of Gold. 

Melt separately, . -j 

x = 210,137. 
y = 62 6', 9Z. 

Gold imitation. . 

71y,9x. 


Type Metals. 

Name. Alwt. 

Smallest type, . 37,1 A. 

Small type, . . 47. 1.4. 

Medium type,. . 5 L,IA. 

Large type, . . 6 L, 1.4. 

Largest type, . . 7 L, 1 A. 

Metal which can be 
forged at red heat, . 

and strong as good 
iron, . . . 38.2Z, 60(7,1.767 


Alloys for Solders. 


Name. 


Allot. 

Melts. 

Newton’s fusible. 

82?, 57,37, 

212° 

Rose’s 

44 

22?, 17,17, 

201° 

A more 

(4 

5 2?, 37,2 7, 

199° 

Still more 

44 

12 7, 257, 502? 




13 cadium, 

’ 155° 

For tin 

solder. 



coarse, 

• • 

17,37, 

500° 

For tin solder, or- 



dinary,. 


27,17, 

360° 

For brass, soft spel- 


ter, . 

• • 

1C, 1Z, 

650° 

Hard, for .iron. 

20,1 Z, 

700° 

For steel. 

• 

19636’, 1Z. 



For fine brass work, 16, 8(7, 8 Z. 
Pewtercr’s soft sol¬ 
der, . . . 22?, 47,37. 

Pewterer’s soft sol¬ 
der, . . 17?, 17,2 7. 

Gold solder, . 24(7,26,1(7. 

Silver solder, hard, 46,10. 

“ soft, 26, l brass wire. 
For Lead, . 167,337. 


Tempering of Steel. 

The ability of heat to color steel or iron can be applied for ascertaining the tem¬ 
perature in flues and chimneys of steam-boilers, and for other temperatures between 
430° and 600° Fall. 

Yellow, very faint, for lancets,. 430 ° 

“ pale straw, for razors, scalpels, .... 460° 

full, for penknives and chisels for cast iron, . . 470° 

Brown, for scissors and chisels for wrought iron, . . 490° 

Red, for carpenters’ tools in general,.510° 

Purple, for fine watch-springs and table-knives, . . 630° 

Blue, bright, for swords, lock-springs,. 550° 

“ full, for daggers, fine saws, ueedles, .... 660° 

“ dark, for common saws. 600° 


When tempering a tool heat to a dull rod; plunge the point to be hardened into 
water half an mch; withdraw immediately. Emery point, so as to watch point for 
color from heat of shank ; at color drop into tub altogether. 










Relative Hardness of Substances. 


465 


Relative Hardness, H, of Substances. 


Minerals. 

H. 

Metals. 

H. 

Woods, Dry. 


H. 

Diamond. Ormuz, 

100 

Cast steel, hardened, 

65 

Chonta, S. Am. 



28 

Diamond, Pink, 

97 

Cast steel, unhard., 

40 

Lignum vitae, 



25 

Diamond, Yellow, 

94 

Cast iron, . 

Iron, hammered, . 

38 

Ebony, . 



24 

Diamond, Cubic, 

92 

37 

Pomegranate, 

• 


23 

Sapphire, 

90 

Pure iron,. 

35 

Boxwood, 



22 

Topaz, 

80 

Antimony, ham., . 

36 

Oak, very old, 

• 


22 

Garnet,... 

72 

Antimony, cast, 

32 

Oak, ordinary, 


• 

21 

Agate, 

71 

Platinum, cast, 

40 

Mulberry. . 

• 


20 

Amethyst, . 

71 

Platinum, ham., 

45 

Cedar, India, 


• 

20 

Quartz, 

70 

Brass, common 

32 

Beech, 



19 

Ruby, pal“- Brazil, 

65 

White metal, hard, 

38 

Ash, 


• 

18 

Ruby, 

64 

Gold, hammered, . 

30 

Alder, . 

• 


18 

Iron pyritO, 

63 

Gold, cast,. 

Copper, ham., . 

26 

Apple tree, . 


• 

17 

Opal, . 

62 

34 

Plum tree. 

• 


16 

Felspar, . . 

60 

Copper, cast, . 

29 

Yew, 


• 

15 

Fluorspar, . 

40 

Silver, ham.,. . 

32 

Maple, 

• 


14 

Copper pyrites, . 

38 

Silver, cast, . 

27 

Pine, yellow, 


• 

14 

Calcareous spar, 

30 

Zinc, • • • 

26 

Hazel, 

• 


13 

Anthracite coal, . 

28 

Aluminum, . 

24 

Cedar, wild, . 


• 

13 

Galena, 

27 

Tin, ham., . 

24 

Birch, 

• 


12 

Amber, . . . 

23 

Tin, cast, . 

Babbitt’s metal, . 

20 

Fir, 


• 

12 

Granite, . 

22 

20 

Pine, white, 

• 


11 

Gypsum, 

20 

Silenium, . 

22 

Spruce,. 


• 

10 

Bituminous coal, 

16 

Bismuth, 

20 

Sassafras, . 

• 


9 

Chalk, . . 

15 

Lead, ham., . 

18 

Hemlock, 


• 

8 

Talc, . . 

10 

Lead, cast, . 

15 

Cork,. 

• 


5 


Mr. Chapman has arranged a scale for the hardness of minerals, as follows: 

1 yields easily to the nail. 

& yields with difficulty to the nail, or merely receives an impression from it. 
Does not scratch a copper coin. 

3 scratches a copper coin, but is also scratched by it, being of about the same 
hardness. 

4: not scratched by a copper coin. Does not scratch glass. 

5 scratches glass, though rather with difficulty, leaving its powder on it. Yields 

easily to the knife. 

6 scratches glass easily. Yields with difficulty to the knife. 

7 does not yield to the knife. Yields to the edge of a file, though with difficulty. 
8, 9 and 10, harder than flint. 

The numbers in Chapman’s scale multiplied by 10 will correspond with the 
hardness in the preceding table. 


Charcoal from 1000 Weights of Dry Wood. 


Oak, . 

. 226 

Beech, . . 

200 

Ash, . 

179 

Chestnut, 

. 232 

Fir, 

156 

Norwegian Pine, 

192 

Mahogany, 

. 254 

Cedar, . . 

198 

Sallow, . 

184 

Walnut, 

. 206 

Pine, 

200 

Birch, 

174 

Elm, 

. 195 

Scotch Pine, . 

164 

Sycamore, . 

197 






















466 


Hydrometer. 


HYDROMETER. 

A roby wholly immersed in a liquid will lose as much of its weight, as the 
weight of the liquid it displaces. 

A floating body will displace its own weight of the 
liquid in which it floats. 

A cylindrical rod of wood or some light materials, 
being set down in two liquids, A and B, of different 
specific gravities, when in equilibrium it will sink to 
the mark a in the liquid A, and to b in the liquid B; 
then the specific gravity of A : B — b, c : a, c, or in¬ 
verse as the immersed part of the rod. This is the 
principle upon which a hydrometer is constructed. 


270. 


n 



r- 

P 

• * 



ir 



r j^ 




- = : 



- - 


run 


Er~: 

A 


_ 


B 


of. 

- - 


c~ 




— 

— 




— 


— 


■a 


Table showing the comparative Scales of Gay I/ussac and Baumc, with the Specific 
Gravity and Proof \ at the temperature of 00° Fahr. 



Gay Lussadi. 

Bau?ne s 

Specific Grav. 

100 

46 

•796 

95 

40 

•815 

-S 90 

36 

•833 

ja 85 

33 

•848 

S 80 

31 

•863 

*» 75 

28 

•876 

2 70 

26 

•889 

o. 05 

24 

•901 

co 

23 

•912 

£ 55 

21 

•923 

3 1 50 

19 

•933 

•3 45 

18 

*942 

g 40 

17 

•951 

n 35 

16 

*958 

£ 30 

15 

*964 

25 

14 

•970 


Proof. 


100 
92 
82 
72 
62 i 
52 
42 
32 
22 
12 , 

0 Proof. 

8 ' 

18 
29 
35 
48 


a> 

>■ 

O 

« . 

t£«w 

a o 
-j o 
= g* 

<X> F-t 

o 

t- 

<D 


®o 

•3 O 

C n 

£3 


HYDROSTATICS. 

Notation. 

A and a = areas of the pressed surfaces in square feet. 

1 and p = hydrostatic pressure in pounds. 

d = depth of the centre of gravity of A or a under the surface of the liquids 
In feet. •. 

S = specific gravity of the liquid. 

Example 1. Fig. 272. The plane A = 3*3 square feet, at a depth of d = 6 feet 
under the surface of fresh water. Required the pressure P = ? Specific gravity 
of fresh water S =» 1. 

P — 62-5 A d = 62*5X3-3X6 = 1237 5 pounds. 

Example 2. Fig. 275. The area of the pistons A = 8 5 square feet, a = 0 02 
square feet, l = 4 feet, c = 9 inches, and ^=18 pounds. Required the pres¬ 
sure P — ? 


P = 


FI A 18X4X8-5 


•e a 


= 40800 pounds. 


0-75X0*02 

Tt must be distinguished that the centre of pressure and centre of gravity of 
the planes, are two different points; the centre of pressure is below the ?entro 
of gravity, when the plane is inclined or vertical. 






















































Hydrostatics. 


467 


272. 


P = 62-5 SAd, 
P 




62-5 Sd, 

d= _ i> _ 

62-5 S A. 


273. 


The Hydrostatic paradox . 


The pressure P is independent of the 
width of column C. 


P — 62-5 5 A k. (same as above.) 


^(62-5SA + 

a ^-62-5 Sh^ 

P a — p A 
62 *5 SA a 


275. Bramah's Hydraulic Press. 


p = 

FI A 

A - P J a 

.e a * 

FI 

F — 

Pea 

F A l 

a = — . 

P e 


A 9 




276. Centre of Pressure of a rectangle y 
the upper edge at the surface 
of the liquid d = § h. 




277. Centre of Pressure of a triangle y 
the base being at the surface of 
the liquid, d *= 4 A. 


278. Centre of Pressure of a 
triangle , the vertex being at the surface 
of the liquid.d = \ h. 







































































































468 


Hydraulics. 


HYDRAULICS. 

Let the vessel A, Fig. 284, be kept constantly full of water up to the water 
linete. In two horizontal faces lower than the water line w, are made orifices 
a and a', through which the water will pass up vertical nearly to the water 
line w. Omitting the resistance of air, &c., the jet should theoretically reach 
the water line to; practically it reaches 0*967/t. 

It is evident that the velocity of the jet through the orifices, must be the ve¬ 
locity due to a body falling the height h, according to the law of force of 
gravity. 

Notation. 

Q = actual quantity of water discharged per second or in the time t, in cubic 
feet. 

h = head, or height of water over the orifice. 

t = operating time in seconds. 

a = area of the orifice in square feet. 

m = the coefficient for contraction. (See Fig. 299 ) 

G = gallon of 231 cubic iuches discharged in the time t. 

V = velocity through the orifice in feet per second. 

Example 1. Fig. 284. How many gallons of water will be discharged in five min¬ 
utes, through uu orifice of 0 025 square feet, applied at 8 feet under the level of 
the water ? 


G = 37-75a t y/h = 37*75X0-025X5X00 y/8 = 800 gallons. 

Eig. 285. The weight P can represent the weight of a column of water whose 
P h' 

height = - ■ — = acting on the area A. 

62*o A 0*9Gj ’ ° 


Fig.2SQ.n = number of down strokes per minute, s = stroke of piston ; tho 
air vessel <7= GA s at the pressure of the atmosphere. 

Example 2. Fig.286. IIow many double strokes must be made per minute by 
the lever of a hand pump, to throw up 22 cubic feet of water 18 feet high, in the 
time of 8 minutes and 15 secouds; the levers 1 = 30 inches, e = 8 inches, 
* = 0*6 feet, F = 20 pounds? 8X60+15 = 495 seconds. 


3630<2 h’ e 3630X22X18X8 
ts FI = 495X0-6X20X30 


64*5 strokes per minute. 


Example 3. Fig. 294. A vessel of rectangular form is of dimensions A = 0 
square feet, the height h — 5 feet. What time will it take the water level to 
sink 2 feet, when the orifice a = 0"212 square feet. 


( _ A (h — ti') = 6(5 — 3) 

2*52a(y7j+f//i/ “ 2-52X0-212(^5+^3) 


Motion of Water in Fipes$ 

Notation. 


L — extreme length of the pipe in feet. 

d = inside diameter in feet, and uniform throughout the length L. 


Example 4. Fig. 287. What will be the velocity of the water through a pipe of 
0 45 feet inside diameter, and L = 68 feet long, the head pressure of water being 
A =-« 8 feet ? 


r= 


48 



0*45X8 

68+50X0-45 


9-6 feet per second. 













































































































Hydraulics. 



Weirs. 

k b t. See Table for Wars. 

. _ Q . Q 

t ~ kb’ b kt' 


Q = 5-35/71 b h t y/ h, 
G = 40m b h ty/ 

t _ _9_l 


Q = 5-35m 5 t(hVIT— K VTY, 
G = 40m 6 t(hSh — /tVT), 

, _ Q 

5*35m A(A\/ A — A'\/ A') * 


. 0-95mA(>/A — 

WIT . ’ 

.4 = area of the vessel in square feet. 
t — time in seconds, in which the water level 
will sink the space h — hf. 


A(h — h') 

4m a (yA+y/ A'), 

Q = 4m a t(V h+ y/ A'), 


_ 

"A—A- 

’ A 



1 ^ 




t 


t - 


3 85a m 


(y* - ys 7 ), 


3-85^' m’ 3*85 am 


A y/h 










































































ITvim utlios 


471 






























































































472 


Hydraulics. 


Example 5. Fig. 2S9. Required 
in a long pipe or hose of L = 135 
pump of D — 0.2 feet in diameter 
h = 20 feet above the piston D ? 

F — 6 86 < 49 X 0-2 2 X 20 ) __ ^95 f ce t p,, r second. 

\ 0.2(135 +50X0.17) . . 

Q = 1.95 X 5.38 X 0.22 = 0.042 per second X 60 = 2.52 cubic feet per minute. 
s = 0.8 feet, the stroke of piston, we shall have 

n = -= 100 strokes per minute. 

0.8 X 0.785 X 0.22 


the velocity and quantity of water discharged 1 
feet long, and d = 0.17 loot, attached to a hand- 
p = 44 pounds, and the end of the pipe elevated 


Table for Water flowing 

This table is set up from careful experiments 
on a large scale, and is suited for weirs only. See 
Fig. 290. q = 4.327 b y W. 

Rule. Multiply the width b, in feet, of the 
weir by the coefficient k t and the product is the 
quantity of water discharged per second, in cubic 
feet, h is the height as represented by Fig. 290. 
The width b should be h. 

Example 6 . How much water will flow over a 
weir of 6 = 5 feet, h = 0.5 feet, in one minute? 

Q — kbt= 1.1295 X 5 X 60 = 338.35 cubic feet. 


over W+irs. 


h. inches. 

h. feet. 

m. 

k. 

0.4 

0.033 

0.424 

0.01365 

0.8 

0.066 

0.417 

0.05452 

1.2 

0.100 

0.412 

0.10592 

1.6 

0.133 

0.407 

0.16616 

2.4 

0.200 

0.401 

0.29171 

3.2 

0.266 

0.397 

0.44480 

4. 

0.333 

0.395 

0.63111 

6. 

0.500 

0.393 

1.1295 

8. 

0.666 

0.390 

1.7464. 

9. 

0.750 

0.385 

2.0331 

12. 

1.000 

0.376 

3.1350 


On tire Velocity of Water in Rivers. 

Notation. 


F = fall of the river in feet per mile. 

R = hydraulic radius in feet, or the area of the cross-section of the river in 
square feet, divided by the wet perimeter in feet. 

V = mean velocity of the water in inches per second. 

M = mean velocity in miles per hour. 


V = 10.9 y F jR, 
31= o.6i9 i/nr, 



F = 


118.8 R 
3P 


3.83 R 


The mean velocity of the water throughout the whole section of the river is to 
the velocity at the surface in the middle of the river as 84:100, or as 100 : 120 . 

Example 1. The cross-section of a river is measured to be 560 square feet, and 
the wet perimeter 196 feet; the foil of the river is 5 feet per mile. Required, the 
hydraulic radius and the mean velocity of the water in miles per hour? 


Hydraulic radius B, = 


560 

196 


= 2.86 feet. 


Mean velocity 31 = 0.619 i/5 X 2.86 = 2.34 miles per hour. 


& 


Example 2. The velocity of the surface in the middle of a river is 36 inches per 
second : hydraulic radius R = 2 feet. Required, the mean velocity and the fall of 
the river per mile? 


Mean Velocity V = 36 X 0-84 = 30.24 inches per second. 
30 342 

Fall F = —-— L ‘ :; -= 3.8487 feet per mile. 

118.8 X 2 * 

























Hydraulics. 


473 


Obstruction in Itivers. 

R = rise of water in feet caused by obstruction. 

A = sectional area in square feet of river unobstructed, and a = that when ob- 
| structed. V = velocity in feet per second of the water without obstruction. 



Resistance to a Plane Facing a Current of Water 

or Moving in Still Water. 

A = area of the plane in square feet. 

R = resistance in pouuds. 

V = velocity in feet per second. 

R = A F 2 , in fresh water. 

R — 1.032 A F 2 , in salt or sea water. 

When the plane is set at an angle of less than 90° to the direction of motion, the 
resistance will be, when <£ = angle of the plane, 

R = A (F sin.</>) 2 , in fresh water. 

R — 1.032 A (Fsin.<£) 2 , in salt water. 

Theoretical Velocity of Water, due to Head or Fall. 

See table for falling bodies, page 308, in which the column S represents the 
head of fall iu feet. 


To find the Number of Gallons of Water G which can be 
raised per Hour from a Well of Depth D, 

'By a Suitable Double-action Force-an d-lift-pump. 

D may also denote the height to which water may be raised in water-works. 


t; 



G 

18000 

One man working a crank, 

D 

\ 

G 

36000 

A donkey, 

D 


G 

126000 

A horse, 

- - 

Per steam horse-power, 

G 

I) 

190000 


1) 


or 0.8 of tho natural effect. 

Example 1. How many gallons of water can be raised per hour from a well 
150 feet deep by a horse? 


^ 126000 


= 840 gallons, the answer. 


Example 2. IIow many gallons of water can be raised per hour to a height of 
D — 150 feet by a steam-engine of 120 actual horse-power? 


0 = 


190000 X 120 


= 152,000 gallons, the answer. 


150 





















474 


Hydraulics. 


MOTION OF WATER IN PIPES. 


Notation. 

Q = cubic feet of water passed through the pipe per minute. 
D = inside diameter of the pipe in feet. 

L = length of the pipe in feet, increased by 50 diameters. 

H = head or fall in leet. 

V = velocity of the water iu the pipe in feet per minute. 


Q = 2356 


T7TTF 

y—JT' 


D = —— 

22.329 V H 


V - = 3000 


HOT 

Mir 


Example 1. A water-pipe of D — 1.75 feet in diameter, L = 36,000 + 50 X 1.75 
= 360S7.5 feet long, head pressure II = 390 feet. Required, how much water it 
cau discharge per minute? 

Q = 2356 J— 0 X _ 992 . 2 6. 
v \ 36087.5 

Example 2. At a distance of 27960 feet from a water-work is required Q — 664 
cubic feet of water per minute, head pressure being II = 256 feet. Required, the 
diameter of the pipe ? L = 27 960 + 50 = 2S010 feet. 


D = 


22.329 




564 2 X 28010 _ 


256 


= 1.4436 feet. 


Example 3. A water-pipe of D = 0.75 feet in diameter, L = 8650 -{- 50 = 8700 
feet, has a head pressure of H = 128 feet. Required the velocity v of the dis¬ 
charge. 

F=3000 - = 315.13 feet per minute. 

V 8700 ^ 


Consumption of AVater in Cubic Feet per bead of Population, 

Including aU Uses, as for Manufactories, Fires, etc., in 24 hours. 


January, 

2.58 

April, 

2.73 

July, 

4.58 

October, 

4.46 

February, 

2.40 

May, 

3.37 

August, 

4.75 

November, 

4.12 

March, 

2.64 

June, 

3.50 

Sept., 

4.61 

December, 

3.61 


On the Flow of AVater in Bends of Pipes. 

Notation. 

L = the whole length of pipe in feet, straight and curved or bent, increased 
by 50 D. 

It = radius in feet of the bend of the centre-line of the pipe. 

<t> = angle of deflection or bend of pipe in degrees. Should the pipe have several 
bends, add all the angles to <f). 

Sin. <f> to be used only up to 90°, and disappears iu the formulae for greater 
angles. 

D = inside diameter of the pipe in feet; V = velocity of the water in feet per 
minute; and H = head or fall in feet; Q — cubic feet of water dis¬ 
charged per minute. 


Q = 2356 
V = 3000 


/7/D 5 / 90 \ /, , D Sin. <p\ 

yl~(i+To) l 1 + T+To)' 

\jun( 90 ' \/ , E Sin. .n 


The formula! will answer for a pipe of the form of a screw-spiral. 




































The Hydraulic Ram. 


475 



THE HYDRAULIC RAM. 


This Hydraulic motor appears to be too little known in many parts or the world. 
The author of this book has been in the interior of many countries where water 
is raised in a very rude and laborious way, and where the hydraulic ram would be 
of great utility. The useful effect of the ram, like that of water-wheels and tur¬ 
bines, depends much upon its construction. In ordinary cases it returns about 50 
per cent, of the natural effect. That is, the quantity of water ( q ) multiplied by 
the height ( h) of the delivery above the ram will be about 50 per cent, of the 
quantity of water (Q) working the ram, multiplied by the head of fall ( F ), in the 
fame unit of time. 


qh = 0.5 Q F. 


9 = 


0.5 QF 
h 


Q = 


2 qh 

~Y' 


Q and q can be expressed in any unit of volume or weight. 
jPand li can be expressed in any unit of length. 

But let us assume Q and q to be cubic feet per minute, 
ft and h = fall and height in feet, 


L = length in feet and D = diameter in inches of the supply-pipe S, 
l = length and d = diameter of the delivery-pipe d. 


Then i> = , and * = \p3EEF>. 


Description of tlie Hydraulic Ram. 

Reference to the figure above. 

The water working the ram is supplied through the pipe S, and escapes through 
an opening at o, until it has gained a velocity sufficient to raise the valve or ball 
I B, which suddenly stops the current, and causes an excessive pressure in the ram 
; R, which opens the valve or ball C\ the water is forced into the vessel and air- 
'chamber A. and finally through the delivery-pipe d to its destination. When 
equilibrium of pressure is restored between S and R, the ball B falls, and the 
* operation is repeated. The ram can make as much as 200 strokes per minute, 
depending upon its size. 

The length of the supply-pipe S should not be less than five times the height 
of the fall l\ because it is the dynamic action (see page 310) of the water in the 
pipe which works the ram. The delivery-pipe may be made 10 or more times the 
height of tlw fall. 

, . * . * 


1 
























476 


Hydrodynamics. 


HYDRODYNAMICS. 

Water Power. 

The natural power contained in a fall of water is equal to the weight of the 
quantity of water passing over per second, multiplied by the vertical space through 
which it falls. 

Fie/. 291. Let Q be the quantity of water which passes through the orifice a in 
the time t = 1" second, in cubic feet of G2.5 pounds each. 

h = the vertical space the water falls; then the value or natural effect of the fall 
is at the orifice a. 

P = 62.5 Q h, 

Q = 5.06 aV'h ; 

p=315.5 
J/= 0.573 ahVh, 

IP 

> 


Example. 1. In a creek passes 13 cubic feet of water per second, 
must that creek be dammed up to produce an effect of ten horses ? 

h = -= 4.9 feet, the answer. 

0.1134 X 18 


But, 

Then we have 

This will be in horse power. 

3 

h = 




h = 


H 


0.1136 Qh, 


0.1136 Q 


How high 


Comparison of Columns of Water in Feet. 

Mercury in inches , and pressure in pounds, per square inch. 


Pounds 1 

Water. 

VIerc’ry. 

Water. 1 

Merc’ry. 

Pounds. 

Merc'ry 

Water. 

Pounds. 

Pr.sq.iu. 

Feet. 

Inches. 

Feet. 

Inches. 

Pr. sq. in. 

Inches. 

Feet. 

Pr. sq. in. 

1 

2.311 

2.046 

1 

0.8853 

0.4327 

1 

1.1295 

0.4887 

2 

4.622 

4.092 

2 

1.7706 

0.8651 

2 

2.2590 

0.9775 

3 

6.933 

6.138 

3 

2.6560 

1.2981 

3 

3.3885 

1.4662 

4 

9.244 

8.184 

4 

3.5413 

1.7308 

4 

4.5181 

1.9550 

5 

11.555 

10.230 

5 

4.4266 

2.1635 

5 

5.6476 

2.4437 

6 

13.8G6 

12.2276 

6 

5.3120 

2.5962 

6 

6.7771 

2.9325 

7 

16.177 

14.322 

7 

6.1973 

3.0289 

7 

7.9066 

3.4212 

8 

18.4SS 

16.368 

8 

7.0826 

3.4616 

8 

9.0361 

3.9100 

9 

20.800 

1S.414 

9 

7.9680 

3.8942 

9 

10.165 

4.3987 

10 

23.111 

20.462 

10 

8.8533 

4.3273 

10 

11.295 

4.8875 

11 

25.422 

22.508 

11 

9.7386 

4.7600 

11 

12.424 

5.3762 

12 

27.733 

24.554 

12 

10.624 

6.1927 

12 

13.554 

5.8650 

13 

30.044 

26.600 

13 

11.509 

5.6255 

13 

14.683 

6.3537 

14 

32.355 

28.646 

14 

12.394 

6.0582 

14 

15.813 

6.8425 

15 

34.GG6 

30.692 

15 

13.280 

6.4909 

15 

16.942 

7.3312 

16 

30.977 

32.738 

16 

14.165 

6.9236 

16 

18.072 

7.8200 

17 

39.288 

34784 

17 

15.050 

7.3563 

17 

19.201 

8.30S7 

18 

41.599 

36.830 

18 

15.936 

7.7890 

18 

20.331 

8.7975 

19 

43.910 

38.87G 

19 

16.821 

8.2217 

19 

21.460 

9.28C2 

20 

4G.221 

40.922 

20 

17.706 

8.6544 

20 

22.590 

9.7750 

21 

48.532 

42.968 

21 

18.591 

9.0871 

21 

23.719 

10.264 

22 

50.843 

45.014 

22 

19.477 

9.5198 

22 

24.849 

10.752 

23 

53.154 

47.060 

23 

20.362 

9.9525 

23 

25.978 

11.241 

24 

65.465 

49.106 

24 

21.247 

10.385 

24 

27.108 

11.7300 

25 

57.776 

51.152 

25 

22.133 

10.818 

25 

28.237 

12.219 

26 

60.087 

53.198 

26 

23.018 

11.251 

26 

29.367 

12.707 

27 

62.398 

55 244 

27 

23.903 

11.683 

27 

30.496 

13.196 

28 

64.709 

57.290 

28 

24.789 

12.116 

28 

31.626 

13.685 

29 

67.020 

59.336 

29 

25.674 

12.549 

29 

32.755 

14.174 

30 

69.331 

61.886 

30 

26.560 

12.981 

30 

33.885 

14.662 





























Water-Wheel^. 


477 


WATER-WHEELS. 

Water-wheels are of two essential kinds, namely, Vertical and Horizontal. 

The Vertical are subdivided into 

Over shot-wheels, Undershot-wheels, Breast-wheels, and High-breast and Low-bread 
wheels. 

The Horizontal are with Floats, Screw-wheels, Turbine , Reaction-ivheds, <£c. 

Waterwheels do not transmit in full the natural effect concentrated in a fall 
of water; under most favourable circumstances SO per cent, has been utilized, 
but with poor arrangements only 20 per cent, may be expected. 

Example 1. Fig. 302. The vertical section of the immersed floats of an under¬ 
shot-wheel in a mid-stream is a = 27 square feet, velocity of the stream V— 8 - G, 
and v =» 4 feet per second. .Required the horse-power of the wheel H ? 

*-- w < 8 ' 6 - 4 >* - 11,4 

Example 2. Fig. 307. On a breast-wheel is acting Q = 88 cubic feet of water 
per second, the head h = 8 feet, velocity of the wheel at the centre of the 
buckets v = 5 feet per second; the water strikes the buckets at an angle u = 8° 
and velocity V — 7 feet per second. Required the horse-power of the wheel, 
if? 

j3 ' = ll^( 8+ i (7XCOS - 8 °“ 5) ) = 65£P * 

Example 3. Required the effect of Poncelet’s wheel, the head h = 4 feet, and 
the orifice a = 5 square feet, the velocity of the wheel at the centre of pressure 
of the floats is v = 6 - 78 feet per second ? 

V = 6-91 ylc = 13-82 feet per second. 

Q == 6-5X5XJ/4 = 65 cubic feet per second. 

H = - 5X f' 8 (13-82 — 6-78) = 15-8 EP. 

i 

Example 4. Fig. 309. A saw-mill wheel is to be built under a fall of h == 18 
feet, and to make n — 110 revolutions per minute. Required the proper diam¬ 
eter of the wheel. 

D = ^ 3-857 feet, 

at the centre of pressure of the buckets. 

Telocity V — 8^18 = 33-94 feet per second. 

Velocity v = ?24X3^O<L10 _ 22 . 2 feet per g^ond. 

The fall discharged 30 cubic feet of water per second. Required the horaa 
power of the wheel. HI 

30 X2 gj ^ 33 . 94 _ 22 . 2 ) = 39 jp 
200 

How many square feet of dry Pine can it saw per hour ? 

See page 264. 30X39 -=1170 square feet. 

The saw is meant to be applied direct on the wheel shaft. 











niDRAULICS. 


302. 

Undershot wheel in a mid-strecrm. 


//= 

200 ' 

When V = 2v about, the efFect will bo, 

H= a = area of float. 

1000 


303. 


Undershot-Wheel. 

H =U V -^ 


H= UlltV- V )y/X 
56-6 v 1 


When V = 2v, about, -hOL. 

3-9 

T 

PonceleCs Wheel. 

H =» ^ — v), when h > 5 feet, 

.rfS.ZO 

Q v 

iz = W V — u ) when h < 5 feet, 

Q = &mas/h, V = 6-91 n/7jT 

305. 

■* 

Breast-Wheel with Parabolic drain . 

Q = S-^sTYT 















































Hydraulics. 


479 



20G. Low-breast Wheel. 


//= *TlV[ A+ ~W'( v cosu ~ v )] 

Q = kb. V = See table for weirs, 

a 

See page 283, Weisbacli’s Hydraulic 
Motors. 


•% 





308. 


Over-shot Wheel. 


n=St[ h + 2h> {Ycosu - v) ] 


Proper velocity about n = 


35 D + 100, 
D 


revolutions per minute. 

See page 247, Weisbach’s ITyd. Motors. 

309. Saw-Mill Wheel. 

\ 

H- 5-L (V-v) 

“ 200 v ’ 

Proper diameter of the Wheel t 

100 /T - . foot 

T) = -- v h , m feet, 

n 

n — revolutions per min. 



































































480 


Turbine Wheels. 


TURBINES. 

Notation. 

Q — cubic feet of water passed through the turbine per second. 
h = height of fall in feet. 

D = diameter ip inches of circle of effort in the turbine. 
a= area in sq. in. of the-conduit passage into the turbine wheel. 

6 = depth in inches of turbine buckets. 
c = depth in inches of leading buckets. 
r= breadth of turbine buckets in inches, 
m = number of buckets in the turbine wheel. 
m'= number of leading buckets, 
n = number ot revolutions of turbine per minute. 

S and s-= height of conduit and discharge in inches. 
t = thickness of steel plate buckets in 16ths of an inch. 

H= actual horse power of the turbine. 

^Eet'er ffU- } «* 

d' = diameter in inches of the discharge pipe. 

W= Hydraulic pressure on the turbine wheel bearing on the end of the 
shaft. 


kyh 

.mm 


n 


D= 


0-436 r 
m kyh 

ft = ——, 

D 

20 k a 

ft = -- 

a D ’ 
a 

r " 0-436D’ 
46 * Q 

r MT’ 

r=°to° 
6 8 ’ 

m 

< = io’ " • 


- 2 


• - - 3 


- 4 


- - 6 


- - 7 


• • 8 


a = 


a = 


20 Q 
y h ’ 

20 fc Q 


Dn ' 
a = 0-436 Dr, 
a = m'rs, - - 
a'=* to r s, - - 
c'=0*98 a, 


Q _ a\fh ^ 
20 


0. 


a D n 
20 k~ 


- 9 

- 10 

- 11 
- 12 

- 13 
• 14 

- 15 

- 16 


to = 6 D, - 
m'= 4’5/D, - 
0-626 D 


b = 


c = 


y m 
0-78 D 


- 17 


18 


- 19 


20 


y to' ’ 

s = 0-S6S, - - 21 

d = H-fr-}- sTl, 22 
d'= D+2 r, - o 23 


W= 


DVi 


H = 0-1134 Q h natural effect of the fall, 
H = 


30 Q 3 ''j 
■> 


a J 


. h = ® 

267-5 ’ 


actual horse power, 

66 per cent of the natural. 


- 24 


25 


26 

27 


The coefficient k can vary from 800 to 1200 without seriously affecting 
the percentage of the utilized power, hut it is best between 900 and 1000. 
These formulae cannot apply to all forms of turbines. Work out the entire 
theory of particular wheel, Weisbach’s Hydraulic Motors. —W. D. M. 

























Turbine Wheels. 


481 


Jonval’s Turbine has so many advantages above other hydraulic mo¬ 
tors that it is considered sufficient to describe the construction of that 
one only, but the principal formulas will answer for any kind of turbines. 

On the accompanying plate is a drawing of a Jonval Turbine such as 
the Author of this Pocket Book has built in Russia. The buckets are not 
supported by concentric rings, but are fastened only on one side, which 
is considered more simple and convenient for replacing new buckets. For 
falls over 30 feet it may be better to make it with concentric rings. 
When a turbine is to be constructed wc have on the one side given the 
natural eli'ect of the fall, and on the other side the actual work to be 
done, which latter should not exceed 66 per cent, of the former. Between 
these two points the turbine is to be so proportioned as to utilize the 
greatest possible effect with smallest expense of Machinery. 

Jonval’s turbine in good condition generally utilizes 60 to 80 per cent. 
Suppose a fall of =25 feet, discharging Q=12 cubic feet of water per 
second, the natural effect will be, 

H = 0-1134X12X 25 = 34 horses, 

of which 34X‘66=22-4 horses to be counted upon as the actual effect of 
the turbine. 

Turbine shaft to make n=200 revolutions per minute with the assumed 
coefficient ft =960. From these dates we will obtain all the principal 
dimensions of the turbine, namely, 


_ 960 y 25 . , 

D= -'-= 24 inches. 

200 


- 1 


r = 


48 


0-436X24 


= 4-6 in. 


a 


20X960X12 
~ 24X200 


= 48 sq. in. 


10 


m — 6/24=24-6 say 25. - - 17 

m' = 4-5y 24 = 22 buckets. - 18 


. 0-625X24 o . 

b — -—— = 3 in. - - 

Y25 

4 inches. 

y 22 

t— — == 2'5, 16ths. - . 
10 


6 

19 

20 

8 


In calculating the breadth r from formula 5, it must come inside of 
formula 7, if not the diameter D must be altered. 

Now proceed with the construction as shown at the bottom of the plate, 
which represents a section of the buckets through the circle of effort of 
the turbine. 

The drawing of the turbine is .? of an inch to the foot, and the construc¬ 
tion of the buckets 3 inches to the foot. 

Draw the base line AB, set off the angle of the leading buckets=10°. 
The distance between the leading buckets will in this case be 24X3-14:22= 
3-43 inches, set off this from S towards A, draw the straight part of the 
second bucket parallel to the first one, draw from S the line A d at right 
angle to the buckets, and e will be the centre for the curved part. From 
the centre of S draw the line o to the end of the second buckets, divide 
this line into eight equal parts take five of them as radus and draw from 
the end of the second bucket a circlearc of about 60°, which will be the 
propelling part of the turbine wheel bucket. 

Distance between the wheel buckets will be 24X3-14:25=3-02 inches, set 
off this from A towards S, draw the second propelling arc. Set off from A 
the depth of the wheel buckets 5 = 3 inches, set off 2 b to s, which will be 
the length of the first wheel bucket. Set off from s to u the distance 
between the buckets 3-02 inches. Make s 0-86 S. Draw from u a curved 
line in the form of a parabola that will leave the space s and tangent the 
propelling circlearc somewhere about x. Care must be taken that the 
discharging area a' of all the wheel buckets will be about 2 per cent, less 
than the conduit area a of all the leading buckets. The surface of the. 
buckets should be made as smooth as possible, or even polished. 

For very high falls the Hydraulic pressure W becomes very considerable 


31 























4S2 


Turbines. 


and may necessitate another arrangement, namely, to lay the shaft horizontally 
and place on it two turbines, so that the leading buckets are either between or 
outside of the wheels; but then comes another disadvantage, namely, that the 
number of revolutions will be greatly increased and may be required to gear it 
down 10 to ‘20 times to the proper speed of the-main shaft. 

To avoid this as much as possible, take k = 800, and make r = —. 


One great advantage with Jonval’s turbine is that it can be placed almost any¬ 
where between the high and low levels to suit the location, though it should not 
be more than 20 feet above the lower level; then, in order to utilize the whole tall, 
care must be taken to make the discharge-pipe perfectly air-tight. It is not neces¬ 
sary to make the discharge straight down from the turbine: it can be carried hori¬ 
zontally or inclined, as may suit the location. The author has built turbines 
similar to that represented on the accompanying plate, at General Maltzof’s es¬ 
tablishment, Kaluga, Russia. 




Approximate or Proportionate Price of Turbines, 

as fitted and delivered at the foundry, without shaftings or gearings, is — 

. 400/HT 

4P -- t 

fF 

in which H= horse power of the turbine and F the height of fall in feet. 

Example,. Required the price of a turbine, II = 100 horses, to work under a fall 
of F= 25 feet. 

$ = 40VM = 4_00X10 = 1375 

f25 2 - 92 

Price I.ist of Turbines in Dollars. 


Head of fall in feet, F. 


power. 

5 

10 

15 

20 

30 

40 

50 

75 

100 

150 

II 

$ 

$ 

% 

$ 

$ 

$ 

$ 

$ 

$ 

$ 

1 

234 

186 

163 

148 

130 

117 

110 

95 

86 

76 

2 

330 

263 

231 

209 

183 

167 

154 

134 

122 

107 

4 

467 

372 

326 

295 

258 

235 

218 

190 

172 

151 

6 

552 

455 

400 

262 

316 

288 

266 

232 

211 

185 

8 

660 

526 

462 

418 

305 

332 

308 

269 

244 

213 

12 

808 

642 

565 

510 

447 

405 

377 

329 

300 

261 

16 

935 

742 

654 

590 

516 

468 

434 

380 

345 

302 

20 

1015 

830 

730 

660 

577 

524 

485 

425 

385 

338 

30 

1280 

1020 

894 

810 

705 

642 

595 

520 

472 

414 

40 

1480 

1180 

1035 

932 

815 

740 

686 

600 

545 

476 

50 

1650 

1320 

1155 

1(145 

913 

828 

768 

671 

610 

532 

60 

1810 

1440 

1264 

1140 

1000 

908 

840 

735 

668 

584 

80 

*2090 

1645 

1460 

1320 

1155 

1050 

975 

848 

770 

674 

100 

2340 

1860 

1630 

1480 

1295 

1175 

1090 

<948 

860 

753 

125 

2620 

2080 

1820 

1650 

1440 

1310 

1220 

1060 

984 

845 

150 

2860 

2280 

2060 

1810 

1580 

1440 

1330 

1170 

1060 

924 

175 

3100 

2460 

2150 

1950 

1700 

1550 

1440 

1260 

1140 

1000 

200 

3310 

2630 

2300 

2090 

1825 

1655 

1540 

1350 

1220 

1066 

225 

3510 

2790 

2440 

2215 

1935 

1755 

1630 

1430 

1300 

1130 

250 

3700 

2940 

2570 

2335 

2035 

1850 

1720 

1500 

1365 

1195 

275 

3890 

;'090 

2700 

2450 

2135 

1945 

1800 

1580 

1430 

1255 

300 

4060 

3225 

2820 

2560 

2235 

2030 

1890 

1650 

1500 

1300 

350 

4380 

3480 

3035 

2760 

2410 

2190 

2035 

1780 

1610 

1410 

400 

4680 

3720 

3250 

2950 

2580 

2310 

2175 

1900 

1725" 

1510 

450 

5080 

3950 

3450 

3130 

2740 

2480 

2310 

2015 

1825 

1600 

500 

5240 

4160 

3610 

3300 

2890 

2620 

2535 

2125 

1925 

1SS5 















































/V,//r7If 





_ 














































































































































49 H 

I 




■ 
























Weir Measurement. 


483 


Weir Measurement, of "Water-Flow. 

Q — cubic feet of water flowing over the weir per second. 

G — gallons of 231 cubic inches each flowing over the weir per hour. 

h= height of water-level over the weir in inches. 

L = wid i h of weir in feet. 

Q = Theoretic formula, Q— G —1898.15 h \/Ti. 

14.137 7.7b 

When the water-level over the weir is fluctuating during the experimental 
measurement, it will not be correct to take the average of h for the calcu¬ 
lation, but the average of hy h must be used._ 

The following table gives the value of h ]/ h up to 21 inches and for every 
10 th of an inch. 

Example,. How many cubic feet of water will flow per second over a weir 
of L*= '6 feet wide and /i = 6.3 inches of water over the comb? 

Q = " — 3.3435 cubic feet. 

l4.1o7 


"Value of h \/h for Weir Measurements. 


h 

0 

1 

Te 

a 

nths of 

3 

an Inc 

4 

) of the 

5 

Heigh 

G 

t A. 

7 

8 

9 

0 

0 . 

0.0316 

0.0894 

0.1643 

0.2530 

0.3536 

0.4647 

0.5858 

0.7155 

0.8531 

1 

1 . 

1.1537 

1.3145 

1.4830 

1.6565 

1.8371 

2.0238 

2.2164 

2.4149 

2.6189 

2 

2.8284 

3.0432 

3.2631 

3.4882 

3.7180 

3.9526 

4.1927 

4.4365 

4.6868 

4.9385 

3 

5.1961 

5.4581 

5.7243 

5.9947 

6.2692 

6.5479 

6.8305 

7.1171 

7.4074 

7.7020 

4 

8.0000 

8.3018 

8.6074 

8.9166 

9.2285 

9.5459 

9.S658 

10.189 

10.516 

10.846 

5 

11.180 

11.520 

11.858 

12.196 

12.550 

12.897 

13.258 

13.610 

13.975 

14.331 

6 

14.697 

15.065 

15.437 

15.S12 

16.192 

16.575 

16.955 

17.343 

17.734 

18.120 

7 

18.522 

18.924 

19.324 

19.724 

20.133 

20.540 

20.950 

21.367 

21.875 

22.210 

8 

22.624 

23.056 

23.488 

23.910 

24.350 

24.781 

25.222 

25.663 

26.156 

26.558 

9 

27.000 

27.455 

27.910 

28.361 

28.820 

29.282 

29.747 

30.211 

30.678 

31.151 

10 

31.623 

32.111 

32.570 

33.060 

33.538 

34.027 

34.513 

35.000 

35.495 

35.985 

11 

36.483 

36.985 

37.483 

37.999 

38.494 

39.000 

39.500 

40.018 

40.548 

41.040 

12 

41.569 

42.090 

42.610 

43.145 

43.675 

44.192 

44.725 

45.255 

45.800 

46.333 

13 

46.872 

47.418 

47.958 

48.506 

49.055 

49.600 

49.934 

50.707 

51.266 

51.824 

14 

52.383 

52.944 

53.508 

54.075 

54.644 

55.219 

55.786 

56.368 

56.936 

57.514 

15 

58.094 

58.676 

59.262 

59.846 

60.439 

61.022 

61.614 

62.206 

62.803 

63.400 

16 

64.000 

64.600 

65.200 

65.809 

66.413 

67.024 

67.632 

68.245 

68.853 

69.473 

17 

70.092 

70.714 

71.333 

71.954 

72.584 

73.210 

73.835 

74.463 

75.095 

75.730 

18 

76.367 

77.005 

77.643 

78.284 

78.930 

79.570 

80.216 

80.865 

81.514 

82.169 

19 

82.819 

83.472 

84.130 

84.789 

85.450 

86.110 

86.772 

87.440 

87.970 

88.771 

20 

89.442 

90.114 

90.790 

91.463 

92.140 

92.819 

93.496 

94.180 

94.862 

95.548 

21 

96.234 

96.922 

97.614 

98.303 

99.000 

99.690 

100.39 

101.09 

101.79 

102.49 

22 

103.19 

103.89 

104.60 

105.30 

106.00 

106.70 

107.41 

108.15 

108.85 

109.59 

23 

110.30 

111.04 

111.73 

112.48 

113.19 

113.90 

114.63 

115.38 

116.12 

116.80 

24 

117.57 

118.35 

119.06 

119.79 

120.51 

121.25 

122.00 

122.75 

123.50 

124.25 






































484 


Light and Colors. 


LIGHT AND COLORS. 

Iitjflit is the sensation transmitted to the eye, and produces the sense of seeing. 
Light is a component part of heat, and a compound imponderable substance whose 
ingredients depend upon the composition of the burning substance; or, burning 
substances can be analyzed by decomposition of its light in a spectrum. 

Decomposition of L.i"lit in tlie Spectrum. 

Colors. 

Maximum ray. 

Combination of Colors. 

Violet. 

Indigo. 

Blue. 

Green. 

Yellow. 

Orange. 

Red. 

Chemical. 

ElectricaL 

Light. 

ITeat, 

Primary. 
Blue. 1 

Yellow. J 
Blue. > 

Red. / 

Yellow. 1 
Red. / 

Secondary. 

Green. > 

Purple. I 
Orange. 1 

Tertiary. 

Dark 

Green. 

Brown. 


All the colors of the spectrum mixed together make white, which is proved by 
the decomposition of white light, which makes the seven colors. 


The velocity of light in planetary space is 192500 miles per second. The velocity 
of light through transparent bodies is not known, but probably varies inverse as 
the square root of the specific gravity of the transparent substance. 

Light passes from the sun to the earth, 95000000 miles, in eight minutes, at 
which rate of velocity light can pass around the earth in one-eighth of a second. 

The intensity of light is inversely proportional to the square of the distance from 
the luminous body. 

The standard unit for measuring the intensity of light is assumed to bo that 
produced by a sperm candle, “short 6,” burning 120 grains per hour. 

A spermatic candle 0.85 in diameter burns about 1 inch per hour. 


MOTION OF GAS IN PIPES. 


Notation. 


Q — cubic feet of gas passed through the gas-pipe per hour. 

L = length in feet, D = diameter in inches, of the pipe. 

II = head of water in inches which presses the gas through the pipe. 
S = specific gravity of the gas, air being 1. 

n = number of candles required for giving the same light as Q cubic 
per hour, (lludely approximate for 16-candle gas). 


Q 


= 780 ttD 2 


III D 

VszT’ 


Q=Vn+A D = J_ 6 
n= (Q — l) 2 22.65\/ 


feet of gas 


SLQ 2 


Example. At a distance of L = 64oO feet from the gas-work is required 0 — 
940 cubic feet of gas per hour. Head of water being II=l inch, specific grav ity 
s =0 5. Required, the diameter of the pipe D = ? J 


D = 


1 5 /0.5 X 6450 X 940 2 _ \ , 

22^5 \-I- =3 * mches ' 


Each light in a room consumes about 4 cubic feet of gas per hour, and ordinarv 
street-lights 5 cubic feet. J 

Differing formulae are given by authorities. See Clegg on Coal-Gas. 
























The Atmosphere. 


485 


THE ATMOSPHERE. 

The mean height of the atmosphere is about 302 feet greater at the equator than 
at the poles, which is caused by the difference of the earth’s attraction at the two 
places, and also by centrifugal force. 

The mean height of the atmosphere in 45° latitude is 60158.6 feet; at the poles, 
60007.6 feet, and at the equator, 60309.6 feet. 

The temperature of the atmosphere is greatest at the surface of the earth, and 
decreases with the height above the surface. The compression of the air by the 
upper layers of the atmosphere generates heat in the lower layers, as explained in 
the article on Air and Ileat. The rays of light from the sun, passing through a 
denser air near the surface of the earth, also generate more heat by friction, as it 
were. The temperature of congelation of water is 32°, which is marked by the per¬ 
petual snow-line on high mountains, as shown in the accompanying table. 

Heights of Snow-Line in Different Latitudes. 


Latitudes of snow-line on high mountains. 


5° 

15,210 


15° 

25° 

35° 

40° 

45° 

55° 

65° 

14,760 

12,560 

10,290 

9,000 

7,670 

5,030 

2,230 


75° 

1,016 


85° 

120 


Heights of snow-line in feet above the sea. 


<i 

2^2 


a' 

3g§S 


New-fallen snow occupies eight times its volume in water. 

Heat is constantly absorbed from the atmosphere by evaporation of water on the 
surface of the seas, which heat is carried up and warms the atmosphere above ; heat 
is also absorbed by support of the growth of vegetation on land. It is this opera¬ 
tion of consuming and generating heat which causes the winds and difference of 
weather. 

As the atmosphere is a material substance, it is subject to the action of the force 
of gravity, which causes a pressure of 14.75 pounds to the square inch at the level 
of the sea; or a column of air one square inch base and of the height of the atmo¬ 
sphere weighs 14.75 pounds, which balances an equal weight of a column of mer¬ 
cury 30 inches high at the temperature of 60° Fahr., or a column of water of 34 
feet high. 

Columns of Air, Mercury and Water. 

A is a vessel full of mercury, in which is placed verti¬ 
cally a glass tube about 3 feet high above the surface l; 
in the glass tube is fitted an air-tight piston a, just one 
square inch area, which can be moved by the piston-rod c. 
Now let the piston stand at a on the level l, and in contact 
with the mercury in the tube. Raise the piston by the pis¬ 
ton-rod and handle c. The mercury in the tube will follow 
until the height of 30 inches is reached, and there remain 
even if the motion of the piston is continued. Now it may 
be supposed that it is some force of the piston that draws 
the mercury up in the tube. If so, why did it separate at 30 
inches? If the column becomes too heavy, it could separate 
at l, and the 30 inches of mercury follow the piston. But 
this is not the case. The weight of the atmosphere pressing 
on the surface l forces the mercury up the tube until the 
weight of the column of mercury and the external pressure 
are in equilibrium, which occurs when the column has 
reached a height of 30 inches. The piston only served to 
overcome the atmospheric pressure in the tube. We have 
then the weight of a column of atmospheric air with one 
square inch base equal to the weight of a column of mer¬ 
cury 80 inches high and one sq. in. base. One cubic inch 
of mercury at 60° Fahr. weighs 0.941 pounds; this, multi- 
30 inches, gives 14.73 pounds, the weight of the column of 
mercury or atmosphere; this is generally termed “the atmospheric pressure por 

square inch.' 1 , , 

The specific gravity of mercury at 60° l'ahr. is 13.58, and 

13.o 8 X 30 _ 33 95 feet 
12 

the height of a column of water required to balance the atmosphere. 


1 : 




plied by the height, 










































4SG 


Wind, Aerodynamics. 


WIND, AERODYNAMICS. 

The motions and effects of gases by the force of gravity are analagons to that 
of liquids. (See Hydraulics.) (See Weisbacli’s Mechanics of Engineering, 1st vol.) 

The altitude or head of the atmosphere at uniform density will be the altitude of 
a column of water 33.95 feet, divided by the specific gravity of the air, 0.0U12046, or, 

... 33 _ 5 — = 2S133 feet. 

0.0012046 

The velocity due to this head will be— 

V = 8.021 / 28183 = 1346.4 feet per second, the velocity with which the air will pass 
into a vacuum. 

Velocity of Wind. 

When air passes into an air of less density, the velocity of its passage is meas¬ 
ured by the difference of their density. 

H and h = density of the air in inches of mercury; t = temperature at the time 
of passage ; and V = velocity of the wind in feet per second. 


V= 1346.4 


M 


IT 


h 


( 


1 + 0.00208* 


)’ * 


6 . 


The force of wind increases as the square of its velocity. 

a —area exposed at right angles to the wind in square feet; F== force of the 
wind in pounds; H = liorse-power, and v = velocity of the plane a in direction 
of the wind, + when it moves opposite, and — when it moves with the wind. 

F = 0.002288a F 2 , when v = o, . . 7. 

F = 0.00228Sa( V ± v) 2 , 8. I/= — 9. 

v ' ’ 240384.6 * 

Example. A rail-train running ENE 25 miles per hour exposes a surface of 
1000 square feet to a pleasant brisk gale NE by E. Required the resistance to the 
train in the direction it moves, and the horse-power lost. 

ENE — N E bv N — 3 points = 33° 45'; V — 14 feet per second, a brisk gale ; 
v = 25 X 1-467 = 36.6 feet per second, and F= 0.002288 sin. 233° 46' X 1000 (14 + 
cos. 33° 45' X 36.6) 2 = 305.1 pounds. 

g= 305-1 X36 _ 6 
550 

Table of Velocity and Force of Wind, in Pounds per Square 

Tncli. 


Miles 

Feet 

Force 

per 

Common Appellations of 

Miles 

per 

hour. 

Feet 

Force 

per 

per 

hour. 

per 

second 

sq. ft. 
pound. 

the force of Winds. 

per 

second 

sq. ft. 
pound. 

i 

1.47 

0.005 

1 

Hardly percept¬ 

18 

26.4 

1.55 

l 

1 

ible. 

20 

29 34 

1.968 

2 

3 

2.93 

4.4 

0.020 

0.044 

I 

Just perceptible. 

25 

30 

36.67 
44 01 

3.075 

4429 

4 

5.87 

0.079 

1 


35 

51.34 

6.027 

5 

7.33 

0.123 


Gentle pleasant 

4d 

58.68 

7.873 

6 

8.8 

0.177 


wind. 

45 

66.01 

9.963 

►f 

I 

10.25 

0.241 


• 

50 

73.35 

12.30 

8 

11.75 

0.315 

■* 


55 

80.7 

14.9 

9 

13.2 

0 400 



60 

88.02 

17.71 

10 

12 

14.67 

17.6 

0.492 

0.708 


Pleasant brisk 
gale. 

■ 66 
70 

95.4 

102.5 

20.85 

24.1 

14 

20.5 

0 964 


75 

110 

27.7 

15 

22.00 

1.107 



80 

117.36 

31.49 

16 

23.45 

1.25 



100 

140.66 

50. 


Common Appellations of 
the force of Winds. 


-Very brisk. 

-Iligh wind. 

j-Very high. 

Storm. 

Great storm. 
| Hurricane. 
Tornado. 




























The Barometer. 


487 


THE BAROMETER. 

The barometer measures the pressure of the atmosphere, as described in the 
former editions of this Pocket Hook. 

The English have graduated the barometer to indicate weather as follows: 


Barometer in indies. 


Weather. 


At 28.3 

= 

Stormy. 

At 28.7 

= 

Much rain. 

At 29.1 

= 

Bain. 

At 29.5 

= 

Change of weather. 

At 29.9 

= 

Fair weather. 

At 30.3 

= 

Set fair. 

At 30.7 

— 

Very dry. 

5 in predicting 

weather-changes are selected from 



tf: 




the “ Barometer Manual” of the Loudon Board of Trade: 

I. If the mercury, standing at thirty inches, rises gradually while the 
thermometer falls,and dampness becomes less, N.W., N.or N.E. wind; less 
wind or less snow and rain may be expected. 

II. If a fall take place with a rising thermometer and increasing damp¬ 
ness, wind and rain may be expected from S.E., S. or S.W. A fall iu 
winter with a low thermometer foretells snow. 

III. An impending north wind, before which the barometer often rises, 
may be accompanied with rain, hail or snow, and so forms an apparent 
exception to the above rules, for the barometer always rises with a north 
wind. 

IV. The barometer being at 29£ inches, a rise foretells less wind or a 
change of it northward, or less dampness. But if at 29 inches, a fast first rise 
precedes strong winds or squalls from N.W.. N. or N.E., after which a gradual rise 
with falling thermometer, a S. or S.W. wind will follow, especially if the rise of 
the barometer has been sudden. 

V. A rapid barometric rise indicates unsettled, and a rapid fall storm}-, weather 
with n»in or 6now; while a steady barometer, with dryness, indicates continued 
fine weather. 

VI. The greatest barometric depressions indicate gales from S.E., S. or S.W.; the 
greatest elevations foretell wind from N.W., N. or N.K., or calm weather. 

VII. A 6udden fall of the barometer, with a westerly wind, is sometimes followed 
with a violent storm from the N.W., N. or N.E. 

VIII. If the wind veer to the south during a gale from the E. to S B, the barom¬ 
eter will continue to lull until the wind is near a marked change, when a lull may 
occur. The gale may afterward he renewed, perhaps suddenly and violently; and 
if the wind then veer to the N.W., N. or N.E., the barometer will rise and tile ther¬ 
mometer fall. 

IX. The maximum height of the barometer occurs during a north-east wind, and 
the minimum during one from the south-west; hence these points may he consid¬ 
ered the poles of the wind. The range between these two heights depends on the 
direction of the wind, which causes, on an average, a change of half an inch; on 
the moisture of the air, which produces, in extreme cases, a change of half an inch; 
and on the strength of the wind, which may influence the barometer to the extent 
of two inches. These causes, separately or conjointly with the temperature, pio- 
duce either steady or rapid barometric variations, according to their force. 


i 






















4S8 


IIygromktry. 


HYGROMETRY. 

On the Humidity and other Properties of Air, deduced from Glaisher's Tables 
of the Greenwich Observatory. 

Mason’s hygrometer, consisting of wet and dry bulb thermometers, is considered 
the best for determining the dew-point and the humidity of the air. 

Example. The temperature of the air being 75°, and the wet-bulb thermometer 
showing 63°, or 12°cold; barometer 30 inches. Required, the humidity of the 
air, the dew-point, weight of vapor per cubic foot, and the weight of a cubic foot 
of the air in grains troy ? 

Table I., 75° and 12° cold = 55 per cent, of humidity. 

Table II., “ “ = 57° temperature of dew-point. 

Table III., weight of dry air = 516.7 grains per cubic foot. 

“ ** *• “ saturated air = 511,4 “ “ “ 

Difference = 5.3 X 0.55 = 2.915 grains. 

Weight of the air 511.4 + 2.9 = 514.3 grains per cubic foot. 

Table III., 9.31 0.55 = 5.12 grains of vapor per cubic foot. 

The weight of air of equal temperature and humidity varies inversely as the 
height of the barometer. 

TABLE I. 


Humidity of the Air, or Percentage of Full Saturation, 

At Different Temperatures , indicated by the Dry and Wet Bulbs 
of the Hygrometer!' ( Glaisher). 


Temp, of 
the air, 

Difference 

in 

Temperature, or Cold on the Wet 

-bulb Thermometer. 

Fahr. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

30 

86 

73 

55 





















35 

91 

83 

76 

70 

64 

57 

53 

48 
















40 

93 

86 

79 

74 

68 

63 

58 

53 

50 

46 














45 

93 

86 

80 

74 

69 

64 

59 

55 

51 

48 














50 

93 

87 

81 

76 

71 

66 

61 

57 

53 

49 

46 













55 

94 

88 

83 

78 

75 

69 

<»•') 

GO 

56 

53 

50 

49 

46 

44 

41 

39 

36 







60 

94 

89 

84 

80 

75 

71 

67 

63 

59 

56 

53 

50 

47 

45 

42 

40 

38 

35 

33 





65 

95 

89 

85 

81 

76 

72 

69 

65 

61 

58 

55 

52 

49 

47 

44 

42 

40 

37 

35 

34 

32 

30 

28 

70 

95 

91 

86 

82 

78 

74 

71 

67 

64 

61 

58 

55 

52 

49 

47 

45 

42 

40 

37 

35 

34 

31 

29 

75 

95 

90 

86 

82 

78 

71 

71 

68 

64 

61 

58 

55 

52 

49 

48 

47 

44 

41 

39 

37 

35 

32 

30 

80 

95 

90 

87 

83 

79 

75 

72 

08 

65 

62 

59 

56 

5> 

50 

49 

48 

44 

42 

40 

38 

36 

33 

31 

85 

96 

91 

87 

83 

79 

75 

72 

68 

65 

62 

59 

56 

54 

51 

49 

46 

44 

42 

40 

38 

36 

34 

32 

90 

90 

91 

87 

83 

79 

75 

72 

68 

65 

62 

59 

56 

54 

51 

49 

46 

44 

42 

40 

38 

36 

34 

32 









Percentage of 

' Humidity. 









TABLE II. 

Temperature of the Pew-point, 

At Different States of the Hygrometer. 


Temp, of 
the air, 

Difference 

in 

Temperature, or C 

old on 

the Wet 

-bulb Thermometer. 

Fahr. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

?3 

30 

25 

21 






















35 

32 

30 

27 

25 

22 

20 

17 

15 

13 















40 

37 

35 

33 

31 

29 

27 

25 

22 

20 

18 














45 

43 

41 

39 

37 

34 

32 

30 

28 

26 

24 

22 













50 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 












55 

53 

52 

50 

48 

46 

45 

43 

41 

40 

38 

36 

34 

33 

31 

29 

28 

26 

24 






60 

58 

50 

55 

53 

51 

50 

48 

46 

45 

43 

41 

39 

38 

36 

34 

33 

31 

29 

28 

26 




65 

63 

62 

60 

58 

57 

55 

54 

52 

50 

49 

47 

46 

44 

43 

41 

39 

38 

36 

34 

33 

31 

30 

28 

70 

68 

67 

65 

64 

62 

61 

59 

58 

56 

55 

53 

52 

50 

49 

47 

46 

44 

43 

41 

40 

38 

37 

35 

75 

73 

72 

70 

69 

67 

66 

64 

63 

61 

60 

58 

57 

55 

54 

52 

51 

49 

48 

4G 

45 

43 

42 

40 

80 

78 

77 

75 

74 

72 

71 

69 

68 

66 

65 

63 

62 

60 

59 

57 

56 

54 

53 

51 

50 

48 

47 

45 

85 

84 

83 

82 

81 

80 

79 

78 

77 

76 

75 

74 

73 

72 

71 

70 

69 

68 

67 

66 

65 

64 

03 

62 

90 

89 

88 

87 

86 

85 

84 

83182 

81 

80 

79 

78 

77 

76 

75 

74 

73 

72 

71 

70 

69 

08 

67 


Temperature of Dew-point. 















































































Hygromrtky. 


4S9 


TABLE III. 


Properties of Air, by Gluislier, Greenwich. Observatory. 

Barometer 30 inches , at 60° Fahrenheit. 


Temp. 

Force of 

Weight 
of vapor 

Wt. per cub. ft. 

Temp. 

Force of 

Weight 
of vapor 

Wt. percub.lt. 

of the 
air. 

inches of 

per cub. 
ft. of 

Dry 

1 Satu¬ 
rated 

of the 
air. 

inchesof 

per cub. 
foot of 

Dry 

Sat’d 

mercury 

sat. air. 

air. 

air. 


mercury 

sat. air. 

air. 

air. 

F &hr. 

Inches. 

Grains. 

Grains. 

Grains. 

Fahr. 

Inches. 

Grains. 

Grains. 

Grains. 

10° 

0.089 

1.11 

590.0 

589.4 

52° 

0400 

4.56 

540.5 

537.9 

11 

0.093 

1.15 

5S8.7 

588.1 

53 

0.414 

4.71 

539.4 

536.7 

12 

0.096 

1.19 

587.5 

586.8 

54 

0.428 

4.86 

53S.3 

535.5 

13 

0.100 

1.24 

586.2 

585.5 

55 

0.442 

5.02 

537.3 

534.4 

14 

0.104 

1.28 

584.9 

584.2 

56 

0.458 

5.18 

536.2 

533.2 

15 

0.108 

1.32 

583.7 

582.9 

57 

0.473 

5.31 

535.1 

532.1 

16 

0.112 

1.37 

582.4 

581.6 

58 

0.489 

5.51 

534.1 

530.9 

17 

0.116 

1.41 

581.1 

5S0.3 

59 

0.506 

5.69 

533.0 

529.8 

18 

0.120 

1.47 

579.9 

579.1 

60 

0.523 

5.87 

532.0 

528.6 

19 

0.125 

1.52 

578.7 

577.8 

61 

0.541 

6.06 

530.9 

527.5 

20 

0.129 

1.58 

577.4 

576.5 

62 

0.559 

6.25 

529.9 

526.3 

21 

0.134 

1.63 

576.2 

575.3 

63 

0.578 

5.45 

528.8 

525.2 

22 

0.139 

1.69 

575.0 

574.0 

64 

0.597 

6.65 

527.8 

524.0 

23 

0.144 

1.75 

573.7 

572.7 

65 

0.617 

6.87 

526.9 

522.9 

24 

0.150 

1.81 

572.5 

571.5 

66 

0.638 

7.08 

525.8 

521.7 

25 

0.155 

1.87 

571.3 

570.2 

67 

0.659 

7.30 

524.7 

520.6 

26 

0.161 

1.93 

570.1 

569.0 

68 

0.681 

7.53 

523.7 

519.4 

27 

0.167 

2.00 

568.9 

567.7 

69 

0.704 

7.76 

522.7 

518.3 

28 

0.173 

2.07 

567.7 

566.5 

70 

0.727 

8.00 

521.7 

517.2 

29 

0.179 

2.14 

566.5 

565.3 

71 

0.751 

8.25 

520.7 

516.0 

30 

0.186 

2.21 

565.3 

564.1 

72 

0.776 

8.50 

519.7 

514.9 

31 

0.192 

2.29 

564.2 

562.8 

73 

0.801 

8.76 

518.7 

513.7 

32 

0.199 

2.37 

563.0 

561.6 

74 

0.827 

9.04 

517.7 

512.6 

33 

0.207 

2.45 

561.8 

566.4 

75 

0.854 

9.31 

516.7 

511.4 

34 

0.214 

2.53 

560.7 

559.2 

76 

0.882 

9.60 

515.7 

510.3 

35 

0.222 

2.62 

559.5 

558.0 

77 

0.910 

9.89 

514.7 

509.2 

36 

0.230 

2.71 

558.3 

556.8 

78 

0.940 

10.19 

513.8 

508.0 

37 

0.238 

2.80 

557.2 

555.6 

79 

0.970 

10.50 

512.8 

506.9 

3S 

0.246 

2.89 

556.0 

554.4 

80 

1.001 

10.81 

511.8 

505.7 

39 

0.255 

2.99 

554.9 

553.2 

81 

1.034 

11.14 

510.9 

504.6 

40 

0.264 

3.09 

553.8 

552.0 

82 

1.067 

11.47 

509.9 

503.4 

41 

0.274 

3.19 

552.6 

550.8 

83 

1.101 

11.82 

508.9 

502.3 

42 

0.283 

3.30 

551.5 

549.6 

84 

1.136 

12.17 

508.0 

501.1 

43 

0.293 

3.41 

550.4 

548.4 

85 

1.171 

12.53 

507.0 

500.0 

44 

0.304 

3.52 

549.3 

547.2 

86 

1.209 

12.91 

506.1 

498.9 

45 

0.315 

3.64 

548.1 

546.1 

87 

1.247 

13.29 

505.1 

497.7 

46 

0.326 

3.76 

547.0 

544.9 

88 

1.286 

13.68 

504.2 

496.6 

47 

0.337 

3.88 

546.0 

543.7 

89 

1.326 

14.08 

503.2 

495.4 

48 

0.349 

4.01 

544.8 

542.5 

90 

1.368 

14.50 

502.3 

494.3 

49 

0.361 

4.14 

543.7 

541.3 

91 

1.411 

14.91 

501.3 

493.2 

50 

0.373 

4.28 

542.6 

540.2 

92 

1.456 

15.33 

500.4 

492.0 

51 

0.386 

4.42 

541.5 

539.0 

93 

1.502 

15.76 

499.4 

491.9 



































400 


Climate and Seasons. 


MEAN TEMPERATURE AT DIFFERENT SEA¬ 
SONS OF THE YEAR. 


Locations. 

J 

Year. 

ean Temperat 
Spring. Sum. 

ure, Fa 

Autui. 

lr. 

Wint'r. 

Hemis¬ 

phere. 

Height 
ah. sea. 
Feet. 

Algiers, 




03.0 

03.0 

74.5 

70.5 

54.0 

N. 

310 

Berlin, . 




47.5 

46.4 

63.1 

47.8 

30.6 

N. 

128 

Herne, 




40.0 

45.8 

60.4 

47.3 

30.4 

N. 

1918 

Boston, . 




4!t 

48 

00 

53 

28 

N. 

71 

Buenos Ayres, 




02.5 

59.4 

73.0 

64.6 

52.5 

S. 


Cairo, . 


• 


72.3 

71.6 

84 0 

74.3 

58.5 

N. 


Calcutta, . 




78.4 

82.0 

83.3 

80.0 

67.8 

N. 


Canton, .... 


• 


00.8 

09.8 

82.0 

72 9 

54.8 

N. 

10 

Christiania, 




41.7 

39.2 

59.5 

42.4 

25.2 

N. 

74 

Cape of Good Hope, 


• 


06.4 

03.5 

74.1 

60.9 

58.6 

S. 


Constantinople, 




50.7 

518 

73.4 

00.4 

40.6 

N. 

150 

Copenhagen, . 


• 


40.8 

43.7 

63.0 

48 7 

31.3 

N. 

20 

Edinburgh, 




47.5 

45.7 

57.9 

48 0 

38/> 

N. 

288 

Jerusalem, 


• 


02.2 

60.6 

72.0 

06.3 

49.6 

N. 

2500 

Jamaica (Kingston), 




79.0 

78 3 

81.3 

too 

76.3 

N. 

10 

Lima, Peru, . 


• 


00.2 

63.0 

73.2 

09 6 

59.0 

S. 

511 

Lisbon, . 




01.5 

59 9 

71.1 

62.6 

52.3 

N. 

236 

London, 




50.7 

49.1 

62.8 

51.3 

39.6 

N. 

50 

Madeira (Funchal), . 




05.7 

03 5 

70.0 

67.0 

61.3 

N. 


Madrid, .... 


• 


57.0 

57.0 

74.1 

50.7 

42.1 

N. 

2175 

Mexico, City, . 




60.5 

53.6 

63.4 

65.2 

60.1 

N. 

6990 

Montreal, 


• 


43.7 

44.2 

69.1 

47.1 

17.5 

N. 


Moscow, . . 




38.5 

43.3 

62.6 

34.9 

13.5 

N. 

480 

Naples. 


• 


61.5 

59.4 

74.8 

62.2 

49.6 

N. 

180 

New Orleans, 




72 

73 

84 

72 

58 

N. 

20 

New York, . 


• 


53 

50 

72 

50 

33 

N. 

20 

New Zealand, . 




59.G 

00.1 

60.7 

68.0 

53.5 

S. 


Nice. 




00.1 

55.9 

72.5 

63.0 

48.7 

N. 


Nicolaief (Russia), . 




48.7 

49.3 

71.2 

50.0 

25.9 

N. 


Paramatta (Australia), 




04.6 

OG.O 

73 9 

048 

54.5 

S. 

. . . 

Palermo, . 




03.0 

59.0 

71.3 

60.2 

52.5 

N. 

180 

Pekin, China, 


• 


52 6 

50.0 

77.8 

54.9 

29 

N. 

97 

Paris, 




61.4 

50.5 

64.6 

52.2 

37.9 

N. 

210 

Philadelphia, 




55 

52 

76 

57 

31 

N. 

30 

Quito, Ecuador, 




00.1 

00.3 

00.1 

62.5 

59.7 

S. 

9560 

Bio Janeiro, . 




73.0 

72.5 

79.0 

74.5 

68.5 

s. 

10 

Rome, 




59.7 

57.4 

73.2 

61.7 

46.6 

N. 

174 

San Francisco, 


• 


57.5 

58 

59 

60 

53 

N. 

150 

St. Petersburg, . 




38.3 

35.1 

60.3 

40.5 

16.7 

N. 

10 

Stockholm, . 


• 


42.1 

38 3 

61.0 

43.7 

25.5 

N. 

134 

Trieste, 




65.8 

53.8 

71.5 

56.7 

39.4 

N. 

288 

Turin, .... 


• 


53.1 

53.1 

71.6 

63.8 

33.4 

N. 

915 

Vienna, 




50.7 

49.1 

62.8 

51.3 

39.6 

N. 

480 

Warsaw, 


• 


45.5 

41.0 

63.5 

46.4 

27.5 

N. 

397 

Washington, . 




59 

60 

79 

58 

38 

N. 

• • • 


Seasons. 

Southern Latitude. 

December, January, February, 
March, April, May, 

June, July. August, 

September, October, November, 

Seasons. 

Summer. 

Fall. 

Winter. 

Spring. 

Nort 

June, 

September, 

December, 

March, 

hern Lati 

July, 

October, 

January, 

April, 

TUBE. 

August, 

November. 

February. 

May. 






















































Rain and Melted Snow. 491 


Rain and Melted Snow. 

Full in Inches at Different Places. 



Locations. 

Year. 

Spring. 

Stimm’r. 

Fall. 

Winter. 

Albany, North America, 

40.67 

9.79 

12.3 

10.3 

8.30 

Algiers,. 

37.01 

8.34 

0.60 

10.3 

17.8 

Baltimore, North America, 

42.00 

11.2 

11.1 

10.52 

9.31 

Berlin, Prussia, 

23.56 

5.66 

7.21 

5.45 

5.24 

Bergen, Norway, .... 

87.61 

15.7 

18.6 

29.8 

23.5 

Bombay, India. 

110. 

• • • 




Boston, North America, 

44.4S 

10.8 

11.8 

12.57 

9.89 

Buffalo, “ 

27.35 

5.90 

8.45 

7.48 

5.52 

Canton, Chinas .... 

69.3 

LS.8 

27.9 

19.3 

3.3 

Charleston, North America, . 

4S.29 

8.60 

18.7 

11.6 

9.40 

Copenhagen, .... 

13.35 

2.84 

6.86 

5.13 

3.52 

Dover, England, .... 

38. 

• • • 




Dublin, Ireland, 

25. 





Edinburgh, Scotland, . 

28. 

• • • 


• • • 


England,. 






Glasgow,. 

Granada (Colombia), 

28.9 

5.43 

7.13 

8.95 

7.39 

115. 

• • • 

• • • 

• • • 

• • • 

Liverpool,. 

31.1 

6.19 

9.78 

10.8 

7.32 

Lima, Peru, .... 

13.5 

5.1 

0.2 

1.2 

7.0 

London, . 

20.69 

4.09 

6.00 

6.15 

4.45 

Madeira Islands, 

30.87 

5.11 

2.30 

6.96 

16.5 

Manchester, England, . 

36. 

7. 

9. 

11. 

9. 

Milano, Italy, .... 

38. 

9.04 

9.18 

11.7 

8.05 

Mississippi State, .... 

53.00 

10.9 

14.2 

9.50 

18.4 

New York, .... 

42.23 

11.5 

11.3 

10.3 

9.63 

New Orleans, .... 

52.31 

13.3 

16.1 

10.8 

12.6 

Ohio, State, .... 

39.69 

10.4 

10.9 

9.03 

6.91 

Pekin, China, .... 

26.9 

2.67 

20.5 

3.22 

0.53 

Peru (Interior),Carabaya, 

355. 

88. 

120. 

87. 

60. 

St. Petersburg, .... 

17.65 

2.89 

6.73 

5.11 

2.93 

Paris,. 

22.61 

5.53 

5.92 

6.51 

4.68 

Philadelphia, .... 

48.00 

13. 

12. 

11. 

12. 

llio Janeiro, Brazil, . 

• • • 

• • • 

• • • 

• • • 

10.76 

Rome, Italy, .... 

30.87 

7.27 

3.4 

10.9 

9.3 

Stockholm, .... 

Tiflis, Caucasus, .... 

19.67 

2.17 

7.81 

6.94 

2.75 

19.26 

6.25 

7.62 

3.51 

1.88 

Washington, .... 

41.20 

10.4 

10.5 

10.2 

11.1 

San Francisco. California, . 

83. 

22. 

1. 

15. 

45. 

Volume of Evaporation and Rain-Fall. 


Inches X 2,323 200 = cubic feet per square mile. 

Inches X 17,335,019 = gallons per square mile. 

Inches X 3630 = cubic feet per acre. 


Length in Miles of the Principal Rivers. 



Europe. 


North and South 
America. 


Asia and Africa. 


Volga, Russia, . . 

2000 

Missouri, . . . 

2900 

Yang-tse-kiang, . 

2800 

Danube, .... 

1600 

Mississippi, . . 

2800 

Lena, .... 
Obe, Hoangho, . 

2600 

Don and Dnieper, . 

1000 

Mackenzie’s, . . 

2500 

2500 

Bhine, .... 

950 

St. Lawrence, . 

2200 

Yenesei, . . . 

2300 

Dwina,. 

700 

Itio Grande, . . 
Colorado, Cal., . 

1800 

Amor, .... 
Cambodia, . . . 

2200 

Petchora, Elbe, Loire, 

600 

1100 

2000 

Vistula. Tagus, . . 

550 

Alabama, . . . 

600 

Indus, Irrawaddy, 

1700 

Dniester, Guadiana, 

500 

Amazon, . . . 

3600 

Nile,. 

3000 

Rhone, Po, Seine, . 

450 

Rio de la Plata, . 

2250 

Niger or Joliba, 

26,00 

Mezene, Desna, . 

400 

Orinoco, . . . 

1500 

Senegal, . . . 

1200 

Dahl, Bug, .... 

300 

Araguay, . . . 

1100 

Orange, . . . 

1000 

Thames, .... 

233 1 

Magdalena, . . 

900 

Gambia, . . . 

700 















































492 


Evaporation. 


Evaporation on tlic Surface of Water in the Open Air. 

When the surface of water is freely exposed to the atmosphere, the dry air in 
contact with it becomes charged with vapor, and consequently becomes lighter 
(see Table, page 357), rises, and gives place to drier air, which repeats the same 
operation. In this way moisture is constantly carried up into the air from the 
surface of the water. The rate of this evaporation depends upon the temperature 
of the water, the dryness, the temperature and the velocity of the air. 

Evaporation of Water in Decimals of an Inch, per 24 Hours, 

on the surface of fresh-water lakes, rivers and canals, at different temperatures of 

the water and currents of the air. 


Water. 


Velocity of wind in miles per hour on the wffter. 


Temp. 

Calm. 

10 

20 

30 

40 

50 

GO 

32° 

0.012 

0.014 

0.016 

0.017 

0.019 

0.021 

0.023 

35 

0.020 

0.023 

0.026 

0.029 

0.032 

0.035 

0.038 

40 

0.040 

0.046 

0.052 

0.058 

0.064 

0.070 

0.076 

45 

0.068 

0.078 

0.088 

0.098 

0.109 

0.119 

0.129 

50 

0.100 

0.115 

0.130 

0.145 

0.160 

0.175 

0.190 

55 

0.133 

0.153 

0.173 

0.193 

0 213 

0.233 

0.253 

60 

0.177 

0.203 

0.230 

0.256 

0.2S3 

0.310 

C.336 

65 

0.225 

0.259 

0.292 

0.326 

0.360 

0.394 

0.427 

70 

0.278 

0.320 

0.361 

0.404 

0.444 

0.486 

0.527 

75 

0.335 

0.385 

0.435 

0.4S5 

0.535 

0.585 

0.635 

80 

0.400 

0.460 

0.520 

0.580 

0 640 

0.700 

0.760 

85 

0.468 

0 538 

0.608 

0.679 

0.749 

0.819 

0.8S9 

90 

0540 

0.621 

0.703 

0.784 

0.865 

0 946 

1.025 

95 

0.620 

0713 

0.808 

0,900 

0.995 

1.088 

1.180 

100 

0.700 

0.805 

0.912 

1.015 

1.123 

1.225 

1.332 

The evaporation on the surface of salt water on the ocean is about 0.8 of that in 

the table. 








The quantity of water evaporated on the surface of all the waters on the earth 

is equal to the quantity of rain-fall. 





Area in Square Miles of the largest Inland Lakes. 


Lakes. 


Sq. Miles. 


Lakes. 


Sq. Miles. 

Eastern Hemisphere. 


Tonting, China, 
Wenern, Sweden, 

• • 

1200 

2400 

Aral Sea, Tartary, 

• • 

16650 

Wettern, Sweden, 


1045 

Azov Sea, Russia, 

• 

8800 

Zaizan. Mongolia. 


1600 

Baikal Sea, Siberia, 


13000 





Balkash, Mongolia, 

• 

5200 

Western Hemisphere. 


Black Sea, Turkey, 

• • 

113000 

Athabasca. N. America, . 

3200 

Caspian Sea, Russia, 

138000 

Erie Lake, N. America, 

7000 

Constance, Switzerland, . 

456 

Great Bear, N. America, . 

4000 

Dead Sea, Palestine 

• 

370 

Great Slave, N. America. 

12000 

Dembia, Abyssinia, 

• • 

13000 

Great Salt Lake, 


1880 

Enare, Lapland, 

• 

870 

Huron, N. America, . 

22800 

Geneva, Switzerland,. 

400 

Maracaibo, S. America, . 

6000 

Hjelmaren, Sweden 

• 

900 

Michigan, N. America, 

22G00 

Tchad, Africa, . 

• • 

11600 

Nicaragua, Cent. America. 

3905 

Ladoga, Russia, 

• 

6200 

Ontario, 

N. America. . 

4950 

Loch Lomond, Scotland, . 

27 

Otehenantekana, N. Amer., 

2500 

Lough leaugh, Ireland, 

80 

Superior, N. America. . 

80000 

Onega, Russia, . 


3300 

Titicaca, Peru, . 


5400 

Ouroomia, Persia, 

• • 

1000 

Winnipeg, N. America, 

7200 












































Difference of Levels. 


493 


BAROMETRICAL OBSERVATIONS. 

For Determining Difference of Levels. 

Notation for the complete formulx of La Place , in French and English 

measures. 


Lower station 


Ch = 
ion, = 


= height of barometer = h f ' 
temp, of barometer 
temp, of the air 


r = h') 

• = T'y 

= n 


Upper station. 


H — height of barometer at the upper station reduced to the temperature of 
the barometer at the lower station. When the height is read on a brass 6cale, the 
reduction will be in 


French measures. 


English measures. 


II—h' [1+0.0001614 {T— T')].\E:=h' [1+0.00008967 (T-T*)] 

Mean radius of the earth = 6,366,200 metres = 20,886.860 feet. 

Mean height of the atmosphere = 18,336 metres = 60,158.6 feet. 

L = mean latitude between the two stations. 

Z — difference of level between the two stations. 


French measures. 


z=Iog 'TT x 18336 x 


z 


log. 


A 

ii 


X 60158.6 


(■+Sf) x • • 

(l + 0.00251 X cos. 2 L) X. 
/ Z+169B6) _ 

\ 6366200 / 

English measures. 

)x • • 

\ 960 / 

(l + 0.00251 X cos. 2 L) X 

Z + 52252\ 

“ I • • • 


2 . 

3 

4, 


A 


20886860 / 


2 . 

3. 

4. 


The factor (1) gives the difference of level when the observations are made in a 
temperature of 32° Fahr., or 0° Cent., the freezing-point of water, and in latitude 
45°, without the factors of correction (2), (3) and (4). 

The factor (2; is the correction for temperature of the air above or below the 
freezing-point. 

The factor (3) is the correction for latitude above or below 45°. 

The factor (4) is the correction for the decrease of the earth’s attraction. This 
correction is included in the following Table I., to suit any level of the stations. 

There are some other barometrical corrections not included in the above for¬ 
mulae. such as for humidity of the air, capillarity and boiling of the glass tube, 
for the hour of the day and season of the year, all of which are so insignificant, 
uncertain and complicated that they have been omitted here. 

















494 


Difference of Levels. 


Explanation of tlie Barometrical Tables. 

The tables have been calculated in Peru, and practically used by the author. 

Table I. is calculated from the factors (1) and (2), which gives the approximate 
heights above the level of the sea, in English and French measures, for every tenth 
of an inch from 11 to 31 inches. The mean temperature of the air and of the 
barometer is assumed to be 00° Fahrenheit = 15.555 Centigrade, and in latitude 
45°. The barometer is assumed to be 30 inches = 760 millimetres at the level of the 
sea, but when it is observed to be higher or lower, make the corresponding addi¬ 
tion or subtraction for difference of levels in the table. 

Table II. contains the correction for difference of level in feet or metres at dif¬ 
ferent temperatures of the air above or below 60° Fahr. 

Table III. contains the correction for heights in different latitudes above or 
below 45°. 

Tables IV. and V. are logarithmic corrections for temperature and latitude. 

Table VII. gives the height of a column of air in metres, corresponding to a dif¬ 
ference of one milimetre of mercury at different heights of the barometer. 


Table VIII. gives the height of a column of air in feet, corresponding to a differ¬ 
ence of one-tenth of an inch of mercury at different heights of the barometer. 

Tables X. and XI. contain the correction for the mercurial column at different 
temperatures of the barometer above or below 60° Fahr. = 15.555 Cent. This 
correction must be made before the barometrical height is applied to Table I. 

Table XII. contains the approximate mean temperature of the air at the level 
of the sea for every month of the year in different latitudes. This table has been 
deduced from observations of Mr. Dove, Humboldt, Raimondi, and other distin- ' 
guished authors. The table agrees very well with the meat temperatures on the i 
Atlantic and Pacific coasts, but will not answer for the North Sea and the Baltic, 
where the temperatures are much higher. A great deal of inconvenience may arise 
for want of a table of this kind. 


When barometrical observations are made far inland, some means must be 
resorted to for estimating the temperature of the air at the level of the sea in the 
latitude of observation, in order to make proper corrections for difference of level. 
From all the meteorological observations of different authors it appears that the 
mean temperature of 24 successive hours is near 9 o’clock in the morning, and 
that the mean temperature of the day from 9 to 6 p. M. is at noon. 


The variation of temperature throughout the day varies with the latitude, that 
is, the higher the latitude, the greater is the variation. 


Example 1. On the 14th of March, 1869, 2h. 15m. p. m., in Oroya, Peru, lati¬ 
tude 11° 30', the barometer stood 19.46 inches, the temperature of the air 62°, and 
that of the barometer 60°. Required, the height of Oroya above the level of the 
sea in feet ? 


Table I. 


Table XII, 
Table II. 


Table III. 


Barometer 19.4 in., = 12099 6 feet. 

Correction 0.6 X diff. 142.6 feet, = 85.5 feet. 


Approximate height, = 12014.1 feet. 

Temperature at Oroya, 62° Fahr. 

Latitude 11° 30', 14th of March, 79° 


Mean temp, of the column of air, 141 = 

Correct mean temp. 70° = 

of the air, 1 2 ° n = 

I 10 feet = 

f 10000 feet = 
■< 2000 feet = 
(, 10 feet = 


Correct for lat. 11° 30', 


70 5°. 

208.2 feet. 

41.6 feet. 
0.2 feet. 

23.6 feet. 
4.8 feet. 
0.0 feet. 


t, 

! 


278.4 feet. 
12014.1 feet. 

12292.5 feet. 


Sum of corrections, . 
Approximate height, . 

Height of Oroya, 


















Aneroid Barometer. 


495 


Example. 2. In the city of Paucartambo, Peru, the barometer was observed to 
stand 21.272 inches, the temperature of the air 70°, and that of the barometer 69°, 
in latitude 13° 18' south. About three miles from the city, on the mountain 
Huanacaury, the barometer stood 18.224 inches, the temperature of the air 62°, 
and that of the barometer 64°. Required, the height of the mountain above the 
city of Paucartambo? 

Barometer at the lower station, . 21.272 inches. 

Correction for 69°, Table XI., subtract .017 inches. 

Height of barometer at 60°, . . 21.255 inches. 

Barometer at the upper station, . 18.224 inches. 

Correction for 64°, Table XI., subtract .006 inches. 

Height of barometer at 60°, . 18.218 inches. 

Barometer. Heights. 

18.218 13845.0, upper station. 

21.255 9565,3 , lower station. 

Logarithms 3.6314133= 4279.7 feet, approximate height. 

Table IV. 0.0053929 = 66° mean temperature. 

Table V. 0.0009888 = 13° 18' latitude. 

3.6377950 = 4343.1 feet, the height required. 

Aneroid Barometer. 

The aneroids made by Negretti & Zambra, London, are compensated, and show 
the height of a column of mercury at the temperature of the freezing-point of 
water, 32° Fahr., or zero Centigrade. The aneroid is not affected by different 
temperatures. . When the aneroid is used with the accompanying Table I., a 
correction must be made to convert the column of mercury from 32° to 60° Fahr., 
namely: 


Table L 


Height of a column of mercury as indicated by the aneroid. 


16 

17 

18 

19 

20 

21 

22 

23 

24t 

25 

26 

27 

28 

29 

.046 

.048 

.050 

.053 

.056 

.059 

.061 

.064 

.067 

.070 

.073 

.075 

.078 

.081 


30 

.084 


Correction in fraction of an inch, always additive. 


Example. Suppose the aneroid to indicate 25.261 inches. 

Correction from the table, . .070 

Height of a column of mercury, 25.331 inches at 60° Fahr. 

Heights of tlie Principal Mountains and Volcanoes. 


North America. Feet. 


Mount St. Elias, 
Jit. Brown, R. M., 
Sierra N evada, Cal., 
Fremont’s Peak, 
Long’s Peak. R. M., 
Oibao Mt., Hayti, 
Cierra del Cobre, 
Black Mt., S. C., 
Mt. Washington. 
Mansfield Jit., Vt., 
Peak of Otter, Vt.. 

South America. 

rilimani, Bolivia,* 
Ausangati, Peru,* 
Chimborazo, Eq., 
Sorato, Bolivia, 
Tolima, N. Gran., 
Cerro de Potosi, 
Cerro de Pasco, 
Organ Mt., Brazil, 


17,860 

16.000 

15.500 
13,470 

12.500 
8,600 
7,200 
6,476 
6,234 
4,280 
4,260 


24.100 

22.150 
21,960 
21.500 
18,250 

16.150 
13,780 

7.500 


Europe. 

Elbruz, Caucasus, 
Mont Blanc, Alps, 
Malhaven, Spain, 
Mt. Maladetta, Py., 
Mt. Caballo, Alps, 
Mt. Scardus, Tttr., 
Ural Jits., Russia, 
Asia. 

Kunchinginga.Hy. 
Dhawalaghiri, Hy., 
Hindo Koo, Cabul, 
Mt. Ararat, Tur., 
Mt. Lebanon, Syr., 
Africa. 

Abba Yared, A by., 
Pi ton des Neiges, 
'l’alba Waha, Aby., 
Oceanica. 

Mt. Ophir, Sum., 
Mt. Semero, Java, 


Feet. 

17,776 

15,668 

11.678 

11,436 

10,154 

10,000 

5,397 

28,176 

28.000 

20,000 

17,210 

12,000 

15,200 

12.500 

12,000 

13,842 

13,000 


Volcanoes, Active. 
Aconcagua, Cliili, 
Gualatieri, Peru, 
Cotapaxi, Equador, 
Misti, Peru*,. . 
Popocatapetl, 
Pichincha, Eqna., 
Kliutchewaskaja, 
Vo I can de Fuego, 
Mauna Loa, S. Is!., 
St. Helen’s, Oreg’n, 
Indrapma, Sum., 
Teneriffe, Can. Is]., 
Erebus, Vic. Land, 
Cartago, C. Amer., 
Etna, Sicily, . . 
Ilecla, Iceland, . 
Souffriere, Guad., 
Jnrollo, Mexico, 
Vesuvius, Italy, 


Feet. 

23,100 

22,000 

19,500 

18,136 

17,735 

16,000 

15,763 

14,000 

13,440 

13.300 

12.300 
12,182 
12.400 
11,480 
10,874 

5,110 

5,108 

4.205 

3,948 


* Measured by the author of this Pocket-book. 






































Barometric and Atmospheric Heights, 


496 


TABLE I. 

Barometric anti Atmospheric Heights. 


Dif. 


76.59 

75.87 

75.19 

74.56 

73.88 
73.24 
72.63 
71.98 
71.13 
70.77 
70 2! 
69 61 
69.07 
68.18 
67.94 
67.42 
t6 88 
66 38 
65.84 
65.32 
64.80 
64 34 

63.88 
03.36 

62.91 
62 46 
61.97 
61.50 
61.09 
60.65 
6 ).23 
59.76 
£9.34 

58.92 
58.68 
58.12 
57.79 

57.40 
56.90 

56.57 

56.20 
55 84 
55.47 
55.11 
51.74 

54.41 
54.10 
53.70 
53.37 
53.04 


French. 


English. 

Altitude. 

Bar. 

Bar. 

Altitude. 

metres. 

m.m. 

in. 

feet. 

848S09 

279.4 

ii. 

27848.5 

8411.48 

281.9 

.1 

27597.2 

8335.Gl 

284.5 

.2 

27848.3 

82G0 42 

287 0 

.3 

27101.6 

81S5.36 

289.5 

.4 

26857.0 

8111.96 

292.2 

.5 

26614.6 

8038.74 

2947 

.6 

26374.3 1 

7966.11 

297.3 

.7 

26136.0 1 

7894.13 

23.1.8 

.8 

25899.8 

7822.70 

302.2 

.9 

25665.5 

7751.93 

304 8 

13. 

25433.3 

7681.70 

307.3 

.1 

25202.9 

7G12.09 

309.8 

.2 

24974.5 

7543.02 

312.4 

.3 

24747.9 

7474.54 

314.9 

.4 

24523.2 

7406 GO 

317.5 

.5 

24300 3 

7339.18 

320.0 

.6 

24079.1 

7272.30 

322.5 

•7 

23S59.7 

7205.92 

325.1 

.8 

28641.9 

7140 08 

327. G 

9 

23425.9 

7u74.7G 

330.2 

13. 

23211.6 

7003.96 

332.7 

.1 

22999.0 

6945.02 

335.2 

•2 

22787.9 

6881.74 

337.8 

.3 

22578.3 

6818.38 

340.3 

.4 

22370 4 

6755.47 

342.9 

.5 

22164.0 

6693.01 

345.4 

.6 

21959.1 

6631.04 

347.9 

.7 

21755.8 

6569.54 

350.5 

.8 

21554.0 

6508 45 

353.0 

.9 

21353.6 

6447.80 

355.6 

14. 

21154.6 

6387.57 

358.1 

.1 

20957.0 

6327.81 

360.6 

.2 

20760.9 

6268.47 

3(53.2 

.3 

20566.2 

6209.55 

865.7 

.4 

203729 

G150.87 

368.3 

.5 

20180.4 

6092.75 

370.8 

.6 

19989.7 

6034 96 

373 3 

.7 

19800.1 

'.5977.56 

375.9 

.8 

19611.8 

15920 66 

378 4 

.9 

19425.1 

15864.0) 

881.0 

15 

19239.5 

15807.89 

383.5 

.1 

19055.1 

5752.05 

386.0 

.2 

18871.9 

5696.58 

388.6 

.3 

18689.9 

5641.47 

391.1 

.4 

18509.1 

5586.73 

393.6 

.5 

18829.5 

5532 32 

396 2 

.6 

18151.0 

15478 22 

398.7 

.7 

17973.5 

j5424 52 

4013 

.8 

17797.3 

5371.15 

403.8 

.9 

17622.2 


Dif. 

251.3 
248.9 

246.7 

244.6 

242.4 

240.3 

238.3 

236.2 

234.8 

232.2 

230.4 

228.4 

226.6 

224.7 

222.9 

221.2 

219.4 

217.8 
216.0 

214.3 
212.0 
211.1 

209.6 

207.9 

206.4 

204.9 

203.3 
201.8 

200.4 
199.0 

197.6 

196.1 

194.7 

193.8 

192.5 

190.9 

189.6 

188.3 

187.2 

185.6 

184.4 

183.2 
182.0 

180.3 

179.6 

178.5 

177.5 
176.2 
175.1 


Dif. 


French. 

Altitude. 

metres. 


Bar. 

m.m. 


52.72 

52.49 

51.94 
51.78 

51.39 
51.141 

50.82 j 

50.50 
50.20 

49.90 
49.65 

49.34 
49.07 
48.77 

48.47 

48.18 

47.91 
47.64 

47.40 

47.15 
40 85 

46.60 

46.36 
46.11 
45.88 

45.60 

45.35 
45.10 
44 90 
44.6J 
44.44 

44.19 

43.95 

43.74 

43.47 
43 22 
43.00 
42.821 
42.62 
42.42! 
42 24! 
42.04 

41.82 

41.60 
41.39 

41.15 
40 94 

40.75 
40.54 

40.37 


421.6 
4241 

426.7 

429.2 

431.8 

434.3 
436 8 

439.4 

441.9 

444.5 
447.0 


5318.11 406.4 

5265.39 408.9 

5212.90 411.4 
5160.96 414.0 
5109.18 4L0.5 

5057.79 419.1 

5006.65 

4955.83 

4905.33 
4855.13 

4805.23 
4755.58 

4706.24 
4657.17 

4608.40 
4559.93 

4511.75 

4463.84 449.5 

4416.20 452.1 

4368.80 454.6 

4321.65 457.1 

4274.80 409.7 

4228.20 462.2 

4181.84 464.8 
4135.73 467.3 

4089.85 409.9 

4044.25 472.4 

3995.90 474.9 
3953.S0 477.5 

3908.90 480.0 

3861.21 482.6 
3819.77 485.1 
3775 58,487.6 
3731.63 ] 490.2 
3687.89 492.7 
3644.42 495.3 

3601.20 497.8 

3558.20 £00.3 
3515.38 502.9 

3472.76 505.4 

3430.34 508.0 

3388.10 510.5 
3346.06 

3304.24 
3262 64 

3221.25 

3180.10 
3139 16 
3098 4! 
3057.87 


513.0 

516.6 

518.1 

520.7 

523.2 

525.7 

528.3 

530.8 



English 

Bar. 

Altitude. 

in. 

feet. 

1G. 

17448.2 

.1 

17275.2 

.2 

17103.3 

.3 

16982.6 

.4 

16762.7 

.5 

16594.1 

.6 

16426.3 

.7 

16259.6 

.8 

16093.9 

.9 

15929.2 

17. 

15765.5 

.1 

15602.6 

.2 

15440.7 

.3 

15279.7 

.4 

15119.7 

.5 

14960.7 

.6' 

14S02.6 

.7 

14615.4 

.8 

14489.1 

.9 

14333.6 

18. 

14178.9 

.1 

14025.2 

.2 

13872.3 

.3 

13720.2 

.4 

13568.9 

.5 

13418.4 

.6 

13268.8 

.7 

13120.0 

.8 

12972.0 

.9 

12824.7 

19. 

12678.1 

.1 

12532.3 

J) 

12387.3 

.3 

12243.1 

.4 

12099.6 

.5 

11957.0 

.6 

11815.2 

.7 

11674.1 

.8 

11533.6 

.9 

11393.S 

20. 

11254.6 

.1 

11116.0 

.2 

10978.1 

.3 

10840.9 

.4 

10704.4 

.5 

|l05G8.6 

.6 

110433.0 

.7 

10299.3 

.8 

110165.6 

.9 

110032.6 


Dif. 

173.0 
171.9 
170.7 
169 9 
168.6 


161.9 


159.0 

158.1 

167.2 

156.3 

155.5 

154.7 

153.7 
152.9 

152.1 

151.3 

150.5 

149.6 

148.8 
148.0 

147.3 

146.6 

145.8 
145 0 

144.2 

143.5 

142.6 

141.8 

141.2 
1405 

139.8 

139.2 

138.6 

137.9 

137.2 
136.5 
135.8 
135.0 

134.3 

133.7 
133.0 
,132.3 


The columns Bar. is the height of the Barometer in inches and milimetrcs. 

The columns Altitude is the corresponding height of level above the sea in feet 
and metres. 

The altitude in metres can he read from the barometer in inches; or, the altitude 
in feet can be read from the barometer in milimetres. 



























































Barometric and Atmospheric Heights, 


497 


TABLE I. 

Barometric and Atmospheric Heights. 


Dif. 

40.17 

40.09 

39.81 
39.65 
39.44 
39.23 
39.11 
38.99 

38.74 
38.53 

38.37 

38.20 
38.00 
37.88 
37.67 
37.52 

37.37 

37.16 
37.06 

36.82 
36.69 

36.59 
36.39 

36.21 
36.09 
35.93 
35.79 

35.60 
3-5.46 
35=30 

35.21 
35.02 
34.90 

34.75 
34.59 
34.47 
34.32 

34.17 
34.05 
33 92 
33.78 

33.61 
33.50 

33.37 
33.25 
33.13 
33.01 

32.83 
32.74 

32.61 


French. 


English 



French. 


English 

Altitude. 

Bar. 

Bar. 

Altitude. 

Dif. 

Dif. 

Altitude. 

Bar. 

Bar. 

Altitude. 

metres. 

m.m. 

in. 

feet. 

metres. 

m.m. 

in. 

feet. 

3017.50 

533.4 

21. 

9900.1 

131.7 

131.9 

130.6 

130.1 
129.4 

128.7 

128.3 

127.6 

127.1 

126.4 

125.9 

125.3 
1247 

124.3 

123.6 

123.1 

122.6 
1219 
121.6 

120.8 

120.4 
120.0 

119.4 
118.8 

118.4 

117.9 

117.4 
116.8 

116.4 

115.8 

115.5 

114.9 

114.5 
114.0 

113.5 

113.1 

112.6 

112.1 

111.7 

111.3 

110.8 

110.3 

109.9 
109.5 
109.1 

108.7 

108.3 

107.7 

107.4 
107.0 

32.46 

3234 

32.22 
32.09 
31.98 
31.88 
31.76 
31.64 

31.52 
31.39 
31 27 
31.15 
3103 
30.94 
30.81 
3072 
30.60 
30.48 
30.41 

30.24 

30.18 
30.05 
29.93 

29.84 
29.75 
29 63 
29 52 
29 45 
29.32 

29.23 
29.11 
29.02 

28.92 

28.84 
28.71 
28.62 

28.53 
28 43 
28 35 

28.25 

28.19 
28.10 
28.01 

27.92 
27.83 
27 77 
27.67 
27.58 
27.52 
27.43 

1210.61 

660.4 

36. 

3971.9 

2977.33 

535.9 

.i 

9768.3 

1178.15 

602.9 

.1 

3865.4 

2937.34 

538.4 

.2 

9637.1 

1145.81 

665.4 

.2 

3759.3 

2897.53 

541.0 

.3 

9o06.5 

1113.59 

668.0 

.3 

3653.6 

2S57.88 

543.5 

.4 

9376.4 

1081.50 

670.5 

.4 

35483 

2818.44 

546.1 

.5 

9247.0 

1049.52 

673.1 

.5 

3443.4 

2779.21 

548.6 

.6 

9118.3 

1017.64 

675 6 

.6 

3338.8 

2740.10 

551.1 

* .7 

8990.0 

985.888 

678.1 

.7 

3234 6 

2701.21 

558.7 

.8 

8862.4 

954.251 

680.7 

.8 

3130.8 

2662.47 

556.2 

.9 

8735.3 

922.734 

683.2 

.9 

3027.4 

2623.94 

558.8 

22. 

8608.9 

891.341 

685.8 

27. 

2924.4 

2585.57 

561.3 

.1 

8483.0 

860.070 

688.3 

.1 

2821.8 

2547.37 

563.8 

.2 

8357.7 

828.919 

690.8 

.2 

2719.6 

2509.37 

566.4 

.3 

8233.0 

797.891 

693.4 

.3 

2617.8 

2471.49 

668.9 

.4 

81(18.7 

766 953 

695.9 

.4 

25163 

2433 82 

571.5 

.5 

7985.1 

736.140 

698 5 

.5 

2415.2 

2396.30 

574.0 

.6 

7862.0 

705.416 

701.0 

.6 

2314.4 

2358.93 

576.5 

.7 

7739.4 

674.815 

703.5 

.7 

2214.0 

2321.77 

579.1 

.8 

7617.5 

044 335 

706.1 

.8 

2114.0 

2284.71 

581.6 

.9 

7495.9 

613.927 

708.0 

.9 

2014.3 

2247.89 

584.2 

23. 

7375.1 

583.682 

711.2 

28. 

1915.0 

2211.20 

586.7 

.1 

7254.7 

553.503 

713.7 

.1 

1816.0 

2174.61 

589.2 

.2 

7134.7 

523.454 

716.2 

.2 

1717.4 

2138.22 

591.8 

.3 

7015.3 

493 523 

718 8 

.3 

1619.2 

2102.01 

5'i4.3 

.4 

6896.5 

463.683 

721.3 

.4 

1521.3 

2065.92 

696.9 

.5 

6778.1 

433 935 

723.9 

.5 

1423 7 

2029.09 

599.4 

.6 

6660.2 

404.309 

726.4 

.6 

1326.5 

1994.20 

601.9 

.7 

6542.8 

374.785 

728.9 

.7 

1229.6 

1958.60 

604.6 

.8 

6426.0 

345.332 

731.5 

.8 

1133.0 

1923.14 

607.0 

.9 

6309.6 

316.010 

734.0 

.9 

1036.8 

1887.34 

609.6 

24. 

6193.8 

286.781 

736.6 

29 

940.9 

1852.63 

612.1 

.1 

6078.3 

257.672 

739.1 

.1 

845.4 

1817.61 

614.6 

.2 

5963.4 

228.657 

741.6 

.2 

750.2 

1782.71 

617.2 

.3 

5848.9 

199.731 

744.2 

.3 

655.3 

1747.96 

619 7 

.4 

5734.9 

170.898 

746.7 

.4 

560.7 

1713.37 

622.3 

.5 

5621.4 

142.186 

749.3 

.5 

466.5 

1678.90 

624.8 

.6 

5508.3 

113.566 

751.8 

.6 

3726 

1644.58 

627.3 

.7 

5395.7 

85.037 

754.3 

.7 

279 0 

1610.41 

629.9 

.8 

5283.6 

50.600 

750.9 

.8 

185.7 

1576.36 

632.4 

.9 

5171.9 

28.254 

759.4; 

.9 

92.7 

1542.44 

635 0 

25. 

5060.6 

0.0000 

762.0 

30. 

0.0000 

1508.66 

637.5 

.1 

4949.8 

28.193 

704.5 

.1 

92.5 

1475.05 

640.0 

.2 

4839.5 

56.295 

767.0 

.2 

184.7 

1411.55 

642.6 

.3 

4729.6 

84.305 

769.6 

.3 

27G.6 

1408.18 

645.1 

.4 

4620.1 

112.225 

772.1 

.4 

368.2 

1374.93 

647.7 

.5 

4511.0 

140.053 

774.7 

.5 

459.5 

1341.80 

650.3 

.6 

4402.3 

167.820 

777.2 

.6 

550.6 

1308.79 

652.8 

.7 

4294.0 

195.495 

779.7 

.7 

641.4 

1275 96 

655.3 

.8 

4186.3 

223.079 

782.3 

.8 

731.9 

1243.22 

657.9 

.9 

4078.9 

250.601 

278.033 

784.8 

787.4 

.9 

31. 

822.2 

912.2 


Dif. 

106.5 
106.1 

105.7 
105 3 

104.9 
104 6 

104.2 

103.8 

103.4 
103.0 

102.6 

102.2 

101.8 
101.6 
101.1 
100.8 

100.4 
100.0 

99.7 

99.3 

99.0 

98.6 

98.2 

97.9 

97.6 

97.2 

96.9 

96.6 

96.2 

95.9 

95.5 

95.2 

94.9 

94.5 

94.2 

93.9 

93.6 

93.3 
93.0 

92.7 

92.5 

92.2 
91.0 

91.6 

91.3 
91.1 

90.8 
90.5 

90.3 
90.0 


The difference in the Bar. m.m. column is 2.5 milimetres; therefore, multiply the 
difference of altitude in metres by the exceeding milimetres and by 0.4; subtract 
the product from the tabular altitude, and the remainder will be the altitude in 
metres, corresponding to the reading of the barometer in millimetres. 


32 









































498 


Correction for Temperature. 


TABLE II.—Correction for Mean Temperature. 


Temp. 




Ileig 

lit in feet or metres. 



Temp. 

Fahr. 

Cent 

1000 

20001 

3000 

40001 

5000 

6000 1 

7000 

8000 

9000i 

10000 

Cent, i 

Fahr. 

6T 

J 6.1 

2.08 

4.17 

6.25 

S.34 

10.42 

12.50 

14.59 

16.68 

18.76 

20.85 

15.0 

59 

62 

16.6 

4.17 

8.31 

12.49 

16.611 

20.82 

24.98 

29.15 

33.32 

37.48 

41.65 

14.4 58 

63 

17.2 

6.25 

12.49 

18.74 

24.98 

31.23 

37.48 

43.72149.97 

56.21 

62.46 

13.8 

57 

64 

17.7 

8.33 

16.67 

24.99 

33.34 

41.65 

49.98 

58.31 

66.61 

74.97 

83.30 

13.3 

56 

65 

18.3 

10.41 

20.82 

31.24 

41.65 

52.06 

62.48 

72.89183.30 

93.72 

104.1 

12.7 

55 

66 

18.8 

12.49 

24.99 

37.4s 

49.98 

62.47 

74.96 

87.46199.96 

112.4 

124.9 

12.2 

54 

67 

19.4 

14.58 

29.15 

43.43 

58.31 

72.89 

86.86 

102.1 

116.6 

130.3 

145.8 

11.6 

53 

68 

20.0 

16.66 

33.32 

49.61 

66.61 

83.30 

99.22 

116.6 133.3 

148.S 

166.6 

11.1 

52 

69 

20.5 

18.74 

37.49 

56.23 

74.98 

93.71 

112.4 

131.2 

149.9 

168.7 

187.4 

10.5 

51 

. 70 

21.1 

20.82 

41.64 

62.46 

83.28 

104.1 

124.9 

145.7 

166.5 

187.4 

208.2 

10.U 

50 

. 71 

21.6 

22.91 

45.81 

68.72 

91.6; 

114.5 

137.4 

160.3 

183.2 

•106.1 

229.1 

9.4 

49 * 

t 72 

22.2 

24.99 

49.98 

74.77 

99.96 

121.9 

149.5 

174.9 

199.9 

224.9 

249.9 

8.8 

48 5 

1 73 

22.7 

27.07 

54.14 

81.21 

108.3 

135.3 

162.4 

189.5 

216.6 

243.6 

270.7 

8.3 

47 ss 

t 74 

23.3 

29.15 

58*31 

87.46 

116.6 

145.7 

174.9 

204.1 

233.2 

262.4 

291.5 

7.7 

46 «2 

a 75 23.8 

31.24 

62.47 

93.71 

124.9 

156.2 

187.4 

218.6 

249 9 

281.1 

312.4 

7.2 

45 f* 

g ' G 

24.4 

33.32 

66.64 

99 96 

133.3 

166.6 

199.9 

233.2 

266.5 

299.9 

333.2 

6.G 

44 S 

- 77 

25.0 

35.40 

70.80 

106.2 

141.6 

177.0 

212.4 

247.8 

283.2 

318.6 

354.0 

6.1 

43 „ 

•I 78 

25.5 

37.48 

74.98 

112.4 

149.9 

187.4 

224.9 

261.41299.8 

337.3 

374.8 

5.5 

42 2 

72 79 

26.1 

39.57 

79.13 

118.7 

158.2 

197.8 

237.4 

277.0 

316.5 

356.1 

395.7 

5.0 

41 I 

S 80 

26.6 

41.65 

83.30 

124.9 

166.6 

20S.2 

249.9 

291.5 

333.2 

374.8 

416.5 

4.4 

40 2 

a 81 

27.2 

43.73 

87.46 

131.2 

174.9 

218.6 

262.4 

306.1 

349.8 

393.6 

437.3 

3.8 

39 c 
38 

o 82 

27.7 

45.SI 

91.63 

137.4 

183.2 

229.0 

274.8 

320.7 

366.5 

412.3 

458.1 

3.3 

s 88 

28.3 

47.90 

95.79 

143.3 

191.6 

233.5 

286.6 

335.3 383.2 

431.1 

478.9 

2.7 

37 g 

t 84 

28.8 

49.98 

99.96 

149.9 

199.9 

249.9 

299.8 

349.8 399.8 

449.8 

499.8 

2.2 

36 H 

5 85 

29.4 

52.06 

104.1 

156.2 

208.2 

260.3 

312.4 

364.4 410.5 

468.6 

520.6 

1.6 

35 8 

" 86 

30.0 

54.14 

108.2 

162.4 

216.5 

270.7 

324.8 

379.0 433.1 

487.3 

541.5 

1.1 

34 ® 

~ 87 

30.5 

56.23 

112.4 

168.6 

224.9 

281.1 

337.2 

393.6 449.8 

506.0 

562.2 

0.5 

33 -Z 

r 88 

31.1 

58.31 

116.6 

174.9 

233.2 

291.5 

349.8 

408.2 466.5 

524.8 

583.1 

0.0 

32 tS 

? 89 

31.6 

59.37 

118.7 

178.1 

237.5 

296.8 

356.2 

415.51474.9 

534.3 

593.7 

—0.5 

31 t 

< 90 

32.2 

61.77 

123.5 

185.3 

247.1 

308.8 

370.6 

432.4 494.1 

555.9 

617.7 

—i.i 

30 3 

91 

32.7 

64.56 

129.1 

193.7 

258.2 

322.8 

387.3 

451.9 516.5 

581.0 

645.6 

—1.6 

29 £ 
28 

92 

33.3 

66.64 

133.3 

199.9 

266.5 

333.2 

399.8 

466.5 533.1 

599.7 

i 666.4 

—2.2 

93 

33.8 

68.72 

137.4 

206.1 

274.9 

343.6 

412.3 

471.0 

1549.7 

618.5 

1 687.2 

—2.7,27 

94 

34.4 

70.80 

141.6 

212.4 

283.2 

354.0 

424.8 

495.6 566.4 637.2 

708.0 

—3.3 

26 

• 95 

35.0 

72.89 

145.8 

218.7 

291.6 

364.4 

437.3 

510.2 583.1 

656.0 

728.9 

—3.8 

25 

96 

35.5 

74.83 

149.6 

224.5 

299.3 

374.1 

449.0 

523.8 

598.6 

673.5 

748.3 

—4.4 

24 

. 97 

36.1 

77.05 

154.1 

231.1 

30S.2 

385.2 

462.3 

539.3 

616.4 

693.4 

770.5 

—5.0 

23 

' 98 

36.6 

79.13 

158.2 

237.4 

316.5 

395.6 

474.8 

553.9 

633.0 

712.2 

791.3 

—5.5 

22 

99 

37.2 

81.22 

162.4 

243.6 

324.9 

406.1 

487.3 

560.5 

649.7 

731.0 

812.2 

—6.1 

21 

100 

37.7 

83.30 

166.6 

249.9 

333 2 

416.5 

499.8 

583.1 

666.4 

749.7 

833.0 

—6.6 

20 

Fahr. 

Ceut 

1000 

2000 

3000 

4000 

5000 

6000 

7000 8006 

9000 100110 

Cent. 

Fahr. 



TABLE 

III. 

—Correction for Mean 

Latitude 

• 



Mean 

ititude. 

1000 

2000 

I 

3000 

Ieight 
4000 

s in fe 
5000 

et or m 
6000 

etres. 

7000 

8000 

9000 

10000 

Mean 

latitude. 


44 

0.09 

0.18 

0.27 

0.36 

0.45 

0.54 

0.63 

0.76 

0.81 

0.90 

46 . 


42 

0.27 

0.54 

0.81 

1.08 

1.35 

1.62 

1.S9 

2.1 G 

2.43 

2.7 

48 $ 


40 

0.44 

0.88 

1.32 

1.76 

2.20 

2.64 

3.08 

3.52 

3.96 

4.4 

50 s 


38 

0.62 

1.24 

1.86 

2.48 

3.10 

3.72 

4.34 

4.96 

5.58 

6.2 

52 -2 

£ 

36 

0.79 

1.58 

2.37 

3.16 

3.95 

4.74 

5.53 

6.32 

7.11 

7.9 

54 * 


34 

0.96 

1.92 

2.88 

3.84 

4.80 

5.76 

6.72 

7.68 

8.64 

9.6 

56 ® 


30 

1.28 

2.56 

3.84 

5.12 

6.40 

7.68 

8.96 

10.2 

11.5 

12.8 

60 | 

<D 

tf. 

28 

1.43 

2.86 

4.29 

5.72 

7.15 

8.58 

10.0 

11.4 

12.9 

14.3 

62 - 

O 

24 

1.71 

3.42 

5.13 

6.84 

8.55 

10.26 

12.0 

13.7 

15.4 

17.1 

66 is 


20 

1.95 

3.90 

5.85 

7.80 

9.75 

11.7 

13.6 

15.6 

17.5 

19.5 

70 ~ 

<2 

18 

2.06 

4.12 

6.18 

8.24 

10.3 

12.3 

14.4 

16.5 

18.5 

20.6 

7-) -M 

rs 

14 

2.25 

4.50 

6.75 

9.00 

11.2 

13.5 

15.7 

18.0 

2i 1.2 

22.5 

76 c 


10 

2.39 

4.78 

7.17 

9.56 

11.9 

14.3 

16.7 

19.1 

21.5 

23.9 

80 3 

*< 

6 

2.49 

4.98 

7.47 

9.96 

12.4 

14.9 

17.4 

19.8 

22.4 

24.9 

84 CO 


2 

2.54 

5.OS 

7.62 

10.2 

12.7 

15.2 

17.8 

20.3 

22.9 

25.4 

88 * 



























































































































Boiling Water, 


499 


TABLE IV.- 


-Logaritlnnic Correction for Temperature of tlie 

Atmosphere.. Always positive. 


Temp. 

Loga- 

Temp. 

Loga- 

Temp. 

Loga- 

Temp. 

Loga- 

Cent. 

Fahr 

rithms. 

Ceut. 

Fahr 

rithms. 

Cent. 

Fahr 

rithms. 

Cent. 

Fahr 

rithms. 

—2.2 

28 

9.97005 

8.33 

47 

9.98808 

18.8 

66 

0.00539 

29.4 

85 

0.02204 

—1.6 

29 

9.97102 

8.83 

48 

9.9S901 

19.4 

67 

0.00628 

30.0 

86 

0.02290 

— l.l 

30 

9.97198 

9.44 

49 

9.98993 

2 i.O 

68 

0.00717 

30.5 

87 

0.02376 

—0.5 

31 

9.97294 

10.0 

50 

9.99086 

20.5 

69 

0.00806 

31.1 

88 

0.02461 

0.0 

32 

9.97391 

10.5 

51 

9.99173 

21.1 

70 

0.00895 

31.6 

89 

0.02547 

+ 55 

33 

9.97487 

11.1 

52 

9.99270 

21.6 

71 

0.00984 

32.2 

90 

0.02632 

1.11 

34 

9 97682 

11.6 

53 

9.99362 

22.2 

72 

0.01072 

32.7 

91 

0.02717 

1.66 

35 

9.97678 

12.2 

54 

9.99454 

22.7 

73 

0.01160 

33.3 

92 

0.02802 

2.22 

36 

9.97773 

12.7 

55 

9.99515 

23.3 

74 

0.01248 

33.8 

93 

0.02886 

2.77 

37 

9.97868 

13.3 

56 

9.99537 

23.8 

75 

0.01336 

34.4 

94 

0.02971 

3 33 

38 

9.97963 

13.8 

57 

9.99728 

24.4 

76 

0.01423 

35.0 

95 

0.03055 

3.88 

39 

9.98058 

14.4 

58 

9.99819 

25.0 

77 

0.01511 

35.5 

96 

0.03139 

4.44 

40 

9.98152 

15.0 

59 

9.99909 

25.5 

78 

0.01598 

36.1 

97 

0.03224 

5.00 

41 

9.98217 

15.5 

60 

0.00000 

26.1 

79 

0.01685 

36.6 

98 

0.O3307 

5.55 

42 

9.98341 

16.1 

61 

0.00090 

26.6 

80 

0.01772 

37.2 

99 

0.03391 

6.11 

43 

9.98434 

16.6 

62 

0.00180 

27.2 

81 

0.01859 

37.7 

100 

0.03475 

6 60 

44 

9.98528 

17.2 

63 

0.00270 

27.7 

82 

0.01945 

38.3 

101 

0.03558 

7.22 

45 

9.98622 

17.7 

64 

0.00360 

28.3 

83 

0.02032 

38.8 

102 

0.03641 

7.77 

45 

9.98715 

18.3 

65 

0.00450 

28.8 

84 

0.02118 

39.4 

103 

0.03724 


TABLE 

V.- 

-Logaritlianic Correction for Mean Latitude of 

Observation. Always positive. 

Lat. 

I Log. 

Lat 

Log. 

Lat. 

Log. 

Lat. 

Log. 

Lat. 

Log. 

Lat. 

Log. 

0 

0.00111 

15 

0.00096 

30 

0.00055 

45 

0.00000 

60 

9.99944 

75 

9.99904 

1 

0.00110 

16 

0.00094 

31 

0.00052 

46 

9.99996 

61 

9.99941 

76 

9.99902 

2 

0.00110 

17 

0.00092 

32 

0.00048 

47 

9.99992 

62 

9.99938 

77 

9.99900 

3 

0.00110 

18 

0.00089 

33 

0.00045 

48 

9.99988 

63 

9.99935 

78 

9.99898 

4 

0.00109 

19 

0.00087 

34 

0.00041 

49 

9.999S4 

64 

9.99932 

79 

9.99897 

5 

0.00109 

20 

0.00085 

35 

0.00038 

50 

9.99981 

65 

9.99929 

80 

9.99896 

6 

0.00108 

21 

0.00082 

36 

0.00034 

51 

9.99977 

66 

9.99926 

81 

9.99894 

7 

0.00107 

22 

000079 

37 

0.00030 

52 

9.99973 

67 

9.99923 

82 

9.99893 

8 

0.00106 

23 

0.00077 

38 

0.00027 

53 

9.99969 

68 

9.99920 

83 

9.99892 

9 

0.00105 

24 

0.00074 

39 

000023 

54 

9.99966 

69 

9.99917 

84 

9.99891 

10 

O.U0104 

25 

0.00071 

40 

0.00019 

55 

9.99962 

70 

9.99915 

85 

9.99891 

11 

0.00103 

26 

0.00068 

41 

0.06015 

56 

9.99958 

71 

9.99913 

86 

9.99890 

12 

0.00101 

27 

0.00665 

42 

0.00011 

57 

9.99955 

72 

9.99910 

87 

9.99889 

13 

0.00099 

23 

0.00062 

43 

0.00008 

58 

9.99951 

73 

9.99908 

88 

9.99889 

14 

0.00093 

29 

0.00059 

44 

0.00004 

59 

9.99948 

74 

9.99906 

89 

9.99889 


TABLE VI.- 

-Temperature 

of Boiling Water, 



Corresponding to the Height of the. Barometer at 60° Fahrenheit. 



French Measures. 

Height 
Diff. Barotn. 

M. M. 

4:54.07 
44'!.68 
453.45 
463.41 
473.55 
483.86 
494.33 
504 09 


9.C1 
9.77 
9.96 
10.14 
10.31 
10.47 
10 66 
10.85 
11.08 
11.24 
11.46 

11.84 

11.85 
12.07 
12.27 


515.84 

526.92 

53S.16 

54062 

561.26 

573.11 

685.18 


Temp. 

Water. 

Cent. 

85.00 

85.55 
86.11 
86.66 

87.22 

87.77 
88.33 
8S.99 
89.44 
90.00 

90.55 
91.11 
91.66 

92.22 

92.77 


English Measures, 

Temp. Height 
Water Barom. DifF. 
Falir. Inches. 

185 17.090 

186 17.468 

187 17.853 

188 18 245 

189 18.644 

190 19.050 

191 19.462 

192 19.882 

193 20.309 

194 20.745 

195 21.188 

196 21.639 

197 22.097 

198 22.564 

199 23.039 


.378 

.385 

.892 

.399 

.406 

.412 

.420 

.127 

.436 

.413 

.451 

.458 

.467 

.475 

.483 


French Measures. 

Height Temp. 
Barom. Water. 
M. M. Cent. 

597.45 93.33 

609.94 93.88 

622.64 94.44 

635.55 95.00 

648.73 95.55 

662.09 96.11 

675.66 96.66 

689.49 97.22 

703.54 97.77 

717.82 98.33 

732.35 98.88 

747 13 90.44 

762.17 100.0 
777.43 100.5 

792.95 101.6 


Diff. 

12.49 

12.70 

12.91 

13.18 

13.36 

13.57 

13.83 

14.05 

14.28 

14.53 

14.78 

15.04 

15.26 

15.52 


English Measures. 

Temp. 

Water 
Fahr. 


200 

201 

202 

203 

204 

205 

206 

207 

208 

209 

210 
211 
212 

213 

214 


Height 

Barom. 

Inches. 

23.522 

24.014 

24.514 

25.022 

25.551 

26.067 

26.602 

27.146 

27.699 

28.261 

28.833 

29.415 

30.007 

30.608 

31.219 


Diff. 


.192 

.500 

.508 

.519 

.526 

.535 

.544 

.553 

.562 

.572 

.582 

.592 

.601 

.611 






























































































500 


Columns of Air and Mercury, 


TABLE VII_Height of a Column of Air in Metres, 

Corresponding to one milimetre in the barometer, at different temperatures. 

Bar. 

M.M. 

—6 

—3 

Ceri 

0 

tignn 

+3 

le tei 

6 

ii pen 

9 

ittire of th 

12 1 15 

e air 

18 

and t 
21 

he barome 
24 27 

ter. 

30 

33 

1 36 

400 

20.7 

20.8 

20.9 

21.0 

21.1 

21.2 

21.3 

21.4 

21.5 

21.6 

21.7 

21.8 

21.9 

22.0 

22.1 

420 

19.7 

19.8 

19.9 

20.0 

20.1 

20.2 

20.3 

20.4 

20.5 

20.6 

20.7 

20.8 

20.9 

21.0 

21.1 

440 

18.8 

18.9 

19.0 

19.1 

19.2 

19.3 

19.4 

19.5 

19 6 

19.7 

19.8 

19.9 

20.0 

20.1 

20.2 

460 

18.0 

18.1 

18.2 

18.3 

18.4 

1 .4 

18.5 

18.6 

18.7 

18.8 

18.8 

18.9 

19.0 

19.1 

19.2 

480 

17.2 

17.3 

17.4 

17.5 

17.6 

17.6 

17.7 

17.8 

17.9 

18.0 

18.1 

18.1 

18.2 

18.3 

18.4 

500 

16.5 

16.6 

16.7 

16.8 

16.9 

16.9 

17.0 

1.7.1 

17.2 

17.3 

17.3 

17.4 

17.5 

17.6 

17.7 

520 

15.8 

15.9 

16.0 

16.1 

16.2 

16.2 

16.3 

16.4 

16.5 

16.6 

16.0 

16.7 

16.8 

16.9 

17.0 

540 

15.3 

15.3 

15.4 

15.5 

15.6 

15.6 

15.7 

15.8 

15.9 

16.0 

16.0 

10.1 

16.2 

16.3 

10.3 

560 

14.8 

14.8 

14.9 

15.0 

15.1 

15.1 

15.2 

15.3 

15.4 

15.5 

15.5 

15.6 

15.7 

15.8 

15.8 

580 

14.3 

14.3 

14.4 

14.5 

14.6 

14.6 

14.7 

14.8 

14.9 

15.0 

15.0 

15.1 

15.2 

15.3 

15.3 

CIO 

13.8 

13.8 

13.9 

14.0 

14.1 

14.1 

14.2 

14.3 

14.4 

14.5 

14.5 

14.6 

14.7 

14.S 

14.8 

620 

13.3 

13.3 

13.4 

13.5 

136 

13.6 

13.7 

13.8 

13.9 

14.0 

14.0 

14.1 

14.2 

143 

14.3 

040 

12.9 

12.9 

13.0 

13.1 

13.2 

13.2 

13.3 

L3.4 

13.5 

13.6 

13.6 

13.7 

13.8 

13.9 

13.9 

660 

12.5 

12.6 

12.6 

12.7 

12.8 

12.8 

12.9 

13.0 

13.1 

13.2 

13.2 

13.3 

13.3 

13.4 

134 

680 

12.2 

12.2 

12.3 

12 3 

12.4 

12.5 

12.5 

,2.6 

12.7 

12 7 

12.8 

12.9 

129 

13.0 

13.1 

700 

11.9 

11.9 

12.0 

12.0 

12.1 

12.2 

12.2 

12.3 

12.4 

12.4 

12.5 

12.6 

12.6 

12.7 

12.7 

720 

11.5 

11.5 

11.6 

11.6 

11.7 

11.8 

11.8 

11.9 

11.9 

12.0 

12.1 

12.1 

12.2 

12.3 

12.3 

740 

11.2 

11 2 

11.3 

11.3 

11.4 

115 

11.5 

11.6 

11.7 

11.7 

11.8 

11.9 

11.9 

12.0 

12.0 

760 

10.9 

10.9 

11.0 

11.1 

11.1 

11.2 

11.2 

11.3 

11.4 

11.4 

11.5 

11.6 

11.0 

11.7 

11.7 

780 

10.6 

10.6 

10.7 

10.8 

10.8 

10.9 

10.9 

11.0 

11.1 

11.1 

11.2 

11.3 

11.3 

11.4 

11.4 


TABLE VIII.- 

-Height of 

a Column of Air in 

Feet, 


Corresponding to one-tenth of 

an inch in 

the barometer at different tem])e 

■atures. 

Bar. 



Fahrenheit temperature of the air 

and the barometer. 



In. 

30° 

35° 

40° 

45° 

50° 

55° 

60° 

65° 

70° 

75° 

80° 

85° 

90° 

95° 

100° 

16 

163 

165 

167 

168 

170 

172 

174 

176 

178 

179 

181 

183 

185 

187 

188 j 

17 

153 

155 

156 

158 

159 

161 

163 

165 

166 

168 

170 

171 

173 

175 

117 

18 

145 

157 

158 

160 

161 

162 

154 

156 

157 

159 

160 

162 

163 

165 

106 J 

19 

135 

137 

138 

140 

142 

144 

146 

148 

149 

151 

152 

153 

155 

157 

158 

20 

130 

132 

133 

135 

136 

137 

139 

140 

142 

143 

145 

146 

148 

149 

151 

21 

124 

126 

127 

128 

130 

131 

132 

133 

135 

136 

137 

139 

140 

142 

143 

22 

118 

120 

121 

123 

124 

125 

126 

127 

129 

130 

131 

132 

134 

135 

136 1 

23 

112 

114 

115 

116 

117 

118 

120 

121 

122 

124 

125 

126 

127 

129 

130 

24 

l'>8 

110 

111 

112 

113 

114 

115 

116 

117 

119 

120 

121 

122 

123 

125 

25 

104 

106 

107 

108 

109 

110 

111 

112 

113 

115 

116 

117 

118 

119 

120 

25.5 

102 

104 

105 

106 

107 

108 

109 

110 

112 

113 

114 

115 

116 

117 

118 

26 

109 

102 

103 

U4 

105 

106 

107 

103 

109 

111 

112 

113 

114 

115 

116 

26.5 

98 

109 

l‘Ul 

102 

103 

101 

105 

106 

108 

109 

110 

111 

112 

113 

114 i 

27 

96 

97 

98 

100 

101 

102 

103 

104 

105 

107 

108 

109 

110 

111 

112 , 

27.5 

95 

96 

97 

93 

99 

100 

101 

102 

103 

104 

105 

106 

107 

108 

ll i9 

28 

93 

94 

95 

96 

97 

98 

99 

100 

101 

102 

103 

104 

105 

106 

107 

28.5 

91 

92 

93 

94 

95 

96 

97 

98 

99 

100 

101 

102 

103 

104 

105 

29 

90 

91 

92 

93 

94 

95 

96 

97 

98 

99 

100 

101 

102 

103 

104 

29.5 

88 

84 

90 

91 

92 

93 

94 

95 

96 

97 

98 

99 

100 

101 

102 

30 

87 

S8 

89 

90 

91 

92 

93 

94 

95 

96 

97 

98 

99 

100 

101 

TABLE IX.—Mean Height of the Barometer 

in different countries, reduced to the level of the sea, and to 60° Fahr. temperature. 


Africa, Northern, . 

Inches. 

30.26 

China, . . 

Inches. 

30.11 

Peru, .... 

Inches. 

30.09 

Atlantic coast,, N. A., 


Denmark, . 

29.99 

Prussia, .... 

30.00 

Northern States, 

30.10 

England, . 

30.03 

Scotland, . . . 

29.93 

Southern States, 

30.17 

France, . . 
Greenland, 

30.00 

Sicily. 

30.11 

Australia, .... 

30.00 

29.75 

Spitsbergen, . . 

29.87 

Frazil. 

30.15 

Italy, , . 

30.00 

Sweden, .... 

29.96 

Canary Islands, . . 

30.16 

Iceland, 

29.70 

Venezuela, . . . 

3001) 

Cape Good Hope, . 

30.11 

Norway,. . 

29.89 

West In. Islands, 

30.02 










































































































Correction for the Barometer, 


501 


TABLE X.—Correction for tlie Mercurial Column 

_ in Millimetres at Different Temperatures of Barometer. 


Tem. 




Height of 

>arometer in millimetres. 




Cea. 

415 

440 

465 

490 

515 

540 

565 

| 590 

615 

640 

! 665 

690 

715 

I 7401 765 

Cent. 

. 16 

0.03)0.03 

0.04 

0.04 

0.04 

0.04 

0.04 0.0510.05 

0.05 

0.05 

0.05 

0.06'0.06 0.06 

15 

9 17 

0.10 

0.11 

0.11 

0.12 

0.12 

0.13 

0.14|0.14 

0.15 

0.15 

0.16 

0.17 

0.17 

0.18 0.18 

14 • 

= 18 

0.16 

0.18 

0.19! 0.20 0.21 

0.22 

0.23 0.24 

0.25 

0.26 

0.27 

0.28; 0.29 

.0 30 0.30, 

1 °» ^ 

2 10 

0.24 

0.25 

0.26 

0.28 0 29 

0.30 

0.32,0.33 

0.35 

0.36 

0.38 

0.39,0.40 

0.42 0.43 

12 | 

§•20 

0.35 

0.37 

0.39,0.42 

0.44 

0.46 

0.48.0.50 

0 52 

0 54 

0 56 

0.59 

0.61 

0.63 0.65 

11 | 

5 21 

0.42 

0.45 

0.42 

0.51 

0.53 

0.56 

0.59 0.61 

0.64 

0.C6 

0.69 

0.71 

0.74 0.76 0.78 

10 g 

» 22 0.49 

0.52 

0.55 0.58 

0 60 

0.63 

0.66 

0.69 

0.72 

0.75 

0 78 

0.8 L 0.84 

0.87 jo 90| 

9 § 

“ 23 

|0.5o 

0.5S 

0.62 

0.65 

0.69 

0.72 

0.75 

0.79 

0.82 

0.85 

0.89 

0.92 

0.95 

0.99; 1.02 

8 * 

r 24: 0.62 

0.66 

0.70 0.73 

0.77 

0.81 

0.85 

0.88 

0.92 

0.86 

1.00 

1.03 

1.07 

1.11 1.15 

7 % 

=2 25! 

0.69 

0.73 0.77 

0.81 

085 

0.89 

0.94 

0.98 

1.02 

1.06 

1.10 

1.14 

1.19 

1.23 1.27 

6 t 

g2G 

0.75 

0.8010.84 0.89 

0.93 

0.98 

1.02 

1.07 

1.11 

1.16 

1.20 

1.25 

1.30 

1.34'1.39 

5 <2 

'g 27 

0.82 

0.87 

0.92 0.97 

1.02 

1.08 

1.12 

1.17 

1.22 

1.27 

1.32 

1.37 

1.42 

1.4711.52 

4 g 

£ 28 

0.89 

0.94 

1.00 

1.05 

1.10 

1.16 

1.21 

1.26 

1.32 

1.37 

1.42 

1.48 

1.53 

1.59)1.64 

3 i 

g 20 

0.9S 

1.01 

1.07 

1.13 

1.19 

1.24 

1.30 

1.36 

1.41 

1.47 

1.53 

1.59 

1.64 

1.70 1.76 

2 g 

8 30 

1.02 

1.09 

1.15 

1.21 

1.27 

1.33 

1.40 

1.46 

153 

1.58 

1.64 

1.70 

1.76 

1.83 1.89 

1 O 

- 31 

1.09 

1.16 

1.22 

1.29 

1.35 

1.42 

1.48 

1.55 

1.61 

1.63 

1.74 

1.S1 

1.88 

1.94 2.01 

0 2 

3 52 

1.16 

1.23 

1.30 

1.36 

1.43 

1.50 

1.56 

1.64 

1.71 

1.78 

1.85 

1.92 

1.99 

2.06 2.13 

—1 ~ 

$ 33 

1.23 

1.30 

1.38 

1.45 

1.52 

1.60 

1.67 

1.74 

1.82 

1.89 

1.97 

2.04 

2.11 

2.19)2.26 

—2 « 

134 

1.29 

1.37 

1.44 

1.52 

1.60 

1.68 

1.76 

1.83 

1.91, 

1.99 

2.07 

2.14 

2 22 

2.30 2.38 

—3 

35 

1.36 

1.44 

1.52 

1.60 

1.69 

1.76 

1.85 

1.93 

2.01 

2.10 1 

2.17 

2.25 

2.33 

2.42 2.50 

—4 


TABLE XI.—Correction for the Mercurial Column in Thou- 

sands of an Inch , at Different Temperatures of the Barometer above or bdoio 60°. 


Temp. 



Height of the barometer in inches. 





Temp 

Fahr. 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

Fahr. 

a 62 

003 

003 

003 

003 

003 

003 

004 

004 

004 

0()4 

005 

005 

005 

005 

005 

58 

3 64 

006 

006 

006 

007 

007 

007 

008 

008 

008 

00:i 

009 

009 

010 

010 

010 

56 c 

0 66 

0')9 

009 

009 

010 

on 

Oil 

012 

012 

012 

013 

014 

014 

015 

015 

016 

54 S 

75 68 

011 

012 

013 

013 

014 

015 

016 

016 

017 

018 

018 

019 

020 

020 

021 

52 | 

t 70 

014 

015 

016 

017 

018 

0*9 

02) 

020 

021 

022 

023 

024 

025 

026 

026 

50 " 

1 72 

017 

018 

019 

020)021 

022 023 

024 

024 

027 

028 

029 

030 

031 

032 

48 | 

§ 74 

020 

021 

.022 

024)025 

026 

027 

028 

029 

031 

032 

034 

035 

036 

038 

46 g 

a; 76 

023 

024 

026 

1 027 028 

030 

031 

032 

034 

036 

037 

039 

040 

Oil 

043 

44 £ 

5 78 

026 

027 

029 

030|032 

033 

035 

036 

038 

040 

042 

044 

045 

046 

048 

42 a 

g 80 

029 

03 ) 

032 

034|036 

037 

039 

041 

043 

045 

046 

048 

050 

052 

054 

40 Js 

£ 82 

031 

033 

035 

037(039 

041 

043 

045 

048 

049 

051 

053 

055 

057 

059 

38 0 

3 84 

0‘>4 

036 

039 

041)043 

045 

047 

049 

052 

054 

056 

058 

060 

062 

064 

36 a 

0 8G 

037 

040 

042 

044 046 

049! 

051 

053 

056 

058 

060 

063 

065 

067 

070 

34 s 

h 88 

049 

043 

045 

047 

050 

052 

055 

057 

061 

063 

065 

068 

070 

072 

075 

32 g 

3 90 

043 

046 

048 

051 

054 

056 ! 

059 

061 

065 

067 

070 

072 

075 

077 

080 

30 g 

j} 92 

046 

049 

051 

054 

057 

060 

063 

065 

069 

071 

074 

077 

080 

0S3 

086 

28 2 

- 94 

049 

052 

054 

057 

360 

064 

067 

069 

074 

076 

079 

082 

085 

088 

091 

26 5 

§ 96 

051 

055 

058 

060 < 

363 

067 

071 

073 

078 

080 

084 

087 

090 

093 

097 

24? 

5 98 

054 

0581 

061 

064 067 

071 

075 

077 

082 

0S5 

088 

092 

095 

09S 

102 

22 < 

5 100 

0571 

0611 

064 

068)0711 

075 

079 

082 

086| 

089 

093 

097 

100 

104 

107| 

20 


Heights in 

Feet, of the Principal Waterfalls. 



Gavarny, Pyrenees, 

1260 

Gray Mare’s Tail, 

350 

Rupin, Himalayas, 

120 

Lanterbrun, 

Switz.. 

912 

Hepste, 

# 


300 

Kakabika, 

S. Am., 

115 

Staubbach, Switz., 

900 

Nakchikin. Kamch. 

300 

Lidford, England, 

100 

Riiiean, 

Norway, 


800 

Terni, Italy, 


270 

Genesee. N 

York, 

100 

Seculego, Pyrenees, 

795 

Montmorency, Can., 

242 

O.vapock, S 

Amer., 

80 

bule&, Sweden. 


600 

Foyers, Scotland, 

207 

Rhine Lauffen, 

SwJ. 

65 

Tequendama,Colum. 

540 

Wilberforce 

, N. 

A., 

160 

Ti 

•ollhetta, Sweden, 

60 

Tosa, Piedmont, 

1 

470 

Cetina, Dalmatia, 

150 

Parana, Paraguay, 

.52 

Missouri, N. Amor., 

400 

Niagara Falls, 


145 

Tivoli, 

Ital v. 

• 

50 

Powerseanrt 

Ircl 

*9 

380 

Tendon 

, France, 


125 

Cataracts of Nile, 

40 






























































































































502 


Temperatures. 


'■*— 


TABLE XII.—Mean Temperature of tlie Air 

at the Level of the Sea. 


Months in the 


North Latitude. 



South Latitude. 


Ther. 

year. 


60 

50 

40 

30 

20 

10 

0 

10 

20 

30 

40 


January, 

• 

25 

4G 

62 

72 

78 

80 

80 

80 

77 

75 

72 


February, 


28 

48 

63 

73 

78 

81 

80 

80 

77 

74 

71 


March, 

• 

32 

50 

64 

74 

79 

82 

81 

79 

76 

72 

69 


April, 


38 

55 

67 

76 

81 

83 

82 

79 

75 

70 

64 

4-> 

May, . 

• 

48 

Gl 

72 

78 

83 

84 

83 

78 

74 

68 

61 

a> 

June, 


58 

G7 

75 

80 

84 

85 

84 

78 

72 

66 

65 

c 

July, . 

• 

Gl 

69 

7G 

81 

85 

86 

84 

77 

73 

64 

52 

o 

u 

August, . 


59 

G8 

75 

80 

84 

85 

83 

78 

73 

64 

51 


September, 


52 

G4 

72 

78 

83 

84 

82 

78 

72 

62 

54 

£ 

October, . 


44 

57 

68 

76 

81 

83 

81 

79 

71 

63 

59 


November, 

• 

35 

52 

65 

74 

80 

82 

80 

79 

73 

66 

66 


December, 


28 

48 

63 

73 

79 

81 

80 

79 

75 

71 

71 


January, 

• 

—3.8 

7.7 

16.6 

22.2 

25.5 

26.6 

26.6 

26.4 

25.3 

23.8 

22.2 


February, 


—2.2 

8.8 

17.2 

22.7 

25.8 

27.2 

26.8 

26 6 

25. 

23.3 

21.6 


March, 

• 

0.0 

10. 

17.7 

23 3 

26.1 

27.7 

27.2 

26.1 

24.4 

22.2 

20.5 


April, 


+3.3 

12.7 

19.4 

24.4 

27.2 

28 3 

27.7 

26.1 

23.8 

21.1 

17.7 


May, . 

• 

S.8 

1G.1 

22.2 

25.5 

28.3 

28.8 

28.3 

25.5 

23.3 

20. 

16.1 


June, . 


14.4 

19.4 

23.8 

26.6 

28.8 

29.4 

28.8 

25.5 

22.2 

18.8 

18.3 

j-. 

July, . 

• 

16.1 

20.5 

24.4 

27 2 

29.4 

30. 

28.8 

25. 

22.7 

17.7 

11.1 


August, . 


15. 

20. 

23.8 

26.6 

28.S 

29 4 

28.3 

25.5 

22.7 

17.7 

10.5 

p 

September, 

# 

11.1 

17.7 

22.2 

25.5 

28.3 

28.8 

27.7 

25.5 

22.2 

16.6 

12.2 

o 

October, . 


6.G 

13.8 

20. 

244 

27.2 

28.3 

27 2 

25.8 

21.6 

17.2 

15. 


November, 

# 

1.0 

11.1 

18.3 

23 3 

26.6 

27.7 

26.8 

26. 

22.7 

18.8 

18.8 


December, 


—2.2 

8.8 

17.2 

22.7 

26.1 

27.2 

26.6 

26.1 

23.8 

21.6 

21.0 



Heights of Natural and Artificial Worlts. 


Heights Above Level of the Sea. 

Feet. 

Heights Above the Ground. 

Feet. 

Green in a balloon. 1837, . 

27 000 

Tower of Babel, said to have been 

680 

Gay-Lussac, Paris, 1804, 

22900 

Pyramid Clusops, Egypt, . 

520 

Highest flight of condor, 

21.000 

Tower of Baal bee, Syria, 

500 

Humboldt in the Andes, 

19,500 

St. Peter’s Cathedral, Rome, . 

500 

Growth of vegetation, . 

The author in the Andes,* 

17.000 

Spire of Strasbourg, 

486 

15 120 

Cathedral, Antwerp, 

476 

Lake Manasarooa, Thibet, 

14 500 

St. Stephen’s spire, Vienna, 

465 

Pine and birch grow. 

14.000 

Highest chimney. Glasgow, 

455 

Highest habitation of people,* 

14.000 

Spire of Salisbury, 

450 

Potosi silver mine, Bolivia, 

13 350 

Cathedral, Milan, . . . 

438 

Lake Titicaca, Peru,* . 

13000 

St. Mary, Ltibeck, . 

404 

La Paz, Bolivia,* 

12 400 

Cathedral, Florence,. . 

384 

Poplar grows at . . 

12000 

St. Paul, London, . 

366 

City of Cuzco, Peru,* 

11.500 

Hotel des Invalides, Paris, 

344 

Oak grows at . 

11.000 

Cathedral, New York, . 

Dome of Capitol, Washington, 

325 

Citv iliobatpba, Andes, 

10-800 

287 

Quito, Equador, 

9,560 

Trinity Church, New York, 

286 

City St. Bernard, Switzerland, 

8,600 

Notre Dame, Paris. . 

220 

City Santa Fe do Bogota, 

8.350 

Column City of London, 

202 

Wild monkeys found at* . 

8.000 

Porcelain. China, . . . 

Leaning Tower of Pisa, 

200 

City of Mexico, 

6.900 

188 

St. Gothard, Alps, 

6.900 

Alexander Column. St. Petersb’g, 

175 

Lake Lucon, France, 

6.220 

July Column. Paris. 

157 

Palm and bananas grow at 

2.500 

Column Napoleon, Paris,, 

13S 




* Measured by the author of this Pocket-book. 












































































Heat. 


503 


HEAT. 

Heat resembles light, electricity , and magnetism. It is convertible into dynamic 
work, and can consequently be resolved into the two physical elements, force 
and motion. Temperature is convertible into force, which is only one element 
of heat, and is no measure of quantity of heat. (See Dynamics and Units 
of Heat.) 

The temperature or intensity of heat is measured in various ways, but most 
generally by the expansion of mercury and alcohol, or the thermometer. 


Thermometers. 

There are three differently graduated thermometers in use—namely, Fahrenheit , 
Centigrade and Reaumur. The last named is gradually being abolished, and now 
used only in Peru. 


Graduation. Fahr. Cent. Reau. 


Zero Fahr. = —17.77° Cent.—14.22° Reau. 




- 

- too- 

- 80 

“ 

Freezing-Point of Water. 




/S7- 

- 75 - 

- 60 

- 

Zero Cent. = 32 Fahr. = zero Reau. 




122 - 

- so- 

- 40- 

- 

Boiling-Point of Water. 




212° Fahr. = 100° Cent. = 80° Reau. 77_ 

- 25- 

- 20 

* 

9° Fahr. = 5° Cent. = 4° Reau. 32 - 

- 0 - 

- O 


Formulas. o ~ 

- m 

- 14.2 


Cent. = | (Fahr. =F 32) = \ Reau. 32- 

- 40- 

- 32 

- 

Fahr. = | Cent. ± 32 = f Reau. dt 32. M 

§ 4 

|) ^ 


Reau. = £ Cent. = f (Fahr. ^ 32). 





The accompanying tables give the equivalents of Centigrade’s and Fahrenheit’s 
thermometers. The Jirst numbers in the table of comparison, — 276* and 461*, are 
the absolute zero of temperature. 

Example 1. How many degrees on Fahr. scale is 964.5° Cent. ? 

Table comparison, Cent. 9G0° = 17G0° Fahr. 

Table Centigrade, Cent. 4.5 = 8.1 “ 

The required, Cent. 9G4.5 17G8.1 “ 

Example 2. How many degrees is 2136.7° Fahr. on Centigrade thermometer? 

Table comparison, Fahr. 2120° — 11 GO 0 Cent. 

Table Fahrenheit, “ J6° = 8.90 “ 

“ “ “ 0.7 == 0.389 “ 

The required degrees,‘Fahr. 213G.7 — 11G9.289 “ 

* Deduced by Nystrom. 



















504 Thermometers. 


Comparison of Fahrenheit ami Centigrade Thermometers. 


Falir. 

Centig. 

Falir. 

Centig. 

Falir. 

Centig. 

Falir. 

Centig. 

Fahr. 

Centig. 

— 5 

— 20.55 

57 

13.88 

119 

48.33 

181 

82.77 

243 

117.22 

— 4 

— 20.00 

58 

14.44 

120 

48.88 

182 

83.33 

244 

117.77 

■— 3 

— 19.44 

59 

15.00 

121 

49.44 

183 

83.88 

245 

118.33 

— 2 

—18.88 

60 

15.55 

122 

50.0!) 

184 

84 44 

246 

118.88 

— 1 

—18.33 

61 

16.11 

123 

50.55 

185 

85.00 

247 

119.44 

Zero. 

—17.77 

62 

1C.66 

124 

51.11 

186 

85.55 

248 

120.00 

+ 1 

— 17.22 

63 

17.22 

125 

51.66 

187 

86.11 

249 

120.55 

2 

—16.66 

64 

17.77 

126 

52.22 

188 

86.66 

250 

121.11 

3 

—16.11 

65 

18.33 

127 

52.77 

1S9 

87.22 

251 

121.66 

4 

— 15.55 

66 

18.68 

128 

53.33 

190 

87.77 

252 

122.22 

5 

—15.00 

67 

19.44 

129 

53.88 

191 

88.33 

253 

122.77 

6 

—14.44 

68 

20.00 

130 

54.44 

192 

88.88 

254 

123.33 

7 

— 13.S8 

69 

20.55 

131 

55.00 

193 

89.44 

255 

123.88 

8 

—13.33 

70 

21.11 

132 

55.55 

194 

90.00 

256 

124.44 

9 

—12.77 

71 

21.G6 

133 

56.11 

195 

90.55 

257 

125.00 

10 

—12.22 

72 

22.22 

134 

56.66 

196 

91.11 

258 

125.55 

11 

— 11.66 

73 

22.77 

135 

57.22 

197 

91.66 

259 

126.11 

12 

— 11.11 

74 

23.:i3 

136 

57.77 

198 

92.22 

260 

126.66 

13 

—10.55 

75 

23.88 

137 

58.33 

199 

92.77 

261 

127.22 

14 

— 10.00 

76 

24.44 

138 

58.88 

200 

93.33 

262 

127.77 

15 

— 9.44 

77 

25.00 

139 

59.44 

201 

93.88 

263 

128.33 

16 

— 8.88 

78 

25.55 

140 

60.00 

202 

94.44 

264 

128.88 

17 

— 8 33 

79 

20.11 

141 

60.55 

203 

95.00 

265 

129.44 

18 

— 7.77 

80 

26.66 

142 

61.11 

204 

95.65 

266 

130.00 

19 

— 7.22 

81 

27.22 

143 

61.66 

205 

96.11 

267 

130.55 

20 

— 6.66 

82 

27.77 

144 

62.22 

206 

96.66 

268 

131.11 

21 

— 6.11 

83 

2+33 

145 

62.77 

207 

97.22 

269 

131.66 

22 

— 5.55 

84 

28.88 

146 

63.33 

208 

97.77 

270 

132.22 

23 

— 5.00 

85 

29.44 

147 

63.88 

21)9 

98.33 

271 

132.77 

24 

— 4.44 

86 

30.00 

148 

64.44 

210 

98.88 

272 

133.33 

25 

— 3.88 

87 

30.55 

149 

65.00 

211 

99.44 

273 

133.88 

26 

— 3.33 

88 

31.11 

150 

65.55 

212 

100 00 

274 

134.44 

27 

— 2.77 

89 

31.06 

151 

C6.ll 

213 

100.55 

275 

135.00 

28 

— 2.22 

90 

32.22 

152 

06.66 

214 

101.11 

276 

135.55 

29 

— 1.66 

91 

32.77 

153 

67 22 

215 

101.66 

277 

136.11 

30 

— Ill 

92 

33.33 

154 

67.77 

216 

102.22 

278 

136.66 

31 

— .55 

93 

33.88 

155 

68.33 

2)7 

102.77 

279 

137.22 

32 

Zero. 

94 

34.44 

156 

68.S8 

218 

103.33 

280 

137.77 

33 

+ 0.55 

95 

35.00 

157 

69.44 

219 

103.88 

281 

138.33 

34 

1.11 

96 

35.55 

158 

70.00 

220 

104.44 

282 

138.S8 

35 

1.66 

97 

36.11 

159 

70.55 

221 

1<>5.00 

283 

139.44 

36 

2.22 

98 

30.66 

100 

71.11 

222 

105.55 

284 

140.00 

37 

2.77 

99 

37 22 

101 

71.C6 

223 

106.11 

285 

140.55 

38 

3.33 

100 

37.77 

162 

72.22 

221 

106.66 

286 

141.11 

39 

3.88 

101 

38.33 

163 

72.77 

225 

107.22 

287 

141.66 

40 

4.44 

102 

38.88 

104 

73.33 

226 

K'7.77 

288 

142.22 

41 

6.00 

103 

39.44 

165 

73.88 

227 

108."3 

289 

142.77 

42 

5.55 

104 

40.00 

166 

74.41 

228 

108.88 

290 

143.33 

43 

6.11 

105 

40.55 

167 

75.00 

229 

1(9.44 

291 

143.88 

41 

6 66 

106 

41.11 

16S 

76.55 

230 

110.00 

292 

144.44 

45 

7.22 

107 

41.66 

169 

76.11 

231 

110.55 

293 

145.00 

46 

7.77 

108 

42.22 

170 

76.66 

232 

111.11 

294 

145.55 

47 

8.33 

109 

42.77 

171 

77.22 

233 

111.66 

295 

146.11 

48 

8.88 

110 

43.33 

172 

77.77 

234 

112.22 

296 

146.C6 

49 

9.44 

111 

43.88 

173 

78.33 

235 

112.77 

297 

147.22 

60 

10.00 

112 

44.44 

174 

78.88 

236 

113.33 

298 

147.77 

51 

10.55 

113 

45.00 

175 

79.44 

237 

113.88 

299 

148.33 

52 

11.11 

114 

45.55 

176 

80.00 

238 

114.44 

300 

148.88 

53 

11.66 

115 

46.11 

177 

80.55. 

239 

115.00 

400 

204.44 

54 

12.22 

116 

46.66 

178 

81.11 

240 

115.55 

600 

315.55 

55 

1277 

117 

47.22 

179 

81.06 

241 

116.11 

800 

433.33 

66 

13.33 

118 

47.77 

180 

82.22 

242 

116.66 

1000 

537.77 
































i 


Thermometers, 


505 


Comparison of Centigrade and Fahrenheit Thermometers. 


Cent. 

Fab r. 

Cent. 

Fahr. 

Cent. 

Fahr. 

Cent. 

Fahr. 

Cent. 

Fahr. 

276* 

461* 

16 

60.8 

330 

626 

950 

1742 

1570 

2858 

—260 

—436 

17 

62.6 

340 

644 

9t.O 

1760 

1580 

2876 

—250 

—418 

18 

64.4 

350 

662 

970 

1778 

1590 

2894 

—240 

-^100 

19 

66.2 

360 

680 

980 

1796 

1600 

2912 

—230 

—382 

20 

68.0 

370 

698 

990 

1814 

1610 

2930 

! —220 

—364 

21 

69.8 

380 

716 

1000 

1832 

1620 

294S 

—210 

—346 

22 

71.6 

390 

734 

1010 

1850 

1630 

2966 

—200 

—328 

23 

73.4 

400 

752 

1020 

1868 

1640 

2984 

—190 

—310 

24 

75.2 

410 

770 

1030 

1886 

1650 

3002 

—180 

—298 

25 

77.0 

420 

788 

1040 

1904 

1660 

3020 

—170 

—274 

26 

78.8 

430 

806 

1050 

1922 

1670 

3038 

—160 

—256 

27 

80.6 

440 

824 

1060 

1940 

1680 

3056 

—150 

—238 

28 

82.4 

450 

842 

1070 

1958 

1690 

3074 

—140 

—220 

29 

84.2 

460 

860 

1080 

1976 

1700 

3092 

—130 

—202 

30 

86.0 

470 

878 

1090 

1994 

1710 

3110 

—120 

—184 

31 

87.8 

480 

896 

1100 

2012 

1720 

3128 

— 110 

—166 

32 

89.6 

490 

914 

1110 

2030 

1730 

3146 

—100 

—148 

33 

91.4 

500 

932 

1120 

2048 

1740 

3164 

— 90 

—130 

34 

93.2 

510 

950 

1130 

2066 

1750 

3182 

— 80 

—112 

35 

95.0 

520 

968 

1140 

2084 

1760 

3200 

— 70 

— 94 

36 

96.8 

530 

986 

1150 

2102 

1770 

3218 

— 60 

— 76 

37 

98.6 

540 

1004 

1160 

2120 

1780 

3236 

— 50 

— 58 

38 

100.4 

550 

1022 

1170 

2138 

1790 

3254 

— 40 

— 40 

39 

102.2 

560 

1040 

1180 

2156 

1800 

3272 

— 30 

— 22 

40 

104.0 

570 

1058 

1190 

2174 

1810 

3290 

— 20 

— 4 

41 

105.8 

5S0 

1076 

1200 

2192 

1820 

3308 

— 19 

— 2.2 

42 

107.6 

590 

li'94 

1210 

2210 

1S30 

3326 

— 18 

— 0.4 

43 

109.4 

600 

1112 

1220 

2228 

1840 

3344 

17.77 

Zero. 

44 

111.2 

610 

1130 

1230 

2246 

1850 

3362 

— 17 

+ 1-4 

45 

113.0 

620 

1148 

1240 

2264 

1860 

3380 

— 16 

+ 3.2 

46 

114.8 

630 

1166 

1250 

2282 

1870 

3398 

— 15 

+ 5.0 

47 

116.6 

640 

1184 

1260 

2300 

1880 

3416 

— 14 

+ 6.8 

48 

118.4 

650 

1202 

1270 

2318 

1S90 

3434 

— 13 

+ 8.6 

49 

120.2 

660 

1220 

1280 

2336 

1900 

3452 

— 12 

+10.4 

50 

122.0 

670 

123.8 

1290 

2354 

1910 

3470 

— 11 

+12.2 

60 

140 

680 

1256 

1300 

2372 

1920 

3488 

— 10 

+ 14.0 

70 

158 

690 

1274 

1310 

2390 

1930 

3506 

- 9 

+15.8 

80 

176 

700 

1292 

1320 

2408 

1940 

&524 

— 8 

+17.6 

90 

194 

710 

1310 

1330 

2426 

1950 

3542 

— 7 

+19.4 

100 

212 

720 

1328 

1340 

2444 

1960 

3560 

— 6 

+21.2 

110 

230 

730 

1346 

1350 

2462 

1970 

3578 

— 5 

+23.0 

120 

248 

740 

1364 

1360 

2480 

1980 

3596 

— 4 

+24.8 

130 

266 

750 

1382 

1370 

2498 

1990 

3614 

— 3 

+26.6 

140 

284 

760 

1400 

1380 

2516 

2000 

3632 

— 2 

+28.4 

150 

302 

770 

1418 

1390 

2534 

2010 

3650 

— 1 

+30.2 

160 

320 

780 

1436 

1400 

2552 

2020 

3068 

Zero. 

+32. 

170 

338 

790 

1454 

1410 

2570 

2030 

3686 

+1 

+33.8 

180 

356 

800 

1472 

1420 

2588 

2040 

3704 

2 

35.6 

190 

374 

810 

1490 

1430 

2606 

2050 

3722 

3 

37.4 

200 

392 

820 

1508 

1440 

2624 

2060 

3740 

4 

39.2 

210 

410 

830 

1526 

1450 

2642 

2070 

3758 

5 

41.0 

220 

428 

840 

1544 

1460 

2660 

2080 

3776 

6 

42.8 

230 

446 

850 

1562 

1470 

2678 

2090 

3794 

7 

44.6 

240 

464 . 

860 

1580 

1480 

2696 

2100 

3812 

8 

46.4 

250 

482 

870 

1698 

1490 

2714 

2110 

3830 

9 

48.2 

260 

500 

880 

1616 

1500 

2732 

2120 

3848 

10 

50.0 

270 

518 

890 

1634 

1510 

2750 

2130 

4166 

11 

51.8 

2S0 

536 

900 

1652 

1520 

2768 

2140 

4184 

12 

53.6 

290 

554 

910 

1670 

1530 

2786 

2150 

4162 

13 

. 52.4 

300 

572 

920 

1688 

1540 

2804 

2 ICO 

4180 

14 

57.2 

310 

590 

930 

1706 

1550 

2822 

2180 

4216 

15 

59.0 I 

320 

608 

940 

1724 

1560 

2840 

2200 

4252 


J 
















































506 


Comparison op Thermometers. 


Number of Degrees Cent. = Number of Degrees Falir. 


Degrees 



Tenths of a Degree—Centigrade Scale. 



Cent. 









.8 


.0 

.1 

.2 

.3 

.4 

.5 

.0 

.7 

.9 


Fahr. 

Fahr. 

Fahr. 

Fahr. 

Fahr. 

Fahr. 

Fahr. 

Fahr. 

Fahr. 

Fahr. 

0 

0.00 

0.18 

0.36 

0.54 

0 72 

0.90 

1.08 

1.26 

1.44 

1.62 

1 

1.80 

1.98 

2.16 

2.34 

2.55 

2.70 

2.88 

3.06 

3.21 

3.42 

2 

3.60 

3 78 

3.96 

4.14 

4.32 

4.50 

4.68 

4.86 

5.04 

5.22 

3 

5.40 

6.58 

5.76 

5.94 

6.12 

6.30 

6.48 

6.06 

6.84 

7.02 

4 

7.20 

7.38 

7.56 

7.74 

7.92 

8.10 

8.28 

8.46 

8.64 

8.82 

5 

9.00 

9.18 

9.36 

9.54 

9.72 

9.90 

10.08 

10.26 

10.44 

10.62 

6 

10.80 

10.98 

11.16 

11.34 

11.52 

11.70 

11.88 

12.06 

12.24 

12.42 

7 

12.GO 

12.78 

12 96 

13.14 

13.32 

13.50 

13.68 

13.86 

14.0 4 

14.22 

8 

14.40 

14.58 

14.76 

14.9 4 

15.12 

15.30 

15.48 

15.66 

15.84 

16.02 

9 

16.20 

16.38 

16.66 

16.74 

16.92 

17.10 

17.28 

17.46 

17.64 

17.82 

Number 

of Degrees 

Falir. = Number of Degrees Cent. 

Degrees 



Tenths of a Degree—Fahrenheit Scale. 



Falir. 











.0 

.1 

.2 

.3 

.4 

•5 

.0 

7. 

.8 

.9 


Ceut. 

Cent. 

Cent. 

Cent. 

Cent. 

Cent. 

Cent. 

Cent. 

Cent. 

Cent 

0 

0.00 

0.06 

0.11 

0.17 

0.22 

0.28 

0.33 

0.39 

0.44 

0.50 

1 

0.56 

0.61 

0.67 

0.72 

0.78 

0.83 

0.89 

0.94 

1.00 

1.06 

2 

1.11 

1.17 

1.22 

1.28 

1.33 

1.39 

1.44 

1.50 

1.56 

1.61 

3 

1.67 

1.72 

1.78 

1.83 

1.89 

1 94 

2.00 

2.06 

2.11 

2.17 

4 

2.22 

2.28 

2 33 

2.39 

2.44 

2.50 

2.56 

2.61 

2.67 

2.72 

5 

2.78 

2.83 

2.89 

2. "4 

3.00 

3.<>6 

3.11 

3.17 

3.22 

3.28 

6 

3.33 

3.39 

3.44 

3.5) 

3.56 

3.61 

3.67 

3.72 

3.78 

3.8 3 

7 

3.89 

3.94 

4.00 

4.06 

4.11 

4.17 

4.22 

4.28 

4.33 

4.39 

8 

4.44 

4.50 

4 56 

4.61 

4 67 

4.72 

4.78 

4.83 

4.89 

4.94 

9 

5.00 

5.06 

5.11 

5.17 

5.22 

5.28 

5 33 

5.39 

5.44 

5.59 


LATENT HEAT. 


Latent heat is the number of units of heat required to change a body from one 
state to another whilst the temperature remains constant—that is, the heat required 
to melt a body from solid to liquid, and to evaporate a liquid. In the one osise it is 
called the latent heat of fusion, and in the other, the latent heat of evaporation. 

Latent Units of Heat per Pound of Substance. 


Solids smelted to 

Latent 

Liquids converted to 

Latent 

liquid. 

heat. 

vapor. 

heat. 

Tee to water,. 

Ill 

Water to steam. 

966 

Tin. 

25.6 

Ammonia. 

895 

Zinc, • • • • • • • • 

50.6 

Alcohol, pure,. 

372 

Sulphur,. 

17.0 

Carbonic acid, .... 

298 

Lead, ••••••• 

9.72 

Bisulphide of carbon, . . 

242 

Mercury,. 

5.00 

Ether, sulphuric, . . . 

174 

Beeswax,. 

175 

Essence of turpentine, . . 

137 

Bismuth,. 

550 

Oil of turpeutine, . . -. 

184 

Cnst iron,. 

233 

Mercury. 

157 

Spermaceti,. 

46.4 

Chvmogene,. 

175 

Fusion. 


Eva po rat ion. 


L — latent heat (units) per pound of 

1 = latent units of heat per pound of 

liquid at smelting-point. 

vapor at boiling-point. 


C =■ specific heat of the liquid. 

T= temperature of boiling-point, Fahr. 

c — specific heat of solid. 
t = temperature of fusion, Fahr. 

( liegnauU .) 


L = {C—c)(t + 256). 

__i__ 

1 = 1091.7 — 0.695( T — 32)— 
0.000000103(T— 39.1) 3 . 




















































































507 


Temperature of Boiling and Evaporation. 


Temperature of Boiling or Evaporation under Atmospheric 

Pressure. 


Liquids. 

Fahr. 

Cent. 

Liquids. 

Fahr. 

Cent. 

Wrought iron, . 

6000° 

2760° 

Alcohol, .... 

173 

78 

Cast iron. . . 

3300 

1815 

Ether, .... 

96 

35 

Mercury, . . 

675 

352 

Carbon, bifeulphuretted, 

116 

47 

Whale oil, . 

630 

332 

Water, distilled, . 

212 

100 

Oil of linseed, 

600 

316 

Salt sea water, 

213 

101 

Oil of turpentine, 

357 

ISO 

Water 20 per cent, salt, 

218 

103 

Sulphuric acid, 

593 

312 

“ 30 “ “ . 

222 

105 

Sulphur, 

570 

300 

“ 40 saturated, 

227 

108 

Phosphorus, . 

557 

292 

Ammonia, liquid , . 

140 

60 

Sweet oil, . . 

412 

211 

Water in vacuo, . 

98 

36 

Naphtha, . 

320 

160 

Chymogene, . . 

+ 38 

3.3 

Nitric acid,. 

220 

104 

Carbonic acid, . 

— 112 

— 80 

Milk of cows,. 

Rectified petroleum, . 

213 

101 

Ammonia, 

— 30 

— 34 

316 

158 





Distillation Temperatures of Coal-oils.— (Tissandier.) 


Light Oils. 

Fahr. 

Cent. 

Heavy Oils. 

Fahr. 

Cent. 

Heavy Oils. 

Fahr. 

Cent. 

Amyl&ne, 

102° 

38.9° 

Cnm5ne, 

304° 

151° 

Carbolic acid 

370° 

188° 

Benzine, 

187° 

86.1° 

Lntidine, 

311° 

155° 

Neph tli line, 

422° 

217° 

Toluene, 

226° 

108° 

Etipione, 

338° 

170° 

Quiloneine, 

402° 

239° 

Xvlene, 

271° 

133° 

Cymene, 

347° 

175° 

Anthracene, 

500° 

260° 

Pyridine, 

302° 

150° 

Aniline. 

359° 

182° 

Chrysene, 

572° 

300° 


The temperature of distillation of vapors is equal to that of the boiling-point of 
the liquid of which the vapor is formed. 


Temperature of Fusion, Freezing or Melting-Point. 


Solids. 



Fahr. 

Cent. 

Solids. 


Fahr. 

Cent. 

Platinum, 

• 

• 

3080° 

1093° 

Puddle slag, . 

• • 

2606° 

1430° 

Wrought iron, . 


• 

2912 

1C00 

Sulphur, 

• 

228 

109 

Cast iron, gray. 

• 


2012 

1100 

Beeswax, white, 

• • 

155 

08 

Cast iron, white, . 


• 

1922 

1050 

“ yellow, 

• 

142 

Cl 

Steel, 

• 

• 

2500 

1371 

Spermaceti, 

• • 

142 

Cl 

Gold, pure, . 


• 

2300 

1260 

Potassium, . . 

• 

136 

58 

Gold, money, . 

• 

• 

2192 

1200 

Sodium, . 

• • 

104 

90 

Copper, 


• 

21 CO 

1232 

Olive oil, 

• 

92 

33 

Brass, common, 

• 

• 

1900 

1038 

Tallow, . 

• • 

36 

2.2 

Silver, . . . 


• 

1850 

1021 

Ice of water, 

• 

32 

o.ooo 

Litharge, 

• 

• 

1739 

954 

“ milk, . 

• • 

30 

— 1.1 

Antimony, . 


• 

800 

427 

“ sea water,. 

• 

28 

— 2.2 

Zinc, . 

• 

• 

740 

393 

“ vinegar, 

• • 

28 

— 2.2 

Lead, . 


. L 

COO 

31G 

“ strong wine 

• 

20 

— 6.6 

Bismuth,. 

• 

B 

470 

254 

“ “ brandy, 

7 

— 13.9 

Tin, 


. T 

420 

215 

“ oil of turpentine, 

14 

— lo 

2 Tin, 1 Lead, . 


• 

SCO 

181 

1 snow, 1 salt, 

• • 

0.00 

— 17.8 

1 Tin, 3 Lead, 


• 

500 

2C0 

1 alcohol, 1 water, 

• 

— 7 

— 21.6 

1 Tin. 1 Bismuth, 

• 

• 

283 

140 

Cyangen,. 

• • 

— 30 

— 34.4 

3T+2L + 6B, . 


• 

212 

100 

Mercury, 

• 

— 40 

— 40 

IT -f 1L + 4B, 

• 

• 

200 

93 

Sulphuric ether, 

• • 

— 47 

— 43.9 

2T + 3L -f 2B, . 


• 

199 

92 

Sulphurous acid,. 

• 

—105 

— 76 

Slag of copper. 

• 

• 

24C2- 

1350 

Nitrous oxide, 

• • 

—150 

— 101 

Slag of tin, . 


• 

2402 

1318 

Nitric acid, . . 

• 

— 55 


Nickel, . , 

t 


2800 

1538 


















































Expansion op Bodies by Heat. 


r.os 


EXPANSION OF BODIES BY HEAT. 

Most bodies in nature expand when heated, and contract when cooled. Solids 
vary but little by the difference in temperature; liquids vary more; but gases are 
extremely susceptible to such differences. 

There is a very singular fact connected 
with the expansion and contraction of 
substances at and near the temperature 
of fusion, which may be illustrated in the 
accompanying figure. 

L(^t A B represent the absciss-axis of 
temperature, 0 D the ordinate axis of ex¬ 
pansion or contraction, and the origin 0 
the temperature of fusion, 0 A tlie tem¬ 
perature of the solid, and 0 B that of the 
liquid. 

Let a solid of volume and temperature 
at a be heated, it will expand until it 
reaches a maximum volume at b. after 
which it contracts toward the temperature of fusion 0. The temperature still 
increasing, the liquid will continue to contract until it reaches a minimum 
volume or maximum density at d, after which it. will expand toward e. The lines 
a b 0 and Ode arc parabolas, of which the absciss-axis A B jwsses through the 

foci /. The formula for the parabola is y — x u , in which the exponent n depends 
upon the nature of the substance operated upon, and also whether it is linear or 
volume expansion, x representing the temperature and y the volume. 

Ice melts at 32° Palir., and the water reaches its maximum density at d = 39° (as 
now accepted, but d is nearer 40°). Ice reaches its maximum volume at b — 24°. 
Ice and water are of equal density at the temperatures 1C°, 32° and 48°. Ice 
generally floats in water, because the difference in temperature is less than 32°: 
but if ice of less than 16° is put into water of more than 48°, it will sink. The 
same phenomenon takes place with other substances; for instance, solid cast iron 
put into molten cast iron will float, but if the fluid cast iron is at a white heat, 
like that in a pneumatic furnace (Bessemer), the solid iron will sink. 

The following formulas are deduced from experiments which have not extended 
through the temperatures of fusion, except that for water, page 392. 

• 

Notation. 

L = linear expansion of solids and liquids, per degree Fahr., between any 
temperatures. 

I = linear expansion per degree between 32° and 212°, as contained in the 
• accompanying table. 

D and d — absolute temperature in degrees Fahr. 

n — exponent of expansion, which varies inversely with the rate of expansion 
of bodies. 


Exponent n 

1.04 

2.5 

2.6 

2.77 

14.1 

15.6 

for 

Water. 

Glass. 

Iron. 

Copper. 

Platinum. 

Mercury. 


Linear expansion per degree from 32° to T° will be 


L =- y D. 

10580000 v 

Linear expansion per degree between any temperature is 

L — --— [v D — 1 7d\ 

10580000 V ) 

The linear expansion per degree multiplied by 2 will be the surface expansion. 
The linear expansion per degree multiplied by 3 will be the volume expansion. 



















Dilatation or Expansion of Substanoes. 509 



Dilatation or Expansion of Substances, 



Per Degree of Fahrenheit Scale. 


Tempera¬ 

tures. 

Solids. 


Linear, l. 

Surface, a. 

Volume, v. 

32° to 212° 

1 Glass, 


0.00000478 

0.00000956 

0.00001434 

212 “ 392 

• 

0.00001 >546 

0.00001093 

0.00001639 

392 “ 572 

) 


0.00000660 

0.00001320 

0.00001980 

32 “ 212 

j- Wrought iron, . 


0.00000656 

0.00001312 

0.00001968 

32 “ 572 



0.00000895 

0.00001790 

0.00002686 

32 “ 212 

Soft, good iron, . 

• 

0.00000680 

0.00001360 

0.00002040 

32 “ 212 

Cast iron, . 


0.00000618 

0.00001236 

0.00001854 

32 “ 212 

Cast steel, . . 

• 

0.00000600 

0.00001200 

0.00001800 

32 “ 212 

Hardened steel, 


0.00000689 

0.00001378 

0.00002067 

32 “ 212 

| Copper, 


0.00000955 

0.00001910 

0.00002865 

32 “ 572 

• 

0.00001092 

0.00002184 

0.00003276 

32 “ 212 

Lead, 


0.00001580 

0.00003160 

0.00004740 

32 “ 212 

Gold, pure, . . . 

• 

O.OOOOOS15 

0.00001630 

0.00002445 

32 “ 212 

Gold, hammered, . 


0.00000830 

0.00001660 

0.00002490 

33 “212 

Silver, pure, . . 

« 

0.00001060 

0.00002120 

0.00003180 

32 “ 212 

Silver, hammered, . 


0.00001116 

0.00002232 

0.00003348 

32 “ 212 

Brass, common cast, . 

• 

0.00001043 

0.00002086 

0.00003129 

32 “ 212 

Brass, wire or sheet, 


0.00001075 

0.00002150 

0.00003225 

32 “ 212 
32 “ 572 

| Platinum, pure, 

• 

0.00000491 

0.00000520 

0.00000982 

0.00001040 

0.00001473 

0.00001560 

32 “ 212 

Palladium, 


0.00000555 

0.00001110 

0.00001665 

32 “ 212 

Roman cement, . 
Platinum, hammered, 

• 

0.O0000797 

0.00001594 

0.00002391 

32 “ 212 


0.00000530 

0.00001060 

0.00001590 

32 “ 212 

Zinc, pure or cast, 

• 

0.0> >001633 

0.00003266 

0.00004899 

32 “ 212 

Zinc, hammered, . 


0.00001722 

0.00003444 

0.00005166 

32 “ 212 

Tin, cast, 

• 

0.00001207 

0.00002414 

0.00003621 

32 “ 212 

Tin, hammered, . 


0.00001500 

0.00003000 

0.00004500 

32 “ 212 

Fire brick, . 

• 

0.00000275 

0.00000470 

0.00000705 

32 “ 212 

Good red brick, 


0.00000305 

0.00000610 

0.00000915 

32 “ 212 

Marble, 

• 

0.00000613 

0.00001226 

0.00001839 

32 “ 212 

Granite, . . . 


0.00000138 

0.00000876 

0.00001314 

32 “ 212 

Bismuth, 

• 

0.00000773 

0.00001546 

0.00002319 

32 “ 212 

Antimony, 


0.00000602 

0.00001204 

0.00001806 

32 “ 212 

Palladium, . 

• 

0.00000555 

0.00001110 

0.00001665 

32 “ 212 

) 


0.00003333 

0.00006666 

0.00010000 

212 “ 392 

V Mercury, 


0.00003416 

0.00006833 

0.00010250 

392 “ 572 

j 


0.00003500 

0.00007000 

0.00010500 

32 “ 212 

) 


0.00008806 

0.00017612 

0.00026420 

212 “ 392 

y Water, 

• 

0.(>0017066 

0.00034133 

0.00051020 

392 “ 572 

f 


0.00018904 

0.00037808 

0.00056713 

32 “ 212 

Salt, dissolved, 

• 

0.00009250 

0.00018500 

0.00027780 

32 “ 212 

Sulphuric acid, . 
Turpentine and ether, 

• 

0.00011111 

0.00022222 

0.000333*3 

32 “ 212 

• 

0.00012966 

0.00025933 

0.00038900 

32 “ 212 

Oil, common, 

Alcohol and Nitric Acid, 

• 

0.00011814 

0.00029629 

0.00044444 

32 “ 212 


0.00015151 

0.00030302 

0.00055555 

32 “ 212 

All permanent gases, 

• 

0.00069416 

0.00138S32 

0.00208250 


Force of Temperature. 



It is temperature which expands bodies, and not the quantity of heat. See 

pages 379 and 392. Temperature is 

convertible into pressure. Let P denote the 

pressure in 

pounds per square inch, 

and T temperature Fahr. 


Then 

/r+mn 

r> 

and T~ 202.8t/'P — 

105.1. 

V 202.8 / 





This force 

multiplied by tne space of expansion, is the work done by the heat. 




































510 


Expansion of Cast Iron, 



Linear Expansion or 

Contraction 

in Inches 




of 

Cast Iron, Lengths 

in Feet. 



Length. 



Difference in Temperature.—Fahrenheit 



100° 

150° 

1200° 

1350° 

300° 

400° 

500° 

000° 

800° 

Feet. 

Inch. 

Inch. 

Inch. 

Inch. 

Inch. 

Inch. 

Inch. 

Inch. 

Inch. 

1 

0.0072 

0.0110 

0.0150 

0.0192 

0.0237 

0.0336 

0.0444 

0.0561 

0.0787 

2 

0.0144 

0.0220 

0.0300 

0.0384 

0.0474 

0.0632 

0.0885 

0.1123 

0.1574 

n 

O 

0.0216 

0.0330 

0.0450 

0.0576 

0.0711 

0.1008 

0.1332 

0.1684 

0.2301 

4 

0.02*8 

0.0440 

0.0600 

0.0768 

0.0948 

0.1344 

0.1776 

0.2246 

0.3148 

5 

0.0360 

0.0550 

0.0750 

0.0960 

0.1185 

0.1680 

0.2220 

0 2805 

0.3935 

6 

0.0432 

0.0660 

0.0900 

0.1152 

0.1422 

0.2016 

0.2664 

0.3368 

0.4722 

7 

0.0504 

0.0770 

0.1050 

0.1344 

0.1659 

0.2552 

0 3108 

0.3929 

0.3509 

8 

0.0576 

0.0880 

0.1200 

0.1536 

0.1890 

0.2688 

0.3552 

0.4496 

0.6396 

9 

0.0648 

0.0990 

0.1350 

0.1728 

0.2133 

0.3024 

0.3996 

0.5052 

0.70S3 

10 

0.0720 

0.1102 

0.1502 

0.1926 

0.2376 

0.3360 

0.4440 

0.5616 

0.7872 

11 

0.0792 

0.1214 

0.1652 

0.2125 

0.2615 

0.3096 

0.4S84 

0.6177 

0.8659 

12 

0.0864 

0.1316 

0.1802 

0.2318 

0.2853 

0.4032 

0.5328 

0.6739 

0,9446 

13 

0.0936 

0.1417 

0.1952 

0.2510 

0.3090 

0.4368 

0.5772 

0.7300 

1.0233 

14 

0.1008 

0.1519 

0.2102 

0.2703 

0.3328 

0.4704 

0.0216 

0.7862 

1.1020 

15 

0.1080 

0.1620 

0.2253 

0.2895 

0.3565- 

0.5040 

0.6660 

0.8423 

1.1808 

16 

0.1152 

0.1722 

0.2403 

0.30*8 

0 3803 

0.5376 

0.7104 

0.8985 

1.2595 

17 

0.1224 

0.1823 

0.2553 

0.32*0 

0.4040 

0.5712 

0.7548 

0.9546 

1.3382 

18 

0.1296 

0.1925 

0.2703 

0.3472 

0.4278 

0.6048 

0.7992 

1.0108 

1.4169 

19 

0.1368 

0.2026 

0.2853 

0 3665 

0.4515 

0.6384 

0.8436 

1.0669 

1 4956 

20 

0.1410 

0.2203 

0.3005 

0.3852 

0.4752 

0.6720 

0.8880 

1.1232 

1.5744 

21 

0.1512 

0.2305 

0.3155 

0.4045 

0.4995 

0.7056 

0.9324 

1.1793 

1.6531 

22 . 

0.1584 

0.2407 

0.3275 

0.4238 

0.5228 

0.7392 

0.9768 

1.2394 

1.7318 

23 

0.1656 

0.2508 

0.3425 

0.3430 

0.5465 

0.7728 

1.0212 

1.2915 

1.8105 

24 

0.1728 

0.2610 

0 3575 

0.3623 

0 5703 

0.8064 

1.0656 

1.3477 

1.8892 

25 

0.1800 

0.2711 

0.3725 

0.3815 

0.5940 

0.8400 

1.1100 

1.4038 

1.9679 

26 

0.1872 

0.2813 

0.3876 

0.4008 

0.6179 

0.8736 

1.1544 

1.4600 

2.0467 

27 

0.1944 

0.2914 

0.4026 

0.4200 

0.6415 

0.9072 

1.1988 

15161 

2 1254 

28 

0.2016 

0.3016 

0.4176 

0.4393 

0.6553 

0.9408 

1.2432 

1.5723 

2.2041 

29 

0.2088 

0.3117 

0.4326 

0.4585 

0.6890 

0.9744 

1.2876 

1.6284 

2 2s 29 

30 

0.2160 

0.3304 

0.4507 

0.5778 

0.7128 

1.0080 

1.3320 

1.6848 

2.3616 

31 

0.2232 

0.3405 

0.4657 

0.5970 

0.7365 

1.0416 

1.3764 

1.7409 

2.4403 

32 

0.2304 

0.3607 

0.4807 

0.6163 

0.7603 

1.0752 

1.4208 

1.7971 

2.5190 

33 

0.2376 

0.3608 

0.4957 

0.6355 

0.7841 

1.1088 

1.4652 

1.8533 

25977 

34 

0.2448 

0.3710 

0.5107 

0.6548 

0.8078 

1.1424 

1.5096 

1.9094 

2.0764 

35 

0.2520 

0.3811 

0.5258 

0.0740 

0.8316 

1.1760 

1.5540 

1.9676 

2.7552 

36 

0.2592 

0.3913 

0.5408 

0.6933 

0.8553 

1.2096 

1.5984 

2.0217 

2.8339 

37 

0.2664 

0.4014 

0.5558 

0.7125 

0.8791 

1.2432 

1.042S 

2.0779 

2.9126 

38 

0.2736 

0.4116 

0.5708 

0.7298 

0.9028 

1.2768 

1.0872 

2.1340 

2.9913 

39 

0.2808 

0.4217 

0.5858 

0.7490 

0.9266 

1.3104 

1.7316 

2.1902 

3.0701 

40 

0.2880 

0.4406 

0.60U9 

0.7704 

0.9504 

1.3440 

1.7760 

2.2464 

3.1488 

45 

0 3240 

0.4957 

0.6760 

0.8667 

1.0692 

1.5120 

1.99*0 

2.5272 

3.5424 

50 

0.3600 

0.5508 

0.7512 

0.9030 

1.1880 

1.6800 

2.2200 

2.8080 

3.9360 

55 

0.3960 

0.6059 

0.8263 

1.0593 

1.3068 

1.8480 

2.4420 

3.0888 

4.3296 

60 

0.4230 

0.6610 

0.9014 

1.1556 

1.4256 

2.0160 

2.6640 

3.3696 

4.7132 

65 

0.4680 

0.6665 

0.9765 

1.2519 

1.5444 

2.1840 

2.8869 

3.6540 

5.1068 

70 

0.5010 

0.7711 

1.0517 

1.3482 

l.<632 

2.3520 

3.1080 

3.9312 

5.5104 

75 

0.5400 

0.8262 

1 126S 

1.4445 

1.7820 

2.5200 

3.3300 

4.2120 

5.9040 

80 

0.5760 

0.8813 

1.2019 

1.5408 

1.9008 

2.0*80 

3.5520 

4.4948 

6.2976 

85 

0.6120 

0.9364 

1.2770 

1.6371 

2.0196 

2.7560 

3.7740 

4.7756 

6.6912 

90 

0.6480 

0.9914 

1.3521 

1.7334 

2.1384 

3.0240 

3.9960 

5.0544 

7.0848 

95 

0.6840 

1.0465 

1.4272 

1.8297 

2.2572 

3.1920 

4.2180 

5.3352 

7.4784 

100 

0.7200 

1.1016 

1.5024 

1.9260 

2.3760 

3.3000 

4.4400 

5.6160 

7.8720 

0.00000600 

612 

626 

642 

660 

700 

740 

780 

820 



Expansion per Degree.—Fahrenheit. 



Multiply by 1.1 for wrought iron, 1.5 for copper, 1.6 for brass and 2.6 for zinc. 







































Properties of Heat, 


611 


Conducting Power of Different Substances for Heat and 

Electricity. 


Metals. 

Silver, fine, . 
Gold, “ . . 
Gold, .991, 
Copper, bam’d, 
Copper, cast, 
Mercury, . . 
Aluminium, . 
Zinc, hammered 
Zinc, cast ver¬ 
tical, . . . 

Zinc, cast hori¬ 
zontal, . . 
Lead, cast, 
Cadmium, . . 
Wrought iron, 
Tin, . 

Steel, . . . 
Platinum, . . 
Cast iron, . . 
Antimony, cast 
vertical, . . 
Antimony, cast 
horizontal, 
German silver, 
Bismuth, . . 

Stone <6 Crystals. 
Marble, . . 
Glass, . . . 
Common brick, 
Fire-brick, . 
Fire-clay, . . 
Porcelain, 
Wood-ashes, . 
Coal, anthracite 
Coal, bitum., . 
Coal, charred, 
Coke, . . . 


100 

98 

84 

85 

81 

68 

66 

64 

63 

60 

20 

57 

43 

42 

40 

40 

36 

21 

19 

10 

6 


12.21 

9.65 

8.422 

6.05 

6.61 

7.55 

0.8359 

19.25 

16.84 

0.788 

19.80 


Quartz sand, . . 
Limestone, . . . 
Lime, ; . . . 
Quartz crystals, 

Slate, .... 
Keen’s cement, 

Plaster and sand, 
Plaster Paris, . . 
Roman cement, . 
Asphalt, .... 
Chalk, . • . . 

Woods. * 

Fir, cross grain, . , 

Fir, with the fibre, 

Pine,. 

Oak, with the fibre, 
Him, “ “ 

Ash, “ 

Applo, “ 

Ebony, “ 

Lampblack, . . . 
Cross: wi t h fibre=l: 3, 

Pirch. 

Black oak, .... 
Chestnut, .... 
Spanish mahogany, 
Walnut, . . . . 
Fur. 

Hare’s fur, .... 
Eider down, .... 
Beaver’s fur, . . . 
Raw silk, .... 
Wool, sheep, . . . 

Cotton,. 

Lint,. 

Sewing-silk, . . . 
Flannel, .... 
Horse-hair, .... 


35.56 

19.8 

24.00 

80.0 

10.00 

1.901 

1.870 

2.026 

2.080 

4.52 

5.853 


U 

« 

«( 


1.10 

3.10 
3.90 
3.30 
3.2 

3.1 

2.8 

2.2 
0.112 

4.10 

3.2 
3.0 
2.8 

3.3 

0.0946 

0.0668 

0.0675 

0.0692 

0.0778 

0.0834 

0.0846 

0.0955 

0.395 

0.304 


Liquids. 
Water, . . . 
Mercury. . . . 
Proof spirit, • 
Alcohol, pure, . 
Nitric acid, 

Sulphur, acid, . 
Sulphur, ether, 
Turpentine, . . 

Gases. 

Air, .... 

Radiating Power 
Water, .... 
Lampblack. . . 
Paper, writing, , 
Rosin, .... 
Sealing-wax, . , 
Glass, common, 

India ink, . . , 

Ice,. 

Bed lead, . . , 
Graphite, . . 
Lead, tempered, 
Mercury, . . . 
Lead, polished, , 
Iron, polished, . 

Tin and silver, . 
Copper and gold, 
Reflecting Powers 
Brass, .... 
Silver, .... 
Tinfoil, . . . 

Tin, .... 
Steel, . . . 

Lead, .... 
Glass, . . . 

Glass, oiled or waxed, 
Lampblack, . 


1.000 

2.80 

0.847 

0.931 

1.5 

1.7 

2.1 

3.1 

0.9855 

100 

100 

98 

96 

95 

90 

88 

85 

80 

75 

45 

20 

19 

15 

12 

12 

100 

90 

85 

80 

70 

60 

10 

5 

0 


Miscellaneous Temperatures. 


In the Bessemer furnace, 
Puddling-furnace, 

Cupola,. 

Heat of common fire, . 

Red beat in daylight, 

Iron red in dark, . 

Mean temp, of the earth, 

“ “ “ torrid zone, 

“ “ “ temp. “ 

“ “ “ polar region, 

Temp, of ignition. 

Highest temp, of wind, . 

Temp, of the human blood, 

A comfortable room, 

Mean temp, of ocean, . 


Fahr. 


i ahr. 

4000° 

1 alcohol, 1 water freezes, 

... 7° 

3500 

Mean temp, of the poles, 

- 13 

3000 

Temp, outside atmosphere, 

— 58 

1100 

Greatest natural cold, . 

— 56 

1070 

Vinous fermentation, 

— G5 

752 

Acetous fermentation begins, 

— 78 

50 

Acetifieation ends, . . . 

— 88 

75 

Phosphorus burns, 

— 43 

50 

Greatest artificial cold produced, 

—166 

20 

A t 50°, Mixtures of — 

l’rod. 

636 

117 

Nitrate of ammonia, . 1' 

Water,.1 j 

1 

> 

cold. 

46 

98 

Sulphate of soda, . . 8 


60 

70 

Muriatic acid, . . .5 


62 

Dilate sulphuric acid, . 5 

Snow, . . . . 4 

- 

23 






















































512 


Air and Heat. 


ON AIR AND HEAT, 

Dry air expands or contracts uniformly 0.00204 its volume per degree Fahr. 
in difference of temperature, or 0.00367 per degree Centigrade under constant 
pressure. Assuming the expansion per degree Fahr. as unit, the primitive 
volume will be— 

0.00204 = 490 - 196 ’ 

V and v = volumes of dry air of temperatures t and T Fahr., and pressure 
p and P above vacuum. The volumes and pressures iu the following formulas 
may be expressed in any units of measures. 

Volume and Temperature Variable under Constant Pressure. 

F«c[l K00204(r—0], and {T-t) ^ d0AdQ ^ V ~~~\ 1. 

Example 1. v = IS cubic feet of air, of t = 36°, is to be heated to T — 84° 
under constant pressure. Required, the volume F? 

V= 18 [1 + .00204 (84-30)] = 19.798 cubic feet. 

Volume and Pressure Variable under Constant Temperature. 

( T — t) = 0, the pressures will be — 

r v 


and 


v 


P 

p = p 


p_ 

p 


V 


V ’ 


v 

V* 


and 


« = V 


JL 

P* 


Example 2. V= 150 cubic feet of air, of pressure p — 1475 pounds to the 
square inch, is to be compressed to 50 cubic feet. The heat generated in the 
compression to radiate through the vessel until the temperature of the com¬ 
pressed air is equal to that in V. Required, the pressure P'l 

V 1 ^0 

Formula 3. P=p -= 14.75 x - c/ r- = 44.25 lbs. 

1 v 50 


When the Temperature, Volume, and Pressure 

are all variable , ice have — 

P=p-Z. [l + .00204 {T-t)l and (T-t) = 490.1964. 

Tt must be distinctly understood in all the*e formulas that the volume v 
belongs to the pressure Pand temperature T\ and the volume Fto p and t. 
The primitive quantities are v , P, T. It may happen in the Formula.4 that 
u>F, t>T, and p~>P. 

Example 3. V = 1000 cubic inches of air of p = 14.75 and t = 59°, is to be 
reduced to v = 320 cubic inches, and the temperature increased to T— 369°. 
Required, the pressure P per square inch ? 

Formula 4. P= 14.75 ~~ [1 + .00204 (369 - 59)] = 75.23 lbs. 

Example 4. v = 290, P = 88.5 to be increased to F= 838, and p — 18.4. Re¬ 
quired, the difference of temperature ( T—t)1 

Formula 4. (T-t) = 490.196l) =325.7°. 


- 
















Heat in Permanent Gases. 


513 


HEAT IN PERMANENT GASES. 

S = specific heat under constant pressure. 

5 = mean specific heat under any pressure and volume from 32° to T°. 

P = 14.7 pounds to the square inch pressure of the gas at t° = 32° F. 

pressure of the same gas at the temperature T°. 
v = volume in cubic feet of the gas at 32°. 

F= volume of the same gas, but of pressure Pand temperature T°. 

TF= weight in pounds of the gas experimented upon. 

^ = weight in a fraction of a pound per cubic foot of the gas. 
h = units of heat in W pounds of gas elevated from 32° to T°, or from a 
pressure of 14.7 to P pound. 


h-SSWyJ-^. 

h = y SVv. . 


2 . 


y 


A I v 

~ ~sw\ ~v~ * 


V = 


A 


S'® \/Vv 


3. 


4. 


The value of y is calculated for different temperatures in the following 
tables of physical properties of permanent gases, by the aid of which the 
units of heat in any gas can be found. 

Having given the weight W, volumes V and v, and the units of heat A, in 
any permanent gas, calculate the value of y by Formula 3 or 4, which gives 
the corresponding temperature of the gas in tlie table. 

Example 11. How many units of heat are required to elevate the tempera¬ 
ture of v = 160 cubic feet of air from 32° to T° = 480°, and expand the volume 
to V— 240 cubic feet? 

In the table find y = 324.29 for 480°. 

Units of heat, A = 324.29 X 0.25 X 0.08042;/160 X 240= 1277.6. 

Example 13. What will be the temperature of F=36 cubic feet of car¬ 
bonic acid heated from 32° and volume v = 24 cubic feet when A — 140 units 
of heat has been expended on it? 


y = 


140 


0.221 X 0.1233;/36 X 24 

This corresponds to a temperature T° = 185° in the table. 

Specific Heat, under Constant Pressure anti Temperature 32°. 


= 133.8.* 


Kinds of Gases. 

Pounds 
per cubic 
foot. 

Cubic foot 
per 

pound. 

Specific 

Water=l. 

gravity. 

Air=l. 

Specific 

heat. 


V 

G 



S 

Atmospheric air. 

0.08042 

12.433 

0.00130 

1.000 

0.15 

Oxygen gas. 

0.0S888 

11.251 

0.00143 

1.104 

0.23 

Nitrogen gas. 

0.07837 

12.760 

0.00126 

0.972 

0.275 

Hydrogen gas. 

0.00559 

178.84 

0.00009 

0.069 

3.3 

Carbonic oxide. 

0.07837 

12.760 

0.00126 

0.972 

0.288 

Carbonic acid. 

0.12333 

8.108 

0.00197 

1.527 

0.221 

Steam. 

0.05021 

19.915 

0.00634 

0.488 

0.475 


33 
































514 


Physical, Properties of Permanent Gases. 




Temp. 

Fahr. 

P V 

p V 

T—t 

Temp. 

Fahr. 

P V 

p V 

T—t 

Temp. 

Fahr. 

P V 

p V 

T—t 

V * 

Temp. 

Fahr. 

PV 

T—t 

V * 

V X 

p V 

l/ X 

rpO 

• X 

y 

yo 

X 

y 

yO 

X 

y 

yo 

X 

y 

32 

i .oooo 

0.0000 

92 

1.1217 

• >6.652 

152 

1.2434 107.62 

212 

1.3650 

154.06 

33 

1.0020 

0.9990 

93 

1.1237 

57.555 

153 

1.2454 108.43 

213 

1.3670 

154.80 

34 

1.0040 

1.9960 

94 

1.1257 

58.436 

154 

1.2475 109.23 

214 

1.3691 

1 o5,o4 

35 

1.0061 

2.9909 

95 

1.1277 

59.326 

155 

1.2495 110.04 

215 

1.3711 

156.28 

36 

1.0081 

3.9839 

96 

1.1297 60.214 

156 

1.2515 110.84 

216 

1.3731 

157.02 

37 

1.0101 

4.9750 

97 

1.1318,61-098 

157 

1.2535 111.65 

217 

1.3751 

157.76 

38 

1.0121 

5.9640 

98 

1.1338 61.983 

158 

1.2556 112.45 

218 

1.3772 

158.50 

39 

1.0142 

6.9508 

99 

1.1358 62.867 

159 

1.2576 113.25 

219 

1.3792 

159.24 

40 

1.0162 

7.9360 

100 

1.1378 63.749 

160 

1.2596 114.05 

220 

1.3812 

159.97 

41 

1.0182 

8.9192 

101 

1.1399 64.627 

161 

1.2616 

114.85 

221 

1.3832 

160.71 

42 

1.0203 

9.9000 

102 

1.1419 65.506 

162 

1.2637 

115.64 

222 

1.3853 161.45 

43 

1.0223 

10.880 

103 

1.1439 66.384 

163 

1.2657 

116.44 

223 

1.3873' 

162.19 

44 

1.0243 

11.857 

104 

1.1459 

67.260 

1(44 

1.2677 

117.24 

224 

1.3893 

162.93 

45 

1.0264 

12.834 

105 

1.14S0 

68.132 

165 

1.2697 

118.04 

225 

1.3913 

163.67 

46 

1.0284 

13.805 

106 

1.1500 

69.005 

166 

1.2717 

118.83 

226 

1.3934 

164.41 

47 

1.0304 

14.777 

107 

1.1520 

69.877 

167 

1.2738 119.62 

227 

1.3954, 

165.15 

48 

1.0325 

15.746 

108 

1.1541 

70.745 

168 

1.2758 120.41 

228 

1.3974 

165.88 

49 

1.0345 

16.714 

109 

1.1561 

71.613 

169 

1.2778 121.20 

229 

1.3995' 

166.61 

50 

1.0365 

17.680 

110 

1.1581 

72.481 

170 

1.2798.121.98 

230 

1.4015 

167.25 

51 

1.0385 

18.666 

111 

1.1602 

73.344 

171 

1.2818 122.77 

231 

1.4035 

167.98 

52 

1.0406 

19.606 

112 

1.1622 

74.208 

172 

1.2839 123.56 

232 

1.4056 

168.70 

53 

1.0426 

20.567 

113 

1.1642 

75.072 

173 

1.2859 124.35 

233 

1.4076 

169.42 

54 

1.0446 

21.575 

114 

1.1663 

75.929 

174 

1.2879 125.13 

234 

1.4096 

170.14 

55 

1.0466 

22.482 

115 

1.1683 

76.790 

175 

1.2899 125.91 

235 

1.4116 

17 0.86 

56 

1.0487 

23.463 

116 

1.1703 

77.648 

176 

1.2920 126.69 

236 

1.41371 

171.58 

57 

1.0507 

24.390 

117 

1.1724 

78.502 

177 

1.2940 127.47 

237 

1.4157 

172.29 

58 

1.0527 

25.341 

118 

1.1744 

79.358 

178 

1.2960 128.25 

238 

1.4177 

173.01 

59 

1.0547 

26.290 

119 

1.1764 80.212 

179 

1.2980 129.02 

239 

1.4198 

173.73 

60 

1.0567, 

017.260 

120 

1.17S4 

81.066 

180 

1.30011129.80 

240 

1.4218 

174.44 

61 

1.0588 

28.184 

121 

1.1805 

81.914 

181 

1.3021 130.57 

241 

1.4238 

175.15 

62 

1.0608 

29.128 

122 

1.1825 

82.764 

182 

1.3041 

131.34 

242 

1.4258 

175.86 

63 

1.0628 

■30.070 

123 

1.1845 

83.621 

183 

1.3062 

132.11 

243 

1.4279 

176.57 

64 

1.0649 

31.010 

124 

1.1866 

84.457 

184 

1.3082 132.88 

244 

1.4299 

177.28 

65 

1.0669 

31.949 

125 

1.1886 

85.303 

185 

1.3102 133.65 

245 

1.4319 

177.99 

66 

1.0689 

32.896 

126 

1.1906 

86.148 

186 

1.3122 

134.42 

246 

1.4340 

178.70 

67 

1.0709 

33.822 

127 

1.1927 

86.988 

187 

1.3143 135.19 

247 

1.4360 

179.41 

68 

1.0720 

34.770 

128 

1.1947 

87.830 

188 

1.3163 135.96 

248 

1.4380 

180.12 

69 

1.0740 

35.703 

129 

1.1967 

88.671 

189 

1.3184 136.73 

249 

1.4401 

180.83 

70 

1.0769 

36.633 

130 

1.1987 

89.510 

190 

1.3204 137.50 

250 

1.4421 

181.54 

71 

1.0780 

37.563 

131 

1.2008 

90 374 

191 

1.3224 138.27 

251 

1.4441 

182.24 

72 

1.0811 

38.470 

132 

1.2028 

91.152 

192 

1.32441139.04 

252 

1.4462 

182.94 

73 

1.0831 

39.396 

133 

1.2048 

92.016 

193 

1.3265 139.81 

253 

1.4482 

183.64 

74 

1.0851 

40.320 

134 

1.2069 92.846 

194 

1.3285 140.58 

254 

1.4502 

184.34 

75 

1.0871 

41.289 

135 

1.2089 93.579 

195 

1.3305 

141.35 

255 

1.4522 

185.04 

76 

1.0692 

42.160 

136 

1.2109 94.510 

196 

1.3326 

142.12 

256 

1.4543 

185.74 

77 

1.0912 

43.079 

137 

1.2129 95.340 

197 

1.3346 

1 42.,S9 

257 

1.4563 

1S6.44 

78 

1.0932 

43.995 

138 

1.2150 96.165 

198 

1.3366 

143.66 

258 

1.4583 

187.14 

79 

1.0953 

44.940 

139 

1.2170196.993 

199 

1.3386 

144.42 

259 

1.4604 

187.84 

80 

1.0973 

45.822 

140 

1.2190 97.819 

200 

1.3407 

145.19 

260 

1.4624 

188.54 

81 

1.0993 

46.734 

141 

1.2211 

98.640 

201 

1.3427 

145.95 

261 

1.4644 

189.24 

82 

1.1014 

47.643 

142 

1.2231 

99.463 

202 

1.3447 

146.70 

262 

1.4664 

189.93 

83 

1.1034 48.552 

143 

1.2251 

100.29 

203 

1.34681147.44 

263 

1.4685 

190.62 

84 

1.1054 49.459 

144 

1.2272 

101.10 

204 

1.3488 

148.18 

264 

1.4705 

191.32 

85 

1.1075 50.302 

145 

1.2292 101.92 

205 

1.3508 

148.92 

265 

1.4725 

192.01 

86 

1.1095 

51.266 

146 

1.2312 

102.74 

206 

1.3529 

149.66 

266 

1.4745 

192.70 

87 

1.1115 

52.108 

147 

1.2333 103.55 

207 

1.3549 

150.39 

267 

1.4766 

193.39 

88 

1.1135 53.069 

148 

1.2353 104.37 

208 

1.3569 

151.12 

268 

1.4786 

194.08 

89 

1.1156 

53.966 

149 

1.2373 105.18 

209 

1.3589 

151.85 

269 

1.4806 

194.77 

90 

1.1176 

54.851 

150 

1.2393 106.00 

210 

1.3610 

152.58 

270 

1.4826 

195.46 

91 

1.1196|55.760 

151 

1.2414 106.81 

1 

211 

1.3630 153.32 

271 

1.4847 

196.15 


















































Physical Properties of Permanent Gases. 


515 


Temp 

P V 

T—t 

Temp. 

j P V 

T—t 

Temp. 

Fahr. 

P V 

T—l 

Temp. 

Fahr. 

PV 

-- 

T—t 

Fahr. 

p V 

V x 

Fahr. 

p V 

V X 

pv 

V X 

p V 

V X 

pO 

X 

y 

T° 

X 

y 

rp O 

X 

y 

po 

X 

y 

272 

1.4867 

196.84 

332 

1.6084 

236.55 

424 

1.7950 

292.59 

720 

2.3952 

444.52 

273 

1.4887 

197.53 

333 

1.6104 

237.19 

426 

1.7990 

293.75 

730 

2.4155 

449.1 1 

274 

1.4907 

198.22 

334 

1.6124 

237.83 

428 

1.8031 

294.91 

740 

2.4357 

453.67 

275 

1.4928 

198.90 

335 

1.6144 

238.43 

430 

1.8071 

296.07 

750 

2.4560 

458.15 

276 

1.4948 

199.58 

336 

1.6165 

239.11 

432 

1.8112 

297.23 

760 

2.4763 

462.60 

277 

1.4968 

200.26 

337 

1.6185 

239.75 

434 

1.8152 

298.39 

770 

2.4966 

467.03 

278 

1.4988 

200.94 

338 

1.6205 

240.39 

436 

1.8193 

299.54 

780 

2.5169 

471.44 

279 

1.5009 

201.62 

339 

1.6226 

241.02 

438 

1.8234 

300.68 

790 

2.5371 

475.84 

280 

1.5029 

202.30 

340 

1.6246 

241.65 

440 

1.8274 

301.82 

800 

2.5574 

480.24 

281 

1.5049 

202.98 

341 

1.6266 

242.28 

442 

1.8315 

302.96 

810 

2.5777 

484.56 

282 

1.5070 

203.66 

342 

1.6286 

242.91 

444 

1.8355 

304.10 

820 

2.5980 

488.87 

2S3 

1.5090 

204.34 

343 

1.6307 

243.54 

446 

1.8396 

305.24 

830 

2.6183 

493.14 

284 

1.5110 

205.02 

344 

1.6327 

244.17 

448 

1.8436 

306.37 

840 

2.6385 

497.43 

285 

1.5131 

205.70 

345 

1.6347 

244.80 

450 

1.8477 

307.51 

850 

2.6588 

501.66 

286 

1.5151 

206.37 

346 

1.6368 

245.43 

452 

1.8518 

308.65 

860 

2.6791 

505.88 

287 

1.5171 

207.04 

347 

1.6388 

246.06 

454 

1.8558 

309.79 

870 

2.6994 

510.07 

288 

1.5192 

207.71 

348 

1.6408 

246.69 

456 

1.8599 

310.91 

880 

2.7197 

514.23 

289 

1.5212 

208.38 

349 

1.6429 

247.31 

458 

1.8639 

312.03 

890 

2.7399 

518.36 

290 

1.5232 

209.05 

350 

1.6449 

247.93 

460 

1.8680 

313.15 

900 

2.7602 

522.45 

291 

1.5252 

209.72 

351 

1.6469 

248.56 

462 

1.8720 

314.27 

910 

2.7805 

526.54 

292 

1.5273 

210.39 

352 

1.6490 

249.19 

464 

1.8761 

315.39 

920 

2.8008 

530.61 

293 

1.5293 

211.06 

353 

1.6510 

249.82 

466 

1.8801 

316.51 

930 

2.8211 

534.66 

294 

1.5313 

211.73 

354 

1.6530 

250.45 

468 

1.8842 317.63 

940 

2.8413 

538.71 

295 

1.5334 

212.40 

355 

1.6551 

251.08 

470 

1.8882 

318.75 

950 

2.8616 

542.67 

296 

1.5354 

213.07 

356 

1.6571 

251.70 

472 

1.8923 

319.37 

960 

2.8819 

546.66 

297 

1.5374 

213.74 

357 

1.6591 

252.32 

474 

1.8963 

320.99 

970 

2.9022 

550.60 

298 

1.5395 

214.40 

358 

1.6611 

252.94 

476 

1.9004 

322.09 

980 

2.9225 

554.52 

299 

1.5415 

215.06 

359 

1.6632 

253.56 

478 

1.9044 323.19 

990 

2.9427 

558.45 

300 

1.5435 

215.72 

360 

1.6652 

254.18 

480 

1.9085 

324.29 

1000 

2.9630 

562.36 

301 

1.5455 

216.38 

362 

1.6692 

255.42 

482 

1.9126 

325.39 

1010 

2.9833 

566.24 

302 ■ 

1.5476 

217.04 

364 

1.6733 

256.66 

484 

1.9166 

326.49 

1020 

3.0036 

570.09 

303 

1.5496 

217.70 

366 

1 .(>773 

257.90 

486 

1.9207 

327.59 

1030 

3.0239 

573.94 

304 

1.5516 

218.36 

368 

1.6814 

259.12 

488 

1.9248 

328.69 

1040 

3.0441 

577.73 

305 

1.5537 

219.02 

370 

1.6854 

260.36 

490 

1.9288 

329.78 

1050 

3.0644 

581.53 

306 

1.5557 

219.68 

372 

1.6895 

261.58 

492 

1.9329 

330.88 

1060 

3.0847 

585.32 

307 

1.5577 

220.34 

374 

1.6935 

262.80 

494 

1.9369 

331.98 

1070 

3.1050 

589.08 

308 

1.5597 

221.00 

376 

1.6976 

264.02 

496 

1.9410 

333.06 

1080 

3.1253 

592.82 

309 

1.5618 

221.65 

378 

1.7016 

265.24 

498 

1.9451 

334.14 

1090 

3.1455 

596.54 

310 

1.5638 

222.30 

380 

1.7057 

266.46 

500 

1.9491 

335.22 

1100 

3.1658 

600.24 

311 

1.5658 

222.96 

382 

1.7097 

267.71 

510 

1.9694 

340.60 

1110 

3.1861 

603.92 

312 

1.5678 

223.61 

384 

1.7138 

268.94 

520 

1.9898 

345.95 

1120 

3.2064 

607.62 

313 

1.5699 

224.27 

386 

1.7179 

270.16 

530 

2.0102 

351.26 

1130 

3.2267 

611.27 

314 

1.5719 

224.93 

388 

1.7219 

271.38 

540 

2.0302 

356.53 

1140 

3.2469 

614.92 

315 

1.5739 

225.58 

390 

1.7260 

272.50 

550 

2.0505 

361.75 

1150 

3.2672 

618.52 

316 

1.5759 

226.23 

392 

1.7301 

273.70 

560 

2.0708 

366.93 

1160 

3.2875 

622.13 

317 

1.5780 

226.88 

394 

1.7341 

274.90 

570 

2.0909 

372.06 

1170 

3.3078 

625.73 

318 

1.5800 

227.53 

396 

1.7382 

276.09 

580 

2.1113 

377.16 

1180 

3.3281 

629.32 

319 

1.5820 

228.28 

398 

1.7422 

277.29 

590 

2.1316 

382.28 

1190 

3.3484 

632.90 

320 

1.5840 

228.83 

400 

1.7463 

278.48 

600 

2.1519 

387.20 

1200 

3.3687 

636.38 

321 

1.5861 

229.48 

402 

1.7504 

279.68 

610 

2.1721 

392.18 

1300 

3.5714 

671.08 

322 

1.5881 

230.13 

404 

1.7544 

280.86 

620 

2.1924 

397.13 

1400 

3.7743 

704.74 

323 

1.5901 

230.78 

406 

1.7585 

282.04 

630 

2.2127 

402.03 

1500 

3.9770 737.35 

324 

1.5922 

231.42 

408 

1.7625 

283.22 

640 

2.2329 

406.89 

1600 

4.1798 

766.95 

325 

1.5942 

232.06 

410 

1.7666 

284.40 

650 

2.2532 

411.71 

1700 

4.3826 

797.49 

326 

1.5962 

232.71 

412 

1.7706 

285.58 

660 

2.2734 

416.50 

1800 

4.5854 

826.60 

327 

1.5982 

233.35 

414 

1.7747 

286.76 

670 

2.2938 

421.25 

1900 

4.7882 

854.45 

328 

1.6003 

233.90 

416 

T.7787 

287.94 

680 

2.3141 

425.98 

■ 2000 

4.9910 

880.91 

329 

1.6023 

234.63 

418 

1.7828 289.01 

690 

2.3343 

430.67 

2100 

5.1938 

906.76 

330 

1.6043 

235.27 

420 

1.7868 290.27 

700 

2.3545 

435.34 

2200 

5.3966 931.72 

331 

1.6063 

235.91 

422 

1.7909 291.43 

1 

1 

710 

2.3749 

439.88 

2300’ 

5.5994 957.80 










































516 


Warming and Ventilation. 


WARMING AND VENTILATION. 

The most, comfortable temperature of a habitable room is about 70° F. or 
21° C., and to make it wholesome 3 to 5 cubic feet of fresh air should be ad¬ 
mitted per minute for each individual occupying the room under ordinary 
circumstances, but in warm weather, or when the occupants exert themselves 
with manual labor, double that quantity of fresh air is required. In hospi¬ 
tals about 50 cubic feet of fresh air is required per minute for each patient. 

Respiration.—An adult in good health respires about 600 to 1000 cubic 
inches of air per minute under ordinary circumstances, but under great 
exertion double that quantity may be required. Air becomes vitiated in the 
act of respiration ; that is, its oxygen combines with carbon, forming carbonic 
acid. A man makes 15 to 20 respirations per minute, using about 50 cubic 

inches in each respiration, or-—-- = 3o cubic feet per hour, nearly. 

Warming.—The specific heat of air under atmospheric pressure is 0.25 
of that of an equal weight of water. One pound of air at 32° F. occupies 
12.433 cubic feet ; or 12.433 X 4 = 49.732, say 50, cubic feet of air can be elevated 
1° F. per unit of heat. 

V = volume of air in cubic feet to he heated. 

T° — temperature of the heated air. 

t° = temperature of the cold air. 

h = units of heat required to heat V volumes of air from i° to T°. 

A =^r< ro -‘°>- 

When warming is accomplished by hot water or steam conducted in pipes 
through the room to be wanned, the beat radiates from the pipe and tints 
heals the air, and at the same time heat is conducted through the walls and 
windows of the room. 

The problem before us is to find the quantity of pipes and fuel required 
for heating and maintaining a desired temperature of a given volume 
of air. 

Experiments on this subject have been made by MM. Peclet, Tredgold, 
Hum ford, Hood, and many others, from which the following data are de¬ 
duced : 

L = length of pipe in feet. 

d = diameter (internal) of pipe in inches. 

T° ■= temperature of the, stegm or water in the pipe. 

t" — temperature of the external air. 

V — the required temperature of the room. 

V = volume of air in cubic feet to be warmed per minute—about 5 cubic 

feet for each occupant of the room—to which add 1£ cubic feet 
for each square foot of glass windows. 

C — consumption of coal iu pounds per hour for maintaining the beat 
in the pipe. 

d V{f — l") d L (T° — V) 

L 8 (T° — t') ‘ L ~ 12800 . * 

On an average, one square foot beating-surface is used for every 100 cubic 
feet of air to be kept warm. 

The exhaust steam from steam-engines is better for heating purposes than 
is live steam. When exhaust steam is used, care should be taken to have the 
steam passages large, so as to prevent back pressure. 


Heating Rooms by Stoves ami Open Fires. 

The quantity of beat utilized in warming apartments by open fires is from 10 
to 20 per cent, of the total heat of combustion. 







Compression and Expansion of Air. 


517 


Compression and Expansion of a Definite Weight of Air 

Enclosed in a Vessel. 

In the compression and expansion of air, as given in the following table, it 
is supposed that no heat is transmitted to or from the air operated upon. 
In compression, the temperature of the air rises - and if the heat is allowed to 
be conducted through the sides of the vessel enclosing the air, the pressure 
will not correspond with the table. In expanding the air the temperature 
is lowered, as seen in the table. The primitive volume is assumed to be 
at 32° Falir. 


Compression ami Expansion of Air. 


Compression of Air. 

Expansion of Air. 

Volume. 

Temp. 

Pressure. 

Volume. 

Temp. 

Pressure. 

v = 1. 

Falir. 

Atmos. 

Lbs. p. sq.in. 

v — 1. 

Falir. 

Atmos. 

Lbs. p. sq. in 

V 

rpO 

A 

P 

V 

T 

A 

P 

1. 

32° 

1.0000 

14.7 

1.0 

+ 32° 

1.0 

14.7 

0.95 

42.43 

1.0297 

15.137 

1.1 

+ 13.20 

0.8751 

12.864’ 

0.90 

53.66 

1.159 

17.036 

1.2 

— 3.3 

0.7747 

11.393 

0.85 

65.81 

1.255 

18.456 

1.3 

— 18.06 

0.6926 

10.181 

0.80 

79.01 

1.366 

20.090 

1.4 

— 81.26 

0.6243 

9.1778 

0.75 

93.43 

1.496 

21.991 

1.5 

•— 39.65 

0.58354 

8.5780 

0.70 

109.26 

1.647 

24.215 

1.6 

— 54.06 

0.5179 

7.613 

0.05 

126.77 

1.828 

26.561 

1.7 

— 64.00 

0.4757 

6.9934 

0.60 

146.30 

2.044 

30.054 

1.8 

— 73.16 

0.4391 

6.4556 

0.55 

168.25 

2.309 

33.948 

1.9 

— 82.34 

0.4083 

6.002 

0.50 

193.20 

2.639 

38.792 

2.0 

— 89.47 

0.3789 

5.570 

0.45 

221.96 

3.058 

44.547 

2.25 

— 106.9 

0.3213 

4.7235 

0.40 

245.70 

3.607 

53.020 

2.5 

— 121.83 

0.2779 

4.0851 

0.35 

295.73 

4.348 

63.917 

2.75 

— 134.77 

0.2426 

3.5666 

0.33 

314.10 

4.721 

69.406 

3.00 

— 146.15 

0.2148 

3.1576 

0.30 

344.87 

5.396 

79.313 

3.25 

— 156.27 

0.1920 

2.8228 

0.25 

407.13 

6.964 

102.38 

3.50 

— 167.29 

0 1731 

2.5446 

0.20 

489.91 

9.518 

139.92 

3.75 

— 173.57 

0.1572 

2.3103 

0.15 

606.4 

14.24 

209.31 

4.00 

— 181.00 

0.1436 

2.1111 

0.125 

>691.0 

18.38 

270.17 

4.5 

— 194.18 

0.1218 

1.7900 

0.10 

800.9 

25.12 

369.24 

5 

— 205.4 

0.1051 

1.5444 

0.05 

1213.5 

66.289 

974.45 

6 

— 223.74 

0.0813 

1.1965 

0.04 

1373.2 

90.60 

1331.8 

7 

— 238.20 

0.0656 

0.9642 

o:o3 

1601.7 

135.53 

1992.3 

8 

— 250.03 

0.0544 

0.7998 

0.02 

1973.0 

239.09 

3514.6 

9 

— 259.92 

0.0461 

0.6782 

0.01 

4469.0 

• 7-94.33 

1167.600 

10 

— 268.39 

0.0355 

0.5216 



























518 


Air and Heat. 


On the Compression and Expansion 

of a definite weight of air enclosed in a vessel. 

In this treatment no heat must be lost or gained by radiation from the sides of 
the vessel in which the air is enclosed. Let D and d represent the degrees of 
absolute temperatures of volumes v and V of the air to be experimented upon. 

The absolute zero is 461° below Falir. zero, and 273° Cent, below the freezing- 
point of water. D = 4C1 + 2*, d = 401 -f- t, ayd D — d = T — t, Fahr. scale. 

Volume and Temperature* 

V 


=(ir 

and T=(iP 


/ ^ \ 2.45 

Compression v= j 6. 

2.45 ry 

d v-> 

* V 

2*45 j y 

Expansion d=D -y— 7. 


Expansion V- 
Co m pression I) = 

Example 5. To what fraction must air of t — 65° be compressed, in order to fire 
tinder at a temperature of T— 550°, d = 401 + G5 — 520°, D =■ 550 + 401 = 1011° ? 

Formula 5. — = (-PP*- \ —0.20, the answer. 

V VlOll/ 

Example 6. IIow much must air of T= 80° be expanded to reduce the temper¬ 
ature to t — 32°, or freezing-point of water ? 

V /541 \ 2,45 


Formula 5. 


v 


( 


403/ 


= 1.330S times, the answer. 


Example 7. v = 300 cubic inches of air of temperature T=380°, or Z)=841°, 
is to be expanded until the temperature becomes t = 80° or d = 541°. Required, 
the volume V, corresponding to that temperature? 

( 821 \ 2,45 

—— ] =1025.9 cubic feet. 

541 / 

Example 8. V= 20 cubic feet of air of t = 32°, or d — 403, is to bo compressed 
to v = 12 cubic feet. Required, the temperature T of compression ? 

2-45 "DO 

Formula 7. I) — 493 -*[—=60729°, or T— 146.29°, 

\ 12 


Pressure and Temperature. 

3-42 v 

’ and — 

P 


Compression 


-Hf) 1 

HP 


v 

F—p 


3.42 . p 

Compression D — d -*/—, 

\ p 


HiT 

Expansion p = P 
Expansion d = D 


3.42 


I) 


3-42 


3.42 


P_ 

P 


8 . 


9. 


10 . 


Example 9. A volume of air of pressure p = 15 pounds to the square inch, and 
of temperature ? = 62°, is to be compressed until the temperature becomes 
2’= 120°. Required, the pressure P per square inch at T = 120° ? 

d = 461+62 = 523, and D = 461 + 120 = 581. 

581 \ 3 * 42 


Formula 9. 


’=15 (— V 
\ 523 / 


= 21.49 lbs. pr. sq. inch. 








Air and Heat. 


519 


Example 10. A volume of air of pressure P-= 45 pounds to the square inch 
and of temperature T = 250° or £> = 711°, is to be expanded to a pressure of 
P —2o pounds. Required, the temperature t of the expanded air? 


Formula 10. 


3.42 


<2=711 ./ —= 598.72°, and 
\ 45 


t — 598.72—461 = 137.72°, the temperature required. 

Pressure ami Volume. 

• 41 /T 

\ v 


• 29 / p 


V p 


and 




IJU 11 

\ V \'P • 


1.4 

Expansion V~v —, 

\ P 

Compression P = p^—^ 


Compression v—V 



P_ 

P 


1.4 

, Expansion 



12 . 

13. 


Example 11. A volume i> == 50 cubic inches, and of pressure P= 80 pounds per 
square inch, is to be expanded until the pressure becomes p = 15 pounds. Required, 
the expanded volume VI 


Formula 12. 


V = 5(L /— =165 cubic inches. 
\ 15 


Example 12. What will be the pressure of a volume of air expanded 1.3308 
times ? 

/ ] \ 1*4 

Formula 13. p = [ —— — ) = 0.5324 of the primitive pressure. 

\ 1.3308 / 


Volume and Weight of Dry Air 


At different Temperatures, under a constant Atmospheric Pressure of 29.92 inches 
in the Barometer, the Volume at 32° l'alir. being the unit. 


Temp. 

Kahr. 

Volume. 

Wt. per 
Cub. ft. 
Pounds. 

Temp. 

Pahr. 

Volume. 

Wt. per 
Cub. ft. 
Pounds. 

Temp. 

Fahr. 

Volume. 

Wt. per. 
Cub. ft. 
Pounds. 

0° 

.935 

.0SG4 

162° 

1.265 

.0368 

550° 

2.056 

.0384 

12 

.960 

.0842 

172 

1.425 

.0628 

600 

2.150 

.0376 

22 

.9S0 

.0824 

182 

1.306 

.0618 

650 

2.260 

.0357 

32 

1.000 

.0807 

192 

1.326 

.0609 

700 

2.362 

.0338 

42 

1.020 

.0791 

202 

1.347 

.0600 

800 

2.566 

.0315 

52 

1.041 

.0776' 

‘212 

1.367 

.0591 

900 

2.770 

.0292 

62 

1.0G1 

.0761 

230 

1.404 

.0575 

1000 

2.974 

.0268 

72 

1.082 

.0747 

250 

1.444 

.0559 

1100 

3.177 

.0254 

82 

1.102 

.0733 

275 

1.495 

.0540 

1200 

3.381 

.0239 

92 

1.122 

.07 20 

300 

1.546 

.0522 

1500 

3.993 

.0202 

102 

1.143 

.0707 

325 

1.597 

.0506 

1800 

4.605 

.0175 

112 

1.163 

.0694 

350 

1.648 

.0490 

2000 

5.012 

.0161 

T 09! 

1.181 

.0682 

375 

1.689 

.0477 

2200 

5.420 

.0149 

132 

1.204 

.<'671 

400 

1.750 

.0461 

2500 

6.032 

.0133 

142 

1.224 

.06 9 

450 

1.852 

.0436 

2S00 

6.644 

.0121 

152 

1.245 

.0649 

509 

1.954 

.0413 

3000 

7.051 

.0114 


For Weight and Volume of air at Low Temperature, see Ilygrometry, page 357. 


























520 


Specific Heat. 


SPECIFIC HEAT. 

Different bodies require different quantities of heat to raise them to the same 
temperature. The amount of heat required to raise the unit weight of a substance 
one degree in temperature is called the specific heat of that substance. The specific 
heat of bodies varies nearly inversely as the specific gravity. The specific heat in 
all bodies increases slightly with the temperature. One pound of water raised from 
32° to 212° requires 180.9 units of heat instead of 180. The specific heat increases 
nearly in the same ratio for all solid and liquid bodies. The specific heat of water 
from 32° to T° will be— 

( 7 ^ 32 )- 
• 1167713 


1. 


The specific heat of water between any temperatures T and t will be— 

(T - 32) 1 ’ 67 - (t - 32) 1,67 


S=l + 


1167713 


2 . 


The following table gives the specific heat of different substances between the 
temperature 32° and 212°, compared with water as unit. When the specific heat 
of a body is required between high temperatures, it is necessary to calculate first 
the specific heat of water between such temperatures, which multiplied by the 
number in the table will give the required specific heat of the body. 

Specific Heat of Substances. 


•Water, .... 

1.000 

Lead, .... 

0.030 

Sweet oil, . . . 

0.310 

Ice of water, 

0.513 

Steel,. 

0.118 

Oil of turpentine, 

0.472 

Cast iron, . . . 

0.140 

Diamond, . . . 

0.147 

Gases of constant 


Wrought iron, . 

0.110 

Arsenic, .... 

o.osi 

volume and under 


Cobalt, .... 

0.150 

Iodine, .... 

0.054 

atmospheric pres- 


Nickel, .... 

0.103 

Sulphur, .... 

0.200 

sure. 


Copper, .... 

0.094 

Lime, burned, 

0.217 

Atmospheric air, . 

0.250 

Zinc,. 

0.093 

Glass-crystals, . . 

0.193 

Oxygen, . . . 

0.230 

Tin,. 

0.047 

Glass, common, 

0.177 

Hydrogen, . . . 

3.30 

Antimony, , . 

0.051 

Woods, average, . 

0.500 

Nitrogen, . . . 

0.275 

Bismuth,.... 

0.030 

Brick, common, 

0.200 

Carbonic acid, . . 

0.221 

Tellurium, . . 

0.091 

Firebrick, . . . 

0.220 

Carbonic oxide, . 

0.288 

Gold,. 

0.029 

Coal,. 

0.261 

Olefiant gas, . . 

0.421 

Silver, .... 

0.057 

Beeswax,.... 

0.450 

Nitro-oxide, . . 

0.237 

Platinum, . . . 

0.034 

Alcohol, s. g. 0.81, 

0.700 

Gas of oils, . . 

0.421 

Brass, .... 

0.094 

Sulphuric acid, 

0.335 

Snip, hydrogen, . 

0.242 

Mercury, .... 

0.030 

Nitric acid, . . 

0.661 

Steam of atm. pr., 

0.475 


Let two different substances of known weight or volume and temperature be 
mixed together; the temperature of the mixture will dissolve the relative quantity 
of heat in the ingredients. 

Mixture of the same Substances. 

JT= weight or volume of a substance of temperature T. 
w = weight or volume of a similar substance of temperature t. 
t' = temperature of the mixture IV + w. We shall have— 


t'{ W -f w) = WT-\- Wt, 3. 


W = vgj-J)_ 4 _ 


T — t' * 


, _ WT + \o t 


t' =- 


W w 


T= w<y —^ + 

w ^ 


5 . 

6 . 



























Specific Heat. 


521 


Example 1. Let W = 4.62 cubic feet of water at T^= 150° be mixed with w = 5.43 
cubic feet at t = 46. Required, the temperature of the mixture V = ? 


i' = 


4.62 X 150° + 5.43 X 46< 
4.62 + 5.43 


= 97.6°, the answer. 


Example 2. How much water of T= 107° must be mixed with w = 27.3 gallon 
of t = 58°, the mixture of the water to be To 0 ? 

27.3(75-58) 

107 — 75 

Mixture of Different Substances. 

W and w expressed by weights only. S and s = specific heat as given in the 
accompanying table. We shall hav 


WS{T—t') = ws{t / — t), 7. 
^(TFaS + ws )—wst 


T= 


WS 


8 . 


t' =■ 


WST-\- wst 


W= 


W S -f- ws 
ws(t' — t) 


s{T—ty 


9. 


10 . 


Example^. To what temperature must W= 20 pounds of cast iron be heated 
to raise w = 131 pounds of water of t = 51° to a temperature V = 64° ? T— ? 
from the table we have s — 1, and 6'= 0.14. 

T= 64(20 X 0-14 + 131) - 131 X 1 X 54 = ^ 

20 X 0.14 ’ 

the required temperature, supposing no vapor escapes from the water. 

If any chemical action takes place in the mixture, these formulas will not an¬ 
swer, because part of the sensible caloric may become latent, or latent caloric may 
be set free. 

Example 4. The temperature of 5 pounds of copper is to be elevated from 60° 
to 80°. How many heat units will be required? 

See table for copper 0.094 ( 80 — 60) = 1.88 heat units, the answer. 

Specific Heat of Gases. 

When heat is applied to a constant volume of gas enclosed in a vessel, the 
specitic heat of that gas increases as the square root of the pressure generated by 
the heat. 

8=^. 11. 
Vp 

When the volume, pressure and temperature are all variable, the specific heat 
of air will be— 

• s= 0 . 95 85 __ 12 . 

JvrtT-i y 

V V \ 493 / 

For any permanent gas of s = specific heat under atmospheric pressure, the 
specific heat under any other pressure and volume will be— 

S = - 3 ‘ 8 ^-■ 13 . 


v V \ 493 ' 





















522 


Dynamics and Units of Heat and Work. 


The weight of 320 cubic feet of air at 59° is (see table) 0.076 X 320 = 24.32 
pounds. The heat units required in the operation will be— 

Formula 13. h = 1.08 X 24.32 (369 — 59) = 1050.625 units. 

The specific heat of any other gas, under different volumes, pressures and tem¬ 
peratures, is equal to that of air multiplied by the specific heat in the table and 
divided by 0.25. 

The number of heat units h required to elevate the temperature of \V pounds of 
gas from t° to T°, will be— 

h = SW(T — t) .13. 

When the pressure is constant, and the volume rs increased by heat, and S = 
specific heat of the primitive volume of air, then the heat units will be— 

A = SJF ( r——. .14 

in which the last term expresses the heat units expended in expanding the volume 
under the pressure P. 

Example 6. One cubic foot of air of t = 32° is enclosed in a cylinder of one 
square foot area of piston, and under atmospheric pressure P= 14.76 pounds to the 
square inch. Let heat be applied to the air until T — 511°, when the volume will 
be about double, the piston being well balanced to move with the constant pres¬ 
sure. Required, the number of heat units imparted to the air, and the heat ex¬ 
pended in moving the piston with the pressure 14.75 pounds. 

Formula 14. h = 0.25 X 0.0S07 X (511 —32) + - (- —1\ = 9.65 + 2.75 = 

5.36 l 1 I 

12.4 heat units, of which 2.75 were expended in moving tne piston one foot. 


UNITS OF HEAT AND WORK. 

The ordinary English unit of heat is that required to elevate the temperature 
of one pound of distilled water one degree Fahr. from 39° to 40° (Fahr. &.), and 
called one calorie or heat unit. 

The German unit of heat is that required to elevate the temperature of one 
pound (German pfund) of water one degree Centigrade, from 4° to 5° (Cent, lb.) 
or (Cent, pfd.) 

The French unit of heat (called calorie) is that required to elevate the tempera¬ 
ture of one kilogramme (2.2047 lbs.) of water one degree Centigrade from 4° to 5° 
(Cent. kilo.). 

A combination of French and English units of heat is sometimes expressed by 
Fahr. scale and French weight (Fahr. kilo.). 

Heat is dynamic work, or the product of the two elements, force and motion, 
in which the temperature of the heat represents force, and the cubic con¬ 
tents of the units of heat represent the product of time and velocity, which is 
space. 

The English unit of dynamic work is one pound raised one foot, called footpound 
(Ft. tt>.). One calorie = 772 ft. lbs. 

The French unit of dynamic work is one kilogramme raised one metre, called 
kilomet. 

One horse-power will consume or generate 2564 heat units per hour. 


Comparison of Different Units of Heat and Work. 


English 

Calorie. 

French Calorie. 

Prussian. 

Dynamic Work. 

Fahr. B>. 

Cent. Tb. 

Fahr. kilo. 

Cent. kilo. 

Cent. pfd. 

Ft. Ib. 

Kilomet. 

1 

0.5555 

0.4536 

0.2520 

0.5769 

772 

106.51 

1.8 

1 

08165 

0.4536 

1.0385 

13S9.6 

191.71 

2.2047 

1.2248 

1 

0.555 

1.2719 

1702 

348.066 

3.968 

2.2017 

1.8 

1 

2.2894 

3063.6 

626.52 

1.733 

0.9630 

0.7862 

0.4368 

1 

1368.2 

273.66 

.0012953 

.0007196 

.0005876 

.0003264 

.0007473 

1 

0.13825 

.0093896 

.0005205 

.0004250 

.0002361 

.0005405 

7.233 

1 













jrUNS AND GUNPOWDER. 


523 


Performance, Weight and Dimensions of Heavy Ordnance. 


Diam. 

Length 

Weight in Pounds of 



Description. 

of 

of 







Bore. 

Gun. 

Gun. 

Proj’tile 

Powd. 

Velocity 

Bore. 


Inches. 

Ft. in. 

Pounds. 

Pounds. 

Pounds 

Ft. per sec 


American, rifle, . . 

4i 

7' 4" 

3,089 




Rifle. 

u a 

• • 

6 

9' 6" 

7,970 




Rifle. 

Eng., wrought iron, 

8 

9' 10" 

14.5(50 

180 

30 

1324 

Smooth. 

American, cast iron, 

9 

10' 

9,084 




(S 

English, “ 

1U 

11' 

40.320 

400 

60 

1298 

it 

American, “ 

11 

11' 6" 

1(5.511 




U 

Russian, “ 

11 

11' 6" 

55.800 

496 

82.5 

1362 

u 

English, “ 

12 

12' 

55,800 

600 

67 

1180 

u 

American, “ 

13 

13' 3" 

16,511 




u 

(t 

15 

15' 6" 

42,100 




Ct 

<i a 

20 

20' 3" 

115,000 

936 

120 

1131 

u 

“ mortar, . 

13 

2' 10" 

17.198 




(( 

Rus. brass, Moscow, 

30 

25' 

80.000 

3000 

• • 

• • • 

u 


Effect of Gun powder. 

The dynamic work of different kinds of gunpowder, utilized in heavy ordnance, 
varies between 150,000 and 200,000 foot-pounds per pound of powder. Let K de¬ 
note the dynamic work in a charge of powder, = weight in pounds of the ball 
or projectile, V— velocity of the projectile in feet per second, then 

„ 64A , ^ WV 2 

V=s ^w’ W= ~V^’ and K= ~W‘ 

The length of the gun for these formulas should be at lea-st 12 times the bore. 

Force of Gunpowder. 

The force of gunpowder depends much upon its quickness of burning and resist¬ 
ance to its expansion. Gunpowder enclosed in a strong vessel, and burned in its 
primitive volume, may reach a pressure of 100 tons to the square inch; but when 
the gas of powder is subjected to an excessive pressure, it seems to condense and 
loses the property of expansion due to a permanent gas less strained. This is a 
very important fact in the use of heavy ordnance, where the gun may be double 
strained with a loss of effect in the projectile. 

Quick powder may strain a gun over 30 tons to the square inch, whilst slower 
powder will strain it only 15 tons, and give a greater velocity to the projectile. 

It appears that the charge ought to be so arranged in a gun that a slow powder 
is first ignited, and then a quicker and quicker until the quickest at last, by which 
the gun need not be strained to more than 15 tons to the square inch, with full 
benefit of the expansion property of the gas, greater velocity of the projectile and 
less risk of bursting the gun. The work done by the gas of powder in a gun should 
be treated under the same laws as that of steam in a steam cylinder. 

This special subject is too extensive for proper treatment in this Pocket-book. 

Composition of Gunpowder. 

The composition of gunpqwder varies in all proportions between the limits of 
70 and 78 parts of saltpetre (nitrate of potash, KN0 3 ), 13 and 15 parts of charcoal, 
9 and 20 parts of sulphur in 100 parts of the powder. 

Chinese powder, 62 saltpetre, 23 charcoal, 15 sulphur. 

The different proportions depend much upon the purpose for which the powder 
is used, and also upon the ideas and experience of the manufacturers and users of 
the powder. The quickest powder requires the highest proportions of saltpetre. 

Size of Gunpowder-grains. 


Use of Powder. 
Sporting,. 
Mortar, 

Cannon, . 
Mammoth, . , 


Size of Sieve. 

Size in Inches. 

Composition 

No. 1 to 2. 

0.03 to 0.06 

77.13.10 

No. 2 to 3. 

0.06 to 0.1 

76.13.11 

No. 4 to 5. 

0.25 to 0.35 

75.13.12 

No. 6 to 7. 

0.6 to 0.9 

74.11.12 


Fine gunpowder is also moulded into lumps to fit the chamber of the gun. 
The Russians mould fine powder into hexagon blocks for heavy ordnance. 











































524 


Properties of Water and Steam. 


PROPERTIES OF WATER AND STEAM, 

III Relation to Heat. 


The following six pages of tables for water and steam have been calculated by 
the author whilst stationed in the Bureau of Steam Engineering of the United 
States Navy Department, under the direction of Chief Engineer lsherwood. The 
tables have been improved for this Pocket-book. 

Properties of Water. 

Column h' contains the heat units required to raise each cubic foot of distilled 
water from 32° to temperature T, under the pressure P. 

Column h contains the heat units required to raise each pound of water from 32° 
to T°. This column is calculated from the formula deduced from Reguault’s 
experiments, namely: 


A= T° —32° + 


(T°-32)« 


or h'=T-l'- 


(T— 32)*-« / -32)« 


1 . 


r v' 

in which the last term is a parabola of exponent n = 2.67, and parameter p' 1167713. 
log. pf — 6.0673350. h'= heat units required per pound of water of temperature t', 


and raised to T°. 

Column c contains the fractional cubic feet per pound of water of temperature T. 
Column io contains the weight in pounds per cubic foot of water of temperature 
T. Water of the maximum density at 39° weighs 62.388055 pounds per cubic foot. 

Column v contains the volume of water of temperature T , that at 39° being I 
unit. This column is calculated from the Formula 2, deduced from Kopp’s experi-! 
ments. 


V = 1 -f- 


(T — 39) 5 


2000000 [0.23 + 0.0007 (T — 39)] ’ 


2 . 


Column t contains the temperature of the steam and water, Centigrade scale. 
Columns i and p give the steam-pressure indicated on the safety-valve or 
mercury-gauge. 

+ means pressure above the atmosphere. 

— means vacuum under the atmosphere. 


Properties of Steam. 

Column P contains tho total steam-pressure in pounds per square Inch, in¬ 
cluding tho pressure of the atmosphere. 

Column I is the same pressure in inches of mercury, The specific gravity of 
mercury at 32° Fahr. is 13.5959, compared with water of maximum density at 39°. 
One cubic inch of mercury weighs 0.49086 pounds, of which a column of 29.9218 
inches is a mean balance of the atmosphere, or 14.68757 lbs. per sq. in. 

Column T contains the temperature of the steam on Fahr.’s scale, deduced from 
Regnault’s experiments. 

Column V contains the volume of steam of the corresponding temperature T, 
compared with that of water of maximum density at 39° Fahr. This column is 
calculated from the formula of Fairbairn and Tate, namely: 


V= 25.62 + 


49513 
I + 0.72 ’ 


. 3. 


Column W contains the weight per cubic foot in fractions of a pound ; and 
Column C the cubic feet per pound of saturated steam under the pressure P 
and temperature T. 

Column H contains the heat unitsper pound of steam from 32° to temperature T 
and pressure P, calculated from the formula— 


H = 1081.91 + 0.305 T. . ... 4. 


Column H' contains the heat units per cubic foot of steam from 32° to tempera¬ 
ture T. 

















Properties op Water and Steam. 


525 


The columns H and //' give the heat units required to heat the water from 
32° to the boiling-point ar.d evaporate the same to steam under the pressure 
P and of temperature T. 

Column 7/ contains the latent units of heat per pound in steam of tempera¬ 
ture T and pressure P. The latent heat expresses the work done in the evap¬ 
oration, or the difference between the number of heat units per pound in 
the steam and in the water of the same temperature. 

Column L' contains the latent heat per cubic foot of steam. 

Latent heat L = H — h , the heat units required to evaporate each pound 
of water from ilie boiling-point into steam. 

The maximum work 7T, which can be realized per heat unit in steam with¬ 
out expansion, is— 

r - 144 f( F— 1) ^ 

1 H'V . 5 ' 

Example 1. Required, the maximum work K that can be realized per heat 
unit in steam of P — 50 lbs. per sq. iu. ? V = 508.29 and H' — 143.3. 


IC- 


144x50(508.29-1) . , 

“M5508.29— =5014 f00t P° ,inds - 

or, 50.14 : 772 = 0.0049 of the total power. 

The maximum work which can be realized per heat unit in steam with ex¬ 


pansion will be— 


s 


K= 


144P(F—l)(2.31og. y + 1) 


6 . 


H'V 

in which S= stroke of piston, and l — part of the stroke with full steam. 

N K 

The natural effect of a steam-engine in horse-power is = —. . 7. 

t oo 0 

of which from 50 to 75 per cent, is realized in ordinary practice. N= num¬ 
ber of heat units passed through the engine in the time r in seconds. 

Example 2. Let the steam in Example 1 be expanded S : / = 3 times. We 
have log. 3 — 0.47712, and 2.09737 X 50.14= 105.16 footpounds per heat unit. 
Suppose each stroke of the piston to use 4 cubic feet of steam expanded 3 
times, and making 90 strokes per minute. 

90x4x143.3 ,„ nc , , 

Then-^-= 439.8 heat units per second, 


60 


and the power will be 


439^8 x 105.16 
1 x 550 


= 84 EP. 


This is the effect of steam when raised from water of 32°, but when the 
feed-water is of higher temperature, calculate the heat units from the Formula 
1, />/, and add the latent heat per pound of the steam; the sum will be the 
heat units required in generating the steam. 


The author’s formulas for temperature and pressure of steam are as follows: 


English Measures. 

T= e iwx / T-m. 
/ r+io i\ g 
V 200 / • 


French Measures. 

t = 84.5 V¥- 73.9. 
_ , 6 
P- . ' 


/f + 7^9 V 
\ 84.5 / 


These formulae, as also the others given, do not accord accurately with ex¬ 
periment, but are approximate only. 


























526 

Properties of Water from Freezing to Boiling Point 

• 

Temp. 

Volume 

Units of heat. 

Pounds 

Cubic ft. 

Temp. 

Fahr. 

1 at 39° 

pr. lb. 

pr. cub. ft. 

pr. cub. ft. 

pr. lb. 

Celsius 

jrO 

V 

h 

h' 

10 

c 

t 

32 

1-000109 

0-000000000 

0-00000 

62.387 

0-01603046 

0-000 

33 

1-000077 

1-000000867 

62-383 

62-383 

0-01602994 

0-555 

34 

1-000055 

2-000000545 

124-77 

62-384 

0-01602956 

1-111 

35 

1-000035 

3-00001609 

187-16 

62-385871 

0-01602927 

1-666 

36 

1-000020 

4-00003468 

249-55 

62-386791 

0-01602904 

2-222 

37 

1-000009 

5-00006294 

311-99 

62-387493 

0-01602886 

2-777 

38 

1-000002 

6-00010241 

374-33 

62-387930 

0-01602874 

3-333 

39 

1-000000 

7-00015455 

436-72 

62-388055 

0-01602871 

3-888 

40 

1-000002 

8-00022076 

499-12 

62-387930 

0-01602874 

4-444 

41 

1-000009 

9-00030234 

561-51 

62-387493 

0-016028S6 

5-000 

42 

1-000019 

10-00040056 

623-89 

62-386869 

0-01602902 

5-555 

43 

1-000034 

11-00051663 

686-28 

62-3859:13 

0-01602926 

6111 

44 

1-000053 

12-00065175 

74S-66 

62-084748 

0-01602956 

6-666 

45 

1-000077 

13-00080704 

811-03 

62-383251 

0-01602994 

7-222 

46 

1-000104 

14-00098362 

873-40 

62-381567 

0-01603038 

7-777 

47 

1-000136 

15-001326 

935-70 

62-379571 

0-016030S8 

8-333 

48 

1-000171 

16-0014050 

997-77 

62-3773S8 

0-01603146 

8-SS8 

49 

1-000211 

17-0016518 

1060-0 

62-374893 

0-01603210 

9-444 

50 

1-000254 

18-0019242 

1122-8 

62-372212 

0-01603278 

10000 

51 

1-000302 

19-0022230 

1185-1 

62-369219 

0-01603355 

10-555 

52 

1-000353 

20-0025493 

1248-0 

62-366039 

0-01603437 

11-111 

53 

1-000408 

21-0029241 

13101 

62-362611 

001603525 

11-666 

54 

1-000468 

22-0032880 

1372-3 

62-358871 

001603621 

12-222 

55 

1-000531 

23-0037024 

1434-3 

62-354944 

0-01603723 

12-777 

56 

1-000597 

240041479 

1496-4 

62-350831 

0-01603828 

13-333 

57 

1000668 

25-0046256 

1558-6 

62-346407 

0-01603942 

13-888 

58 

1-000740 

26-0051362 

1620-9 

62-341921 

0-01604057 

14414 

59 

1-000819 

27-0056808 

1683-2 

62-337000 

0-01604184 

15-000 

CO • 

1-000901 

28-0062600 

1745-5 

62-331893 

001604316 

15-555 

61 

1-000986 

29-0068749 

1807-8 

62-326620 

0-01604451 

16-111 

62 

1-001075 

300075263 

1870-1 

62-321059 

001604594 

16-666 

63 

1-001167 

31-0082149 

1932-4 

62-315333 

0-01604741 

17-222 

64 

1-001262 

32-0089416 

1994 4 

62-309420 

0-01604894 

17-777 

65 

1-001362 

33-0097073 

2056-6 

62-303198 

0-01605054 

18-333 

66 

1-001464 

34-010513 

2118-7 

62-296852 

0-01605218 

18-888 

67 

1-001570 

35-011359 

2180-8 

62-290259 

0-01605388 

19-444 

68 

1-001680 

36-012246 

2242-9 

62-283418 

0-01605564 

20-000 

69 

1-001793 

37 013175 

2305-0 

62-276293 

0-01605748 

20-555 

70 

1-001909 

38-014148 

2367-1 

62-269183 

0-01605921 

21111 

71 

1-002028 

39-015164 

2429-2 

62-261788 

0-01606122 

21-666 

72 

1-002151 

40016224 

2491-2 

62-254146 

0-01606318 

22-222 

73 

1-002277 

41017330 

2553-2 

62-246320 

0-01006521 

22-777 

74 

1-002406 

42-018482 

2615-2 

62-23S309 

0-01606728 

23-333 

75 

1-002539 

43-019680 

2677-1 

62-230052 

0-01606941 

23-888 

76 

1-002675 

44-020926 

2739-2 

62-221612 

0-01607158 

24-444 

77 

1-002814 

45-022220 

28010 

62-212987 

001607382 

25-000 

78 

1-002956 

46023563 

2862-8 

62-204179 

0-01607610 

25-555 

79 

1-003101 

47-024956 

2924-6 

62-195187 

0-01607841 

26-111 

80 

1-003249 

48-026398 

2985-4 

62-186012 

0-01608078 

26-666 

81 

1-003400 

49-027893 

304S-2 

62-176654 

0-01608321 

27-222 

82 

1-003554 

50-029438 

31110 

62 167113 

001608567 

27-777 

83 

1-003711 

51-031039 

3172-8 

62-157388 

0-01608820 

28-333 

84 

1-003872 

52-032688 

3234-4 

62-147420 

0-01609077 

28-888 

85 

1-004035 

53-034394 

3296.2 

62-137330 

0-01609338 

29-444 

86 

1-004199 

54-036154 

3358-2 

62-127182 

0-01609601 

30000 

87 

1-004370 

55-037969 

3418-7 

62-116605 

001609875 

30-555 

88 

1*004542 

56-039841 

3480-4 

62-105969 

0-01610151 

31-111 

89 

1-004717 

57-041769 

3542-1 

62-095152 

0-01610432 

31-666 

90 

1-004894 

58-043754 

3G03-8 

62-084214 

0-01610715 

32-2-22 






















Properties of Water from Freezing to Boiling Point. 527 


Temp. 

Volume 

Units of heat. 

Pounds 

Cubic ft. 

Temp. 

Falir. 

1 at 39° 

pr. lb. 

pr. cub. ft. 

pr. cub. ft. 

pr. lb. 

Celsius. 

2 y o 

V 

h 

li' 

w 

c 

t 

91 

1-005094 

59-045797 

3665-0 

62-071860 

0-01611036 

32*777 

92 

1-005258 

60-047899 

3726*6 

62*061734 

0-01611298 

33-333 

93 

1-005444 

61-050061 

37S8-2 

62-050252 

0-01611597 

33-888 

94 

1-005633 

62-052282 

3849-8 

62*038591 

0-01611900 

34-444 

95 

1-005825 

63-054564 

3911-2 

62-026749 

0-01612208 

35-000 

96 

1-006019 

64-056907 

3972-6 

62-014787 

0-01612519 

35-555 

97 

1-006216 

65-059312 

4033-9 

62-002646 

0-01612834 

36-111 

98 

1-006415 

66-0617 SO 

4095-2 

61-990386 

0-01613153 

36-666 

99 

1-006618 

67*064311 

4156-5 

61*977885 

0-01613478 

37*222 

100 

1-006822 

6S-066906 

4217-7 

61-965322 

0-01613806 

37-777 

101 

1-007030 

69-069565 

4278-9 

61-952528 

0-01614140 

38-333 

102 

1-007240 

70-072290 

4340-1 

61-939612 

0-01614475 

38-888 

103 

1-007553 

71-075080 

4401-3 

61-920370 

0-01614977 

39-444 

104 

1-007668 

72-077937 

4462-5 

61-913303 

0-01615161 

40-000 

105 

1-007905 

73-080861 

4523-0 

61-898745 

0-01615541 

40-555 

106 

1-008106 

74-083852 

4585-0 

61*886403 

0-01615863 

41-111 

107 

1-008328 

75-086912 

4645-9 

61-872778 

0-01616220 

41-666 

108 

1*008554 

76-090044 

4706-8 

61*858913 

0-016165S1 

42-222 

109 

1-008781 

77-093239 

4767-7 

61*844994 

001616946 

42-777 

110 

1-009032 

78*096509 

4828-6 

61-829609 

0-01617348 

43-333 

111 

1-009244 

79-099846 

48S9-5 

61-816622 

0-01617677 

43-888 

112 

1-009479 

80-103255 

4950-4 

61-802231 

0-01618064 

44-444 

113 

1-009718 

81-10674 

5011*3 

61-787602 

0-01618447 

45-000 

114 

1-009956 

82-11029 

5072-2 

61-773042 

0-01618829 

45*555 

115 

1-010197 

83-11392 

51330 

61-758305 

0-01619216 

46111 

116 

1-010442 

84-11762 

5193-7 

61-743331 

0-01619608 

46*666 

117 

1-010688 

S5-12140 

5254-3 

61-728302 

0-01620003 

47-222 

118 

1-010938 

86-12525 

5314-9 

61-713037 

0-01620403 

47-777 

119 

1-011189 

87-12918 

5375-5 

61-697719 

0-01620S06 

48-333 

120 

1011442 

88-13318 

5436-1 

61-682286 

0-01621211 

48-888 

121 

1-011698 

89-13726 

5496-6^ 

61-666678 

0-01621621 

49-444 

122 

1-011956 

90-14141 

5557-1 

61-650956 

0-01622034 

50-000 

123 

1-012216 

91-14565 

5617-6 

61-635123 

0-01622451 

50-555 

124 

1-012478 

92-14996 

5678-1 

61-619170 

0-01622871 

61*111 

125 

1-012743 

93-15435 

5738-6 

61-603047 

0-01623296 

51-666 

126 

1-013010 

’ 94-15S82 

5798-9 

61-586810 

0-01623724 

52-222 

127 

1-013278 

95-16338 

5859-2 

61-580516 

0-01624153 

52-777 

128 

1-013550 

96-16801 

5919-5 

61-553998 

0-01624590 

53-333 

129 

1-013823 

97-17272 

5979-7 

61-537423 

0-01625027 

53-888 

130 

1-014098 

98-17752 

6040-0 

61-520735 

0-01625468 

54-444 

131 

1-014358 

99-18239 

6100-2 

61-504966 

0-01625884 

55-000 

135 

1-015505 

. 103-20274 

6340-3 

61-435497 

0-01627724 

57-222 

140 

1-010962 

108-23009 

6639-6 

61*347282 

0-01630064 

60-000 

145 

1-018468 

113-25965 

6937-9 

61-256765 

0-01632473 

62-777 

150 

1*020021 

118-29147 

7215-1 

61-163500 

0-01634961 

65-555 

155 

1-021619 

123-32562 

7531-2 

61-067829 

0-01637523 

68-333 

160 

1-023262 

128-36217 

7826-2 

60-969776 

0-01640156 

71*111 

165 

1-024947 

133-40119 

8098-1 

60-869542 

0-01642857 

"3-888 

170 

1-026672 

138-44273 

8412-8 

60-767270 

0-01645623 

"6-666 

175 

1-028438 

143-48687 

8704-2 

60-662047 

0-01648477 

79-444 

ISO 

1-030242 

148-53666 

8994-9 

60-556699 

0-01651345 

82-222 

185 

1-032083 

153-58316 

9281-9 

60-448679 

0-01654296 

85-000 

190 

1-033960 

158-63545 

9571-6 

60-338944 

0-01657305 

87*777 

195 

1-035873 

163-69057 

985S-5 

60-227513 

0-01660370 

90-555 

200 

1-037819 

168-74858 

10318 

60*114581 

0-01663489 

93-333 

205 

1-039798 

173-80956 

10428 

60*000168 

0-01666662 

96-111 

210 

1-041809 

178-87355 

10712 

59-884350 

0-01679885 

98-888 

212 

1-042622 

180-90000 

18824 

59-837654 

0-01681160 

100-000 
























628 


Properties of Water. 


Temp. 

Fahr. 

Units of heat. 

Wi 

Bulk 

iter. 

Weight 

Volume 

Temp. 

ludic. press. 

Atmos, excluded 

per 

per 

cub. ft. 

lbs. pr. 

wat.=l 

Celsius 

inches 

lbs. pr. 

Scale. 

cub. ft. 

pound. 

per lb. 

cub. ft. 

at 39° 

Scale. 

mercury 

s<i. in. 

T 

h' 

h 

c 

w 

V 

1 

i 

V 

101.36 

4301 

69.430 

.01617 

61.848 

1.0071 

30.83 

— 28.52 

— 14 

126.21 

5631 

94.369 

.01624 

61.583 

1.0130 

41.87 

— 26.48 

— 13 

141.67 

6583 

109.91 

.01630 

61.317 

1.0174 

48.74 

— 24.44 

— 12 

153.27 

7331 

121.58 

.01637 

61.101 

1.0210 

53.90 

— 22.41 

— 11 

162.51 

797 4 

130.89 

.01638 

60.920 

1.0241 

58.00 

— 20.37 

— 10 

170.25 

8421 

138.69 

.01644 

60.7 62 

1.0267 

61.44 

— 18.33 

— 9 

17 (i 

8812 

145.46 

.016 47 

60.657 

1.0288 

64.43 

— 16.29 

— 8 

182.96 

9203 

151.52 

.01652 

60.514 

1.0309 

67.09 

— 14.26 

— 7 

188.36 

9531 

156.97 

.01656 

60.372 

1.0333 

69.49 

—12.22 

— 6 

193.20 

9755 

101.87 

.01659 

60.282 

1.0359 

71.6 4 

— 10.18 

— 5 

197.60 

9975 

166.32 

.01663 

G0.169 

1.0369 

73.60 

— 8.149 

— 4 

201.90 

10183 

170.67 

.01666 

60.072 

1.0385 

* 75.51 

— 6.111 

— 3 

205.77 

10398 

174.59 

.01669 

59.973 

1.0401 

77.23 

— 4.074 

— 2 

209.55 

10613 

178.42 

.01672 

59.896 

1.0416 

78.91 

— 2.037 

— 1 

212.00 

10824 

180.9 

.01674 

59.838 

1.0426 

10().(X) 

0.0000 

0 

213.04 

108 S3 

1S1.95 

.01675 

59.814 

1.0430 

100.58 

0.63(35 

0.3125 

216.33 

11047 

185.29 

.01677 

59.735 

1.0444 

102.45 

+ 2.037 

+ 1 

219.45 

11225 

13S.45 

.01679 

59.659 

1.0457 

104.36 

+ 4.074 

+ 2 

222.40 

11389 

191.44 

.01680 

59.592 

1.0469 

105.78 

+ 6.111 

+ 3 

225.25 

11550 

194.33 

.01081 

59.523 

1.0481 

107.35 

+ 8.149 

+ 4 

227.95 

11718 

197.08 

.01684 

59.459 

1.0492 

10^.86 . 

+ 10.18 

+ 5 

230.60 

11868 

199.77 

.016S6 

59.389 

1.0503 

110.33 

+ 12.22 

+ 6 

233.10 

12012 

202.40 

.01088 

59.329 

1.0514 

111.50 

+14.26 

+ 7 

235.49 

12150 

204.73 

.01690 

59.270 

1.0524 

113.05 

+16.29 

+ 8 

237.81 

12282 

207.10 

.01692 

59.212 

1.0534 

114.00 

+18.33 

+ s 

240.07 

12408 

209.39 

.01693 

59.154 

1.0545 

115.59 

+ 20.37 

+ 10 

242.24 

12528 

211.57 

.01695 

59.097 

1.0555 

116.«0 

+ 22.41 

+ 11 

244.32 

12642 

213.72 

.01696 

59.057 

1.0564 

117.95 

+ 2 4.44 

+ 12 

246.35 

12750 

215.78 

.01697 

59.006 

1.0573 

119.08 

+ 26.48 

+ 13 

248.33 

12852 

217.80 

.01698 

58.953 

1.0589 

120.18 

+ 28.52 

+ 14 

250.26 

12946 

219.76 

.01699 

58i)0l 

1.0590 

121.25 

+ 30.55 

+ 15 

252.13 

13053 

221.67 

.01700 

58.851 

1.0599 

122.29 

+ 32.59 

+ 16 

253.98 

13157 

223.55 

.01701 

58.803 

1.0607 

123.32 

+ 34.63 

+ 17 

255.77 

13258 

225.88 

.01702 

58.757 

1.0615 

124.32 

+ 36.67 

+ 18 

257.52 

13336 

227.16 

.01703 

58.713 

1.0623 

125.29 

+ 38.71 

+ 19 

259.22 

13430 

228.89 

.01704 

58.671 

1.0631 

126.23 

+ 40.74 

+ 20 

260.88 

13520 

230.59 

.01705 

58 631 

1.0639 

127.15 

+ 42.78 

+ 21 

262.50 

13008 

232.2 4 

•01"07 

58.592 

1.0646 

128.05 

+ 44.82 

+ 22 

264.09 

13694 

233.86 

.01708 

58.560 

1.0654 

128.94 

+ 46.85 

+ 23 

265.65 

13778 

235.15 

.01709 

58.517 

1.0661 

129.80 

+ 4S.89 

+ 24 

267.17 

13860 

237.00 

.01710 

58.481 

1.0t568 

130.65 

+ 50.93 

+ 25 

268.66 

13940 

238.52 

.01711 

58.435 

1.0675 

131.48 

+ 52.97 

+ 26 

270.12 

14018 

240.02 

.01712 

58.400 

1.0684 

132.29 

+ 55.00 

+ 27 

271.55 

14094 

241.48 

.01713 

58.366 

1.0688 

133.05 

+ 57.04 

+ 28 

272.96 

14168 

242.92 

.01714 

58.332 

1.0695 

133.86 

+ 59.08 

+ 2.* 

274.33 

14241 

244.32 

.01715 

58.298 

1.0701 

134.63 

+ 61.11 

+ 30 

275.68 

14314 

245.70 

.01716 

58.264 

1.0708 

135.33 

+ 63.15 

+ 

277.01 

14385 

217.06 

.01717 

68.230 

1.0714 

136.12 

+ 65.19 

+ 32 

278.32 

14454 

248.40 

.01718 

58.197 

1.0720 

136.84 

+ 67.23 

+ 33 

279.62 

14522 

24973 

.01719 

58.164 

1.0726 

137.56 

+ 69.20 

+ 34 

280.89 

14592 

251.03 

.01720 

58131 

1.0732 

138.27 

+ 71.30 

+ 35 

282.14 

14659 

252.30 

.01721 

58.098 

1.0738 

138.96 

+ 73.34 

+ 36 

283.39 

14725 

253.58 

.01722 

58.066 

1.0744 

139.66 

+ 75.38 

+ 37 

284.58 

14789 

254.80 

.01723 

58.035 

1.0750 

140.33 

+ 77.41 

+ 38 

285.76 

14852 

256.01 

.01724 

58.004 

1.0756 

140.98 

+ 79.45 

+ 39 

286.96 

14913 

257.24 

.01725 

57.972 

1.0761 

141.64 

+ 81.49 

+ 40 

288.06 

14973 

258.38 

.01726 

57.941 

1.0767 

142.27 

+ 83.52 

+ 41 

289.24 

15032 

259.67 

.01727 

57.910 

1.0773 

142.91 

+ 85.56 

4- 42 

290.37 

15091 

260.71 

.01728 

57.879 

1.0778 

143.5 4 

+ 87.61 

+ 43 

291.48 

15149 

261.87 

.01729 

57. y 48 

1.0783 

144.15 

+ 89.64 

+ 44 

292.58 

15208 

262.99 

.01730 

57.817 

1.0789 

144.76 

+ 91.67 

+ 45 











































Properties of Water. 


529 





Water. 



Indie. 

press. 

Temp. 

Units of heat 

Bulk 

Weight 

Volume 

Temp. 

Atmos, excluded 

Fahr. 

Per 

per 

cub. ft 

lbs. pr. 

wat.=l 

Celsius 

inches 

lbs. 

pr. 

Seale. 

cub. ft 

pound. 

per lb. 

cub. ft. 

at 39° 

Scale. 

mercury 

sq. 

F 

m. 

T 

h' 

h 

c 

w 

V 

t 

i 

n 

293.06 

15205 

204.10 

.01731 

57.786 

1.0794 

145.37 

+ 93.71 

+ 

46 

294.73 

15321 

265.20 

.01732 

57.769 

1.0799 

145.96 

-j- 95.75 

4* 

47 

295.78 

15377 

266.27 

.01733 

57.742 

1.0804 

146.54 

+ 97.78 

-t- 

48 

296.82 

15432 

267.34 

.01734 

57.714 

1.0809 

147.12 

+ 99.82 

+ 

49 

297.84 

15485 

268.39 

.01735 

57.687 

1.0814 

147.69 

+ 101.8 


50 

298.85 

15536 

269.42 

.01735 

57.660 

1.0820 

148.25 

+ 103.9 

+ 

61 

299.85 

155SS 

270.45 

.01736 

57.633 

1.0825 

148.80 

+ 105.9 

+ 

62 

300.84 

15639 

271.46 

.01737 

57.606 

1.0830 

149.34 

+ 108.0 

+ 

63 

301.81 

15690 

272.46 

.01737 

57.580 

1.0835 

149.89 

+110.0 

+ 

54 

302.77 

15789 

273.44 

.01738 

57.554 

1.0840 

150.43 

+ 112.0 

+ 

55 

303.72 

15789 

274.42 

.01739 

57.529 

1.0844 

150.95 

+ 114.1 

+ 

56 

304.09 

15839 

275.40 

.01739 

57.504 

1.0849 

151.48 

+ 116.1 

+ 

57 

305.60 

15888 

276.35 

.01740 

*57.480 

1.0854 

152.00 

+ 118.1 

+ 

58 

306.52 

15936 

277.30 

.01741 

57.456 

1.0859 

152.51 

+ 120.2 

+ 

59 

307.42 

15983 

278.22 

.01741 

57.432 

1.0863 

153.01 

+ 122.2 

+ 

60 

308.38 

16029 

279.14 

.01742 

57.410 

1.08G7 

153.51 

+ 124.3 

+ 

61 

309.22 

16075 

280.07 

.01743 

57.388 

1.0871 

154.01 

+ 1*26.3 

+ 

62 

310.11 

16120 

280.98 

.01743 

57.364 

1.087 5 

154.50 

+ 128.3 

+ 

63 

310.99 

16165 

281.87 

.01744 

57.344 

1.0880 

154.99 

+ 130.4 

+ 

64 

311.86 

16209 

282.78 

.01745 

57.322 

1.0884 

155.48 

+ 132.4 

+ 

(35 

312.72 

16254 

288.66 

.01745 

57.300 

1.0888 

155.95 

+ 134.4 

+ 

66 

313.57 

16298 

284.54 

.01746 

57.278 

1.0892 

15G.42 

+ 136.5 

+ 

67 

31442 

16342 

285.41 

.01746 

57.254 

1.0897 

156.90 

+ 138.5 

+ 

68 1 

31.5.25 

1688 4 

286.27 

.01747 

57.232 

1.0901 

157.36 

+ 140.5 

+ 

69 

316.08 

16426 

287.12 

.01748 

57.210 

1.0905 

157.82 

+ 142.6 

+ 

70 

316.90 

10467 

287.90 

.01748 

57.18S 

1.0909 

158.28 

+ 144.6 

+ 

71 

317.71 

16507 

288.80 

.01749 

57.1G6 

1.0913 

158.73 

+ 146.7 

+ 

72 

318.51 

16547 

289.62 

.01750 

57.144 

1.0918 

159.17 

+ 148.7 

+ 

73 

319.31 

16587 

290.44 

.01751 

57.122 

1.0921 

159.G2 

+ 150.7 

+ 

74 

320.10 

16637 

291.26 

.01752 

57.101 

1.0926 

160.05 

+ 152.8 

+ 

75 

320.88 

16077 

292.06 

.01752 

57.080 

1.0929 

160.49 

+ 154.8 

+ 

76 

321.66 

16717 

2 >2.85 

.01753 

57.059 

1.0935 

160.92 

+ 156.8 

+ 

77 

322.42 

16756 

293.65 

.01753 

57.038 

1.0937 

161.34 

+ 158.9 

+ 

78 

323.18 

16795 

294.43 

.01754 

57.017 

1.0941 

161.76 

+ 160.9 

+ 

79 

823.94 

16834 

295.21 

.01755 

56.996 

1.0945 

162.17 

+ 163.0 

+ 

80 

324.67 

16872 

295.00 

.01756 

56.975 

1.0949 

162.59 

+ 165.0 

+ 

81 

325.43 

16010 

296.75 

.01750 

56.954 

1.0953 

163.02 

+ 167.0 

+ 

82 

326.17 

16947 

297.51 

.01757 

56.933 

1.0956 

163.43 

+ 169.1 

+ 

83 

326.90 

16984 

298.26 

.01757 

56.912 

1.0960 

163.83 

+ 171.1 

+ 

84 

327.63 

17020 

299.01 

.01758 

50.89 i 

1.0964 

161.24 

+ 173.1 

+ 

85 

328235 

17056 

299.75 

.01759 

56.871 

1.0968 

161.64 

+ 175.2 

+ 

86 

329.07 

17092 

300.50 

.01759 

56 862 

1.0972 

165.04 

+ 177.2 

+ 

87 

329.78 

17127 

301.23 

.01760 

56.844 

1.0975 

165.43 

+ 179.2 

+ 

88 

330.48 

17162 

301.95 

.01761 

5G.826 

1.0979 

165.82 

+ 181.3 

+ 

89 

331.18 

17197 

302.67 

.01761 

56.808 

1.0982 

166.21 

+ 183.3 

+ 

90 

331.87 

17231 

303.38 

.01762 

56.790 

1.0986 

166.59 

+185.4 

+ 

91 

332.56 

17205 

304.10 

.01763 

56.772 

1.0989 

166.98 

+ 187.4 

+ 

92 

333.24 

17299 

304.80 

.01763 

56.754 

1.0993 

167.35 

+ 189.4 

4 * 

93 

333.92 

17333 

305.50 

.01764 

56.735 

1.0996 

167.77 

+ 191.5 

+ 

94 

334 59 

17306 

306.19 

.01765 

56.716 

1.0999 

168.10 

+ 193.5 

+ 

95 

335.26 

17399 

306.88 

.01765 

56.699 

1.1003 

168.47 

+ 195.5 

+ 

96 

336.58 

17465 

308.34 

.01767 

56.664 

1.1010 

169.21 

+ 199.6 

+ 

98 

337.23 

17497 

308.91 

.01768 

56.647 

1.1013 

169 57 

+ 201.6 

+ 

99 

337.S9 

17529 

309.60 

.01769 

56.629 

1.1017 

169.94 

+ 203.7 

+ 100 

341.0 

17088 

312.87 

.01772 

56.549 

1.1035 

171.70 

+ 213.9 

+ 105 

344.1 

17840 

316.04 

.01775 

56.469 

1.1050 

173.40 

+ 224.1 

+ no 

347.1 

17993 

319.12 

.01778 

56.389 

1.1065 

175.06 

+ 234.2 

+ 115 

350.0 

18136 

322.13 

.01781 

56.309 

1.1090 

176.68 

+ 244.4 

+ 120 

352.8 

18278 

325.06 

.01784 

56.220 

1.1095 

178.25 

+ 254.6 

+ 125 

355.6 

18413 

327.91 

.01786 

56.146 

1.1100 

179.78 

+ 264.8 

+ 130 

358.4 

18549 

330.75 

.01788 

56.073 

1.1124 

181.35 

+ 275.0 

+ 135 

- - - ■ 


34 































Properties of Steam. 


630 


1 




Steam. 





Press 

Total pressure 

Temp, 

Volume 

Weight 

Bulk 

Units of heat, from 32° to T 

ob. at. 

lbs. pr. 

inches 

Fahr. 

wat.-—1 

lbs. pr. 

cub. ft. 

Total pr. 

Latent pr. 

lbs. pr. 

sq. in. 

mer. 

Scale. 

at 39° 

cub. it. 

pr. lb. 

pound. 

cub. ft. 

pound. 

cub. ft. 

sq. in. 

P 

I 

T 

V 

W 

C 

// 

H 

L 

L' 

V 

1 

2.037 

101.36 

17083 

.00347 

2^8.21 

1112.8 

3.8614 

1043.4 

3.6337 

— 14 

2 

4.074 

126.21 

10353 

.00602 

165.94 

1120.4 

6.7449 

1026.0 

6.1165 

— 13 

3 

6.111 

141.67 

7283.8 

.0 i860 

116.75 

1125.1 

9.6308 

1015.2 

8.6901 

— 12 

4 

8.119 

153.27 

56 >8.4 

.01112 

89.895 

1128.7 

12.551 

1007.1 

11.199 

— 11 

5 

10.18 

162.51 

456 ).6 

.01366 

73.180 

1131.5 

15.456 

1000.6 

13.714 

— 10 

6 

12.22 

170.25 

3851.0 

.01619 

t>1.742 

1133.8 

18.156 

995.17 

16.11:; 

— 9 

7 

14.26 

176.97 

3330.8 

.01872 

53.388 

1135.9 

| 20.846 

990.44 

18.194 

— 8 

8 

16.29 

182.98 

2935.1 

.02125 

47.046 

1137.7 

24.170 

9 >6.22 

20.957 

— 7 

9 

18.33 

188.36 

2024.0 

.02377 

42.059 

1139.4 

27.083 

982.41 

23.352 

— 6 

10 

20.37 

193.20 

2373.0 

.02028 

38.037 

1140.S 

29.980 

978.99 

25.728 

—• 5 

11 

22.41 

197.60 

2166.3 

.0-880 

34.723 

1142.2 

32.895 

97 5.88 

28.099 

— 4 

12 

24.14 

-01.90 

1993.0 

.03130 

31.945 

1143.5 

35.791 

972.84 

34.450 

— 3 

13 

26.48 

205.77 

1845.7 

.03380 

29.584 

1141.7 

38.091 

970.11 

32.789 

_ 2 

14 

28.52 

209.55 

1718.9 

.03629 

27.561 

1145.8 

41.581 

967.43 

35 435 

— i 

14.7 

29.92 

-12.00 

1611.5 

.03800 

36.311 

1146.6 

43.571 

965.70 

36.706 

0 

15 

30.55 

213.01 

1003.6 

.03878 

25.784 

1146.9 

41.476 

964.93 

37.121 

0.3125 

16 

32.59 

216.83 

1511.7 

.04123 

24 230 

1147.9 

47.328 

962.63 

39 690 

+ 1 

17 

34.63 

219.45 

1426.2 

.01374 

22.859 

1148.8 

50 248 

960.49 

42.012 

+ 2 

18 

36.67 

222.40 

1349.8 

.01622 

21.636 

1149.7 

5 5.138 

958.32 

41.393 

+ 3 

19 

38.71 

225.25 

1281.1 

.04868 

20.539 

1150.6 

56.011 

058.30 

40.098 

+ 4 

20 

40.74 

227.95 

1219.7 

.05119 

19.550 

1151.4 

58.894 

951.38 

48.655 

+ 5 

21 

42.78 

230.60 

1163.8 

.05.60 

189 54 

1152.2 

61.758 

952.50 

51.921 

+ 5 

22 

44.82 

233.10 

1112.9 

.05605 

17.838 

1153.0 

64.637 

950.62 

53.282 

+ 7 

23 

46.85 

235.49 

1066.3 

.05851 

17.0J2 

1153.7 

67.503 

949.03 

55.529 

+ 8 

24 

48.89 

237.81 

1023.6 

.06095 

16.407 

1154.5 

70.367 

9 57.37 

57.743 

+ 9 

25 

50.93 

210.07 

984.23 

.06338 

15.776 

1155.1 

73.410 

945.76 

59.942 

+ 10 

26 

52.97 

242.24 

947.86 

.06582 

15.193 

1155.8 

76.074 

944.25 

62.161 

+ 11 

27 

55.00 

244.02 

914.14 

.06824 

14.652 

1156.4 

78/413 

912.74 

64.423 

+ 12 

28 

57.01 

246.35 

8V2.80 

.07067 

14.150 

1157.1 

81.772 

941.29 

66.521 

+ 13 

29 

59.08 

218.33 

853.60 

.07308 

13.6S2 

1157.7 

84.604 

934.88 

68.686 

+ 14 

30 

61.11 

250.26 

82632 

.07550 

13.215 

1158.2 

87.444 

938.50 

7<».857 

+ 15 

31 

63.15 

252.13 

800.79 

.07791 

12.835 

1158.8 

90.166 

937.17 

73.015 


32 

65.19 

253.98 

766.83 

.08031 

12.451 

1159.4 

93.121 

935.45 

75.126 

+ 17 

33 

67.23 

255.77 

754.31 

.08271 

12.090 

1159.9 

95.861 

934.57 

77.298 

+ 18 

34 

69.26 

257.52 

733.09 

.08510 

11.750 

1160.5 

98 782 

933.32 

79.425 

+ 19 

35 

71.30 

259.22 

713.08 

.08749 

11.429 

1161.0 

101.18 

932.10 

81.549 

+ 20 

36 

73.31 

260.88 

691.17 

.089S7 

11.137 

1161.5 

104.38 

930.92 

83.662 

+ 21 

37 

75.38 

262.50 

676.27 

.09225 

10.810 

1162.0 

107.19 

924.76 

85.770 

+ 22 

38 

77.41 

264.09 

659.31 

.09462 

10.568 

11025 

104.98 

928.62 

S7.806 

+ 23 

39 

79.45 

265.65 

613.21 

.09700 

10.310 

1162.9 

112.79 

927.51 

89.968 

+ 24 

40 

81.49 

267.17 

627.91 

.09936 

10.064 

1103.4 

115.59 

926.42 

92.059 

+ 25 

41 

83.52 

268.06 

613.34 

.10172 

9.S3H) 

1163.9 

118.39 

925.35 

94.126 

-i- 26 

42 

85.56 

270.12 

599.46 

.10 707 

9.6086 

1164.3 

121.17 

924.30 

96.192 

+ 27 

43 

87.60 

271.55 

586.23 

.10642 

9.3963 

1164.7 

123.95 

923.28 

98.255 

+ 28 

44 

89.64 

272.96 

573.58 

.1:1877 

971938 

1105.2 

126.74 

922.27 

100.32 

4- 29 

45 

91.67 

274.33 

561.50 

.11111 

9.0002 

1165.6 

129.51 

921.29 

102.36 

4- 30 

46 

93.71 

275.68 

519.9 4 

.11344 

8.814 i 1 

1163.0 

132.29 

9205)2 

104.40 

+ 31 

47 

95.75 

277.01 

538**7 

.11577 

8.6374 

1166.4 

1/.5 07 

919.36 

106.13 

+ 32 

48 

97.78 

278.32 

528.25 

.11810 

8.4673 

1166.8 

137.83 

918.43 

108.46 

-j- 33 

49 

99.82 

279.62 

618.07 

.12042 

8.3010 

1107.2 

110.69 

917.49 

110.18 

+ 34 

50 

101.86 

280.89 

508.20 

.12273 

8.1472 

1107.6 

14 5.30 

916.58 

112.49 

+ 35 

51 

103.90 

282.14 

498.89 

.12505 

7.9966 

1167.9 

146.0 S 

915.68 

114.50 

+ 36 

52 

105.93 

283.39 

489.85 

.12736 

7.85171 

1168.1 

1 18.85 

914.79 

116.51 

+ 37 

53 

107.97 

284.58 

481.15 

.12966 

7.7122 

1168.7 

151.6! 

913.95) 

118.50 

+ 38 

54 

110.01 

2.85.7 6 

472.77 

.13196 

7.5779 

1109.0 

151.48 

913.08 

120.49 

+ .39 

55 

112.04 

286.96 

464.6 » 

• 1 3 li28 

7.4468 

1169.4 

157.02 

912.22 

122.47 

+ 40 

56 

114.08 

288.09 

956.1 0 

.13652 

7.3236 

1164.8 

159.74 

911.42 

124.43 

+ 41 

57 

116.12 

280.2 j 

449.38 

.13883 

7.2030 

1170.1 

162.45 

910.48 

126.40 

+ 42 

58 

118.16 

290.37 

442.12 

.14111 

7.0866 

1170.5 

163.15 

9,59.78 

128.38 

+ 43 

59 

120.19 

291.48 

435.10 

.14338 

6.9741 

1170.8 

167.84 

908.97 

130.33 

+ 44 

60 

122.23 

292.-53 

428.32 

.14566 

6.8654 

1171.2 

170.58 

908.18 

132.28 

+ 45 































Properties of Steam. 


53l 






Steam. 





Press 

Total pressure 

Temp. 

Volume 

■Weight 

Bulk 

Units of heat, from 32° to T 

ob. at. 

lbs. pr. 

inches 

Fahr. 

wat.=l 

lbs. pr. 

cub. ft. 

Total pr. 

Latent pr. 

lbs. pr. 

sq. in. 

mer. 

Scale. 

at 39° 

cub. ft. 

pr. lb. 

pound. 

cub. ft. 

pound. 

cub. ft. 

sq. 

in. 

P 

/ 

r 

V 

IF 

C 

H 

IT 

L 

U 

V 

61 

124.27 

293.66 

421.75 

.14792 

6.7601 

11715 

173.27 

907.40 

134.22 

+ 

46 

62 

126.30 

294.73 

415.40 

.15018 

6.6583 

1171.8 

175.95 

906.63 

136.16 

+ 

47 

63 

128.34 

295.78 

409.25 

.15241 

6.5597 

1172.1 

178.65 

905.87 

138.09 

+ 

48 

6t 

130.38 

296.82 

403.29 

.15469 

6.4642 

1172.5 

181.84 

905.13 

140.01 

+ 

49 

65 

132.42 

297.84 

397.51 

.15694 

6.3715 

1172.8 

181.03 

904.39 

141.93 

+ 

50 

66 

134.45 

298.85 

391.90 

.15919 

0.2817 

1173.1 

186.72 

903.66 

143 85 

+ 

51 

67 

136.49 

299.85 

386.47 

.16130 

6.1994 

1173.4 

189.40 

902.94 

145.64 

+ 

52 

68 

138.53 

800.81 

381.18 

.16366 

6.1099 

1173.7 

192.07 

902.23 

147.66 

+ 

53 

69 

140.86 

301.81 

376.06 

.16 90 

6.0277 

1174.0 

194.74 

901.53 

149.56 

+ 

54 

70 

142.60 

302.77 

371.07 

.16812 

5.9478 

1174.3 

197.42 

900 81 

151.45 

+ 

55 

71 

141.64 

303.72 

366 

.17035 

5.8702 

1174.6 

200.08 

900.15 

153.34 

+ 

56 

72 

146.68 

301.69 

36l.o3 

.17256 

5.7948 

1174.9 

202.74 

899.46 

135.21 

+ 

57 

73 

148.72 

305.60 

356.95 

.17478 

5.7214 

1175.1 

205.40 

898.79 

157.09 


58 

74 

150.75 

306.52 

352.49 

.17' 90 

5.6500 

1175.4 

298.04 

89*.18 

158.88 

+ 

59 

75 

152.79 

307.42 

318.15 

.17919 

5.5S05 

1175.8 

210.67 

897.57 

160.S3 

+ 

60 

76 

154.83 

308.32 

343.93 

.18139 

5.5129 

1176.0 

213.80 

8-16.83 

162.67 

+ 

61 

77 

156.86 

309.22 

339.81 

.18359 

5.4468 

1176.2 

215.93 

896.18 

164.56 

+ 

62 

78 

158.90 

310.11 

335 80 

.is 578 

5.3*25 

1176.5 

218.56 

895.54 

166.37 

+ 

63 

79 

160.94 

310.99 

331.89 

.18797 

5.3190 

1176.8 

221.19 

894.92 

168.22 

+ 

61 

80 

162.98 

311.86 

328.08 

.19015 

5.2588 

1177.0 

223.82 

894.27 

170.04 

+ 

65 

81 

165.01 

312.72 

324.37 

.19233 

5.1992 

1177.3 

226.44 

895.65 

171.87 

+ 

66 

82 

167.05 

313.57 

320.74 

.19451 

5.1410 

1177.6 

224.06 

893.03 

17-1.70 


67 

83 

169.09 

31442 

317.20 

.19668 

5.0843 

1177.9 

231.68 

892.51 

175.52 

+ 

68 

84 

171.12 

315.25 

313.74 

.19885 

5.028-9 

1178.1 

234.28 

891.82 

177.33 

+ 

69 

85 

173.16 

316.08 

310.36 

.20101 

4.9748 

1178.3 

236.89 

891.22 

179.14 

+ 

70 

86 

175.20 

316.90 

307.07 

.20317 

4.9219 

1178.6 

239.50 

890.63 

180.95 

+ 

71 

87 

177.24 

317.71 

303.85 

.20532 

4.8703 

1178.8 

242.10 

800.04 

182.75 

+ 

72 

88 

179.27 

318.51 

300.70 

.20747 

4.8198 

1179.1 

244.09 

889.46 

184.53 

4* 

73 

89 

181.31 

319.31 

297.62 

.20962 

4.7704 

1179.3 

247.29 

888.88 

186.33 

+ 

74 

90 

183.35 

320.10 

294.61 

.21185 

4.7222 

1179.6 

249.88 

888.31 

188.12 

+ 

75 

91 

185.38 

320.88 

291.66 

.21390 

4.6750 

1179.8 

252.45 

887.74 

189.88 

+ 

76 

92 

187.42 

321.66 

288.78 

.21603 

4.G2S8 

1180.0 

255.02 

887.19 

191.66 

+ 

77 

93 

189.46 

322.42 

285.96 

.21816 

4.5836 

1180.3 

257.58 

886.63 

193.43 

+ 

78 

94 

191.50 

323.18 

283.21 

.22029 

4.5394 

1180.5 

260.14 

886.08 

195.19 

+ 

79 

95 

193.53 

323.94 

280.50 

.22241 

4.4961 

1180.7 

262.69 

885.53 

196.94 

+ 

80 

96 

195.57 

324.67 

277.86 

.22453 

4.4537 

1180.9 

265.28 

885.00 

193.71 

+ 

81 

97 

197.61 

325.43 

275.2 < 

.22672 

4.4106 

1181.2 

267.77 

884.45 

200 49 

+ 

82 

98 

199.65 

326.17 

272.73 

.22875 

4.3715 

1181.4 

270.30 

883.91 

202 18 

+ 

83 

99 

201.68 

326.90 

270.24 

.23085 

4.3316 

1181.6 

273.10 

888 38 

203.92 

+ 

84 

100 

203.72 

327.63 

287.80 

.23296 

4.2926 

1181.9 

275.52 

882.85 

205.67 

+ 

85 

101 

205.76 

328.35 

265.41 

.23505 

4.2543 

1182.1 

277.85 

882.33 

207 39 

+ 

86 

102 

207.79 

329.07 

263.07 

.23715 

4.2167 

1182.3 

280.38 

881.81 

209.12 

+ 

87 

103 

209.83 

329.78 

260.77 

.23924 

4.1799 

1182.5 

282.90 

881.29 

210 84 

+ 

88 

104 

211.87 

330.48 1 

258.52 

.24132 

4.1438 

1182.7 

285.42 

880.78 

212 55 

+ 

89 

105 

213.91 

331.18 

256.31 

.21340 

4.1083 

1182.9 

287.93 

880.27 

214.26 

+ 

90 

106 

215.94 

331.87 

251.14 

.24548 

4.0736 

1188.2 

290.45 

879.77 

215.96 

+ 

91 

107 

217.98 

332.56 

252.01 

.24750 

4.0394 

1183.4 

292.94 

879.27 

217.66 

+ 

92 

108 

220.02 

333.24 

249.92 

.24963 

4.0058 

1183.6 

295.41 

879.79 

219.36 

+ 

93 

109 

222.05 

333.92 

247.87 

.25] 69 

3.9731 

1183.8 

297.91 

878.28 

221.05 

+ 

94 

110 

224.10 

334 59 

245.86 

.25375 

3.9408 

1188.9 

300.44 

877.80 

222.74 

+ 

95 

111 

226.13 

335.26 

243.8s 

.25581 

3.9091 

1184.2 

302.93 

877.31 

224.42 

+ 

96 

113 

230.20 

336.58 

240.03 

.25991 

3.8474 

1184.6 

307.90 

876.25 

227.74 

+ 

98 

114 

232.24 

387.23 

238.15 

.26204 

3.8100 

1181.8 

310.36 

875.88 

229.51 

+ 

99 

115 

284.28 

337.89 

236.31 

.26400 

3.7*78 

1185.0 

312.86 

875.40 

231.10 

+ 100 

120 

244.4 

341.0 

227.56 

.27421 

3.6475 

1185.9 

325.20 

873.09 

239.41 

+ 105 

125 

254.6 

344.1 

219.50 

.28422 

3.5184 

1186.9 

337.39 

870.85 

247.51 

+ 110 

130 

26-1.8 

347.1 

212.07 

.2J419 

3.4991 

1187.8 

349.44 

868.68 

255155 

+ 115 

135 

275.0 

350.0 

205.18 

.30406 

3.2880 

1188.7 

361.42 

866.56 

2&.4S 

+ 120 

140 

285.2 

352.8 

198.78 

.31385 

3.1862 

1189.5 

373.34 

864.49 

271.32 

|- 125 

145 

295.4 

355.6 

192.83 

.32354 

3.0308 

1190.4 

385.20 

862.48 

278.97 

+ 130 

150 

305.6 

358.4 

187.26 

.33315 

3.0001 

1191.2 

396.S6 

860.45 

286.66 

4-135 






























532 Properties of Water. 




Water,by the 

Author's Formula. 



f 

Tempt 
ol’ the 

Cent. 

rat tire 
Water. 

Falir. 

Volume. 
Water = 

1 at 40°. 

i 

Weight.! 
,hs. per 
mine ft.j 

1 

Bulk. 

Cubic 

feet per 
pound. 

Units o 

Tot a 
pound. 

f Heat ii 
32° to 2 
per 

cubic ft. 

i Water 
ro 

Latent 

pound. 

from 

per 
cu. ft. 

J*0 

rpQ 

V 

$ 

6 

II 

11' 

L 

L' 


179.2 

354.8 

1.11070 

56.166 

0.01780 

326.73 

18349 

3.927 

220.8 


180.7 

357.4 

1.11208 

56.098 

0.01782 

329.41 

18481 

4.010 

225.0 


182.2 

360.0 

1.11344 

56.031 

0.01784 

332.09 

18607 

4.090 

229.0 


183.7 

362.5 

1.11478 

55.965 

0.01787 

334.67 

18730 

4.168 

233.3 


185.0 

365.0 

1.11613 

55.900 

0.017S9 

337.24 

18850 

4.244 

237.2 


18(5.5 

367.4 

1.11742 

55.834 

0.01791 

339.72 

18966 

4.318 

241.0 


188.0 

369.8 

1.11869 

55.770 

0.01793 

342.19 

19080 

4.390 

244.6 


188.5 

372.0 

1.11993 

55.708 

0.01795 

344.46 

19190 

4.460 

248.5 


190.0 

374.2 

1.12109 

55.648 

0.01797 

346.73 

19296 

4.530 

252.1 


191.2 

376.4 

1.12227 

55.591 

0.01799 

349.00 

19399 

4.598 

255.7 


192.5 

378.5 

1.12343 

55.534 

0.01800 

351.16 

19501 

4.666 

259.1 


193.7 

380.6 

1.12456 

55.477 

0.01802 

353.33 

19602 

4.731 

262.5 


194.4 

382.6 

1.12561 

55.426 

0.01804 

355.39 

19698 

4.794 

265.7 


197.0 

386.6 

1.12783 

1 / 

0.01807 

359.54 

19885 

4.940 

272.8 


199.1 

390.4 

1.13000 

55.211 

0.01811 

363.48 

20068 

5.082 

279.8 


201.1 

394.0 

1.13210 

55.108 

0.01814 

367.20 

20236 

5.200 

286.6 


203.5 

397.6 

1.13301 

55.017 

0.01S17 

370.92 

20402 

5.318 

292.9 


205.0 

401.0 

1.13577 

54.926 

0.01821 

374.44 

20561 

5.437 

299.1 


20(5.8 

404.3 

1.13760 

54.838 

0.01824 

357.86 

20720 

5.558 

305.2 


208.7 

407.5 

1.13944 

54.752 

0.01826 

381.18 

20870 

5.679 

311.2 


210.2 

410.6 

1.11119 

54.670 

0.01829 

384.40 

21015 

5.800 

317.1 


211.9 

413.5 

1.14285 

54.590 

0.01832 

387.40 

21147 

5.903 

324.6 


213.6 

416.5 

1.14441 

54.514 

0.01834 

390.50 

21273 

6.006 

332.0 


215.1 

419.2 

1.14589 

54.440 

0.01837 

393.31 

21364 

6.109 

339.5 


216.7 

422.1 

1.14743 

o4.36 / 

0.01839 

396.31 

21510 

6.212 

346.7 


218.2 

424.8 

1.14897 

54.299 

0.01841 

399.11 

21622 

6.315 

353.8 


219.6 

427.4 

1.15050 

54.230 

0.01844 

401.82 

21751 

6.418 

356.9 


221.1 

430.0 

1.15202 

54.161 

0.01846 

404.52 

21876 

6.521 

359.9 


222.4 

432.4 

1.15339 

54.093 

0.01849 

407.02 

21997 

6.624 

362.8 


223.6 

434.9 

1.15481 

54.024 

0.01851 

409.63 

22114 

6.727 

365.6 


225.1 

437.3 

1.15621 

53.959 

0.01853 

412.13 

22238 

6.830 

368.5 


226.4 

439.6 

1.15764 

53.895 

0.O1856 

414.53 

22347 

6.926 

373.2 


227.7 

441.9 

1.15880 

53.834 

0.01858 

416.92 

22452 

7.020 

377.9 


228.9 

444.1 

1.16003 

53.777 

0.01859 

419.21 

22553 

7.111 

382.5 L 

230.2 

446.4 

1.16127 

53.721 

0.01861 

421.60 

22650 

7.200 

386.9 


231.4 

448.5 

1.16250 

53.667 

0.01863 

423.79 

22744 

7.288 

391.1 

232.5 

450.6 

1.16372 

53.614 

0.01865 

425.97 

22843 

7.374 

395.3 


233.6 

452.6 

1.16494 

53.563 

0.01867 

428.06 

22938 

7.459 

399.4 


231.7 

454.6 

1.16571 

53.513 

0 01869 

430.14 

23029 

7.542 

403.6 


235.9 

456.7 

1.16695 

53.455 

0.01871 

432.32 

23116 

7.623 

407.3 


237.0 

458.7 

1.16818 

53.406 

0.01872 

434.40 

23200 

7.700 

411.2 


238.0 

460.6 

1.16942 

53.352 

0.01874 

436.38 

23282 

7.787 

415.5 


239.0 

462.5 

1.17066 

53.293 

0.01876 

438.39 

23363 

7.893 

423.3 


241.1 

466.1 

1.17274 

5^. 108 

0.01881 

442.21 

23555 

8.113 

433.2 


244.1 

471.5 

1.17598 

53.027 

0.01886 

447.83 

23741 

8.329 

442.9 


246.5 

475.7 

1.17917 

52.900 

0.01890 

452.24 

23923 

8.541 

452.4 


248.8 

479.8 

1.18231 

52.768 

0.01895 

456.55 

74091 

8.747 

461.6 


253.1 

487.6 

1.18531 

52.588 

0.01901 

464.66 

24436 

9.060 

476.5 


257.2 

494.9 

1.18961 

52.430 

0.01907 

472.28 

24762 

9.381 

491.8 


261.0 

501.8 

1.19343 

52.264 

0.01913 

479.51 

25061 

9.710 

507.5 


263.5 

508.4 

1.19472 

52.102 

0.01019 

486.40 

25577 

10.00 

521.0 


268.1 

514.6 

1.20131 

51.943 

0.01925 

492.97 

25606 

10.37 

538.7 


271.9 

521.4 

1.20562 

51.787 

0.01931 

500.14 

25901 

10.74 

556.2 


273.3 

526.0 

1.20812 

51.642 

1.01936 

505.00 

26079 

11.00 

568.1 


277.5 

531.6 

1.21147 

51.498 

0.01942 

510.84 

26307 

11.242 

578.8 

- 






























































Properties op Steam. 


533 


Steam, by tl»e Author's Formula. 


Total 

lbs. 

l>er 

sq. 

inch. 

Pressure. 

Inches 

mer¬ 

cury. 

Temper¬ 

ature 

Pabi*. 

Volume. 
Water 
— 1 at 
40. 

Weight, 
lbs. per 
cubic 
foot. 

Hulk. 
Cubic 
feet per 
pound. 

Units 

Tota 

pound. 

of Heat 1 

1 per 
cubic ft. 

rom 32° 1 

Later 

pound. 

o T°. 

it per 
cubic ft. 

Pres¬ 

sure 

above 

atmo¬ 

sphere. 

P 

i 

J'O 

t 

$ 

€ 

H 

IV 

L 

IJ 

P 

140 

285.2 

354.8 

194.3 

0.3212 

3.1139 

1190.1 

381.88 

863.5 

277.0 

125 

145 

295.4 

357.4 

187.8 

0.3322 

3.0105 

1190.9 

395.16 

861.5 

275.8 

130 

150 

305.6 

360.0 

181.8 

0.3432 

2.9136 

1191.7 

408.38 

859.6 

294.5 

135 

155 

310.8 

362.5 

176.5 

0.3534 

2.8289 

1192.5 

421.54 

857.8 

303.2 

140 

160 

325.9 

365.0 

171.5 

0.3646 

2.7432 

1193.3 

435.08 

856.1 

312.1 

145 

165 

336.0 

367.4 

166.6 

0.3756 

2.6617 

1194.0 

448.64 

854.3 

321.0 

150 

170 

346.3 

369.8 

161.1 

0.0871 

2.5831 

1194.7 

462.22 

852.5 

329.9 

155 

175 

356.5 

372.0 

157.0 

0.3973 

2.5171 

1195.4 

475.80 

851.0 

338.7 

160 

180 

366.7 

374.2 

152.8 

0.4075 

2.4541 

1196.1 

488.96 

849.4 

347.1 

165 

185 

376.9 

376.4 

148.8 

0.4182 

2.3916 

1196.8 

502.10 

847.8 

355.5 

170 

190 

378.1 

378.5 

145.0 

0.4292 

2.3299 

1197.4 

515.20 

846.2 

363.9 

175 

195 

387.3 

380.6 

141.5 

0.4409 

2.2634 

1198.1 

528.27 

844.8 

372.4 

180 

200 

407.4 

382.6 

138.1 

0.4517 

2.2137 

1198.7 

542.07 

843.3 

381.0 

185 

210 

427.8 

386.6 

132.0 

0.4719 

2.1192 

1199.8 

568.40 

840.3 

398.0 

195 

220 

448.2 

390.4 

126.3 

0.4935 

2.0265 

1201.0 

574.70 

837.5 

414.8 

205 

230 

468.5 

394.0 

120.8 

0.5165 

1.9360 

1202.2 

620.96 

835.0 

431.3 

215 

240 

488.9 

397.6 

116.1 

0.5364 

1.8646 

1203.2 

647.41 

832.3 

447.9 

225 

250 

509.3 

401.0 

111.7 

0.5595 

1.7874 

1204.2 

673.85 

829.8 

464.4 

235 

260 

529.7 

404.3 

107.5 

0.4803 

1.7230 

1205.2 

700.28 

827.4 

480.8 

245 

270 

550.0 

407.5 

103.7 

0.6016 

1.6621 

1206.2 

726.66 

825.0 

497.1 

255 

280 

570.4 

410.6 

100.2 

0.6238 

1.6031 

1207.2 

753.04 

822.8 

513.3 

265 

290 

590.8 

413.5 

97.01 

0.6459 

1.5481 

1208.1 

779.40 

820.7 

529.4 

275 

300 

611.1 

416.5 

91.22 

0.6681 

1.4967 

12U9.0 

805.74 

S1S.6 

545.4 

285 

310 

631.5 

419.2 

91.13 

0.6896 

1.4499 

1209.8 

832.96 

816.5 

561.4 

295 

320 

651.9 

422.1 

88.21 

0.7107 

1.4071 

1210.6 

858.36 

814.4 

577.3 

305 

330 

672.3 

424.8 

85.44 

0.7302 

1.3695 

1211.5 

884.63 

812.4 

593.2 

315 

340 

692.6 

427.4 

83.19 

0.7547 

1.3250 

1212.3 

910.89 

810.5 

608.9 

325 

350 

713.0 

430.0 

80.99 

0.7745 

1.2915 

1213.1 

937.13 

808.6 

624.5 

335 

360 

733.4 

432.4 

78.84 

0.7943 

1.2590 

1213.9 

963.34 

806.9 

640.2 

315 

370 

753.8 

434.9 

76.74 

0.8146 

1.2275 

1214.7 

989.51 

805.1 

655.8 

355 

380 

774.1 

437.3 

74.66 

0.8353 

1.1968 

1215.5 

1015.7 

803.4 

671.3 

365 

390 

794.5 

439.6 

72.90 

0.8626 

1.1597 

1216.2 

1041.8 

801.7 

686.7 

375 

400 

814.9 

441.9 

71.19 

0.8745 

1.1434 

1216.8 

1067.9 

800.0 

702.0 

385 

410 

835.2 

444.1 

69.52 

0.8952 

1.1170 

1217.4 

1094.0 

799.4 

717.2 

395 

420 

855.6 

446.4 

67.90 

0.9142 

1.0938 

1218.0 

1120.2 

797.7 

732.4 

405 

430 

876.0 

448.5 

66.34 

0.9400 

1.0634 

1218.7 

1146.3 

795.0 

747.6 

415 

440 

896.4 

450.6 

64.91 

0.9599 

1.0417 

1219.4 

1172.3 

793.5 

762.8 

425 

450 

916.7 

452.6 ‘ 

63.55 

0.9804 

1.0201 

1220.1 

1198.3 

792.0 

777.9 

435 

460 

937.1 

454.6 

62.22 

1.0007 

0.9993 

1220.7 

1224.3 

790.5 

792.9 

445 

470 

957.5 

456.7 

60.94 

1.0211 

0.9793 

1221.3 

1250.4 

789.0 

807.8 

455 

480 

977.8 

458.7 

59.72 

1.0446 

0.9573 

1221.9 

1:276.5 

787.5 

822.7 

465 

49 ) 

998.2 

460.6 

58.54 

1.0652 

0.9388 

1222.5 

1302.3 

786.1 

837.4 

475 

500 

1018.6 

462.5 

57.45 

1.0859 

0.9209 

1223.0 

1328.1 

784.7 

852.1 

485 

525 

1069.5 

466.1 

54.81 

1.1381 

0.8786 

1224.5 

1392.6 

782.3 

881.8 

510 

550 

1120.4 

471.5 

52.47 

1.1890 

0.8410 

1225.8 

1456.9 

778.0 

921.3 

535 

575 

1171.4 

475.7 

50.32 

1.2397 

0.8066 

1227.2 

1521.0 

775.0 

960.4 

560 

600 

1222.3 

479.8 

48.35 

1.2901 

0.7751 

1228.3 

1584.8 

771.8 

1000 

585 

650 

1324.2 

487.6 

44.75 

1.3943 

0.7172 

1230.6 

1709.5 

766.0 

1082 

635 

700 

1426.0 

494.9 

41.70 

1.4961 

0.6684 

1232.7 

1933.8 

760.4 

1157 

685 

750 

1527.9 

501.8 

39.05 

1.5977 

0.6259 

1234.9 

2057.7 

755.4 

1234 

735 

800 

1629.8 

508.4 

36.73 

1.6986 

0.5887 

1237.0 

2101.2 

750.6 

1307 

785 

850 

1731.6 

514.6 

34.6S 

1.7989 

0.5554 

1238.9 

2228.3 

745.9 

1374 

835 

900 

1833.5 

521.4 

32.87 

1.8979 

0.5269 

1241.0 

2355.4 

740.0 

1435 

885 

950 

1935.5 

526.0 

31.21 

1.9992 

0.5002 

1242.4 

2482.5 

737.4 

1490 

935 

1000 

2037.2 

531.6 

29.73 

2.0986 

0.4765 

1243.5 

2609.6 

732.3 

1538 

985 





































534 


Mean Pressure. 


Mean Pressure above Vacuum of Expanding Steam. 


Absolute 

Steam 

Pressure. 

P 

1.333 

3 

i 

Grade 

1.5 

Steam 

% 

3 

of Ex pa 

Cut-off 

5 

S 

nsion of 

2 

at l froi 

h 

Steam, 

2.666 

t Begin i 

3 

5 

denoted 

3 

ling of 8 

4 

by X. 

4 

>troke. 

\ 

8 

1 

5 

25 

24.130 

23.481 

22.938 

21.164 

18.567 

17.488 

19.913 

9.6232 

30 

2S.956 

28.100 

27.521 

25.396 

22.280 

20.986 

17.897 

11.548 

oo 

33.782 

32.874 

32.110 

29 630 

25.992 

24.484 

20.880 

13.472 

40 

38.608 

37.468 

36.700 

33.862 

28.964 

27.982 

23.862 

15.396 

45 

43.474 

42.151 

41.287 

38.095 

32.677 

31.479 

26.845 

17.320 

50 

48.262 

46.835 

45.875 

42.328 

37.133 

34.977 

29.828 

19.246 

55 

53.088 

51.518 

50.462 

46.561 

40.846 

38.474 

32.811 

21.170 

60 

57.914 

56.202 

55.050 

50.794 

44.559 

41.972 

35.794 

23.095 

Go 

62.740 

60.885 

59.637 

65.027 

48.273 

45.470 

38.777 

25.020 

70 

67.566 

65.569 

64.225 

59.260 

51.986 

48.967 

41.760 

26.944 

75 

72.393 

70.252 

68.812 

63.493 

55.700 

52.465 

44.743 

28.S69 

80 

77.216 

74.936 

73.400 

67.726 

59.413 

55.963 

47.726 

30.794 

85 

82.042 

79.619 

77.987 

71.959 

63.126 

59.461 

50.709 

32.718 

90 

86.866 

85.303 

82.574 

76.192 

66.840 

62.958 

53.692 

34.643 

95 

91.699 

89.986 

87.163 

80.425 

70.553 

66.456 

56.675 

36.568 

100 

96.524 

93.670 

91.750 

84.657 

74.267 

69.954 

59.657 

38.493 

105 

101.35 

98.353 

96.337 

88.S90 

77.981 

73.451 

62.640 

40.417 

110 

106.17 

103.04 

100.92 

93.123 

81.694 

76.949 

65.622 

42.342 

115 

111.00 

107.72 

105.51 

97.356 

85.407 

80.447 

68.606 

44.267 

120 

115.83 

112.40 

110.10 

101.59 

89.121 

83.944 

71.589 

46.191 

125 

120.65 

117.08 

114.68 

105.82 

92.834 

87.442 

74.572 

48.116 

130 

125.48 

121.77 

119.27 

110.05 

96.548 

90.940 

77.555 

50.041 

135 

130.30 

126.45 

123.86 

114.28 

100.26 

94.437 

80.53S 

51.966 

140 

135.13 

131.13 

128.45 

118.52 

103.97 

97.935 

83.520 

53.890 

145 

139.96 

135.82 

133.03 

122.75 

107.68 

101.43 

86.502 

55.S15 

150 

144.78 

140.50 

137.62 

126.98 

111.40 

104.93 

89.485 

57.739 

155 

149.60 

145.18 

142.20 

131.22 

115.11 

108.42 

92.468 

59.663 

160 

154.43 

149.87 

146.79 

135.45 

118.82 

111.92 

95.451 

61.588 

165 

159.26 

154.55 

151.38 

139.68 

122.54 

115.42 

98.434 

63.513 

170 

164.08 

159.23 

155.97 

143.92 

126.25 

118.92 

101.41 

65.437 

175 

168.91 

163.92 

160.55 

148.15 

129.96 

122.42. 

104.40 

67.362 

180 

173.73 

168.60 

165.14 

152.38 

133.68 

125.91 

107.38 

69.287 

185 

178.56 

173.28 

169.73 

156.61 

137.39 

129.41 

110.36 

71.212 

190 

183.39 

177.97 

174.32 

160.85 

141.10 

132.91 

113.35 

73.136 

195 

188.21 

182.65 

178.90 

165.08 

144.S2 

136.41 

116.33 

75.061 

200 

193.04 

187.34 

183.50 

169.31 

148.53 

139.91 

119.31 

76.986 

210 

202.69 

196.71 

192.68 

177.78 

155.96 

146.90 

125.27 

80.835 

220 

212.34 

205.08 

201.85 

186,25 

163.39 

153.90 

131.24 

84.684 

230 

221.99 

215.45 

211.03 

194.71 

170.82 

160.89 

137.20 

88.534 

240 

231.65 

224.81 

220.20 

203.18 

178.23 

167.89 

143.17 

92.383 

250 

241.30 

234.18 

229.38 

211.64 

185.67 

174.88 

149.13 

96.232 

200 

250.96 

243.55 

238.55 

220.11 

193.18 

181.88 

155.11 

100.08 

270 

260.61 

252.91 

247.73 

228.57 

200.52 

188.87 

161.07 

103.93 

280 

270.26 

262.28 

256.90 

237.04 

207.95 

195.87 

167.04 

107.78 

300 

289.56 

281.00 

275.24 

253.96 

222.80 

209.86 

178.97 

115.48 































Mean Pressure. 


535 


Mean 

Pressure for High-Pressure Engines Above Atmosphere. 



Grade of Expansion of Steam, 

denoted by X. 


Pressure 

above 

U 

14 

| 1.6 

2 

| 2 § 

3 


8 

Atnio- 









sphere. 


Steam Cut-off at l from Beginning of Stroke. 


P 

3 

2 

6 

1 

3 

1 

X 

1 

4 

3 

8 

5 

8 

3 

4 

5 

25 

23.908 

22.768 

22.000 

19.162 

14.264 

13.282 

9.162 

0.696 

30 

28.774 

27.451 

26.587 

23.395 

17.977 

16.779 

12.145 

2.620 

35 

33.562 

32.135 

31.175 

27.628 

22.433 

20.277 

15.128 

4.546 

40 

38.388 

36.818 

35.762 

31.861 

26.146 

23.774 

18.111 

6.470 

45 

43.214 

41.502 

40.350 

36.094 

29.859 

27.272 

21.094 

8.395 

50 

48.040 

46.185 

44.937 

40.327 

33.573 

30.770 

24.077 

10.320 

55 

52.866 

50.869 

49.625 

44.560 

37.286 

34.267 

27.060 

12.244 

60 

57.693 

55.552 

54.112 

48.793 

41.000 

37.765 

30.043 

14.169 

65 

62.516 

60.236 

58.700 

53.026 

44.713 

41.263 

33.4126 

16.094 

70 

67.342 

64.919 

63.287 

57.259 

48.426 

44.761 

36.009 

18.018 

75 

72.166 

70.603 

67.874 

61.492 

52.140 

48.258 

38.992 

19.943 

80 

76.999 

75.286 

72.463 

65.725 

55.853 

51.756 

41.975 

21.868 

85 

81.824 

78.970 

77.050 

69.957 

59.567 

55.254 

44.957 

23.793 

90 

86.650 

83.653 

81.637 

74.190 

63.281 

58.751 

47.940 

25.717 

95 

91.470 

88.340 

86.220 

78.423 

66.994 

62.249 

50.922 

27.642 

100 

96.300 

93.020 

90.810 

82.656 

70.707 

65.747 

53.906 

29.567 

105 

101.13 

97.700 

95.400 

86.890 

74.421 

69.244 

56.889 

31.491 

no 

105.95 

102.38 

99.980 

91.120 

78.134 

72.742 

59.872 

33.416 

115 

110.78 

107.07 

104.57 

95.350 

81.848 

76.240 

62.855 

35.341 

120 

115.60 

111.75 

109.16 

99.580 

85.560 

79.737 

65.838 

37.266 

125 

120.43 

116.43 

113.75 

103.82 

89.270 

83.235 

68.820 

39.190 

130 

125.26 

121.12 

118.33 

108.05 

92.980 

86.730 

71.802 

41.115 

135 

130.08 

125.80 

122.92 

112.28 

96.700 

90.230 

74.785 

43.039 

140 

134.90 

130.48 

127.50 

116.52 

100.41 

93.720 

77.768 

44.963 

145 

139.73 

135.17 

132.09 

120.75 

104.12 

97.220 

80.751 

46.888 

150 

144.56 

139.85 

136.68 

124.98 

107.84 

100.72 

83.734 

48.813 

155 

149.38 

144.83 

141.27 

129.22 

111.85 

104.22 

86.710 

50.737 

160 

154.21 

149.22 

145.85 

133.45 

115.26 

107.72 

89.700 

52.662 

165 

159.03 

153.90 

150.44 

137.68 

118.98 

111.21 

92.680 

54.587 

170 

163.86 

158.58 

155.03 

141.91 

122.69 

114.71 

95.660 

56.812 

175 

168.69 

163.27 

159.62 

146.15 

126.40 

118.21 

98.650 

58.436 

180 

173.51 

167.95 

164.20 

150.38 

130.12 

121.71 

101.63 

60.361 

185 

178.34 

172.64 

168.80 

154.81 

133.83 

125.21 

104.61 

62.286 

190 

183.16 

177.32 

173.39 

158.81 

137.54 

128.71 

107.59 

64.210 

195 

187.99 

182.01 

177.98 

163.08 

141.26 

132.20 

110.57 

66.135 

200 

192.81 

186.69 

182.58 

167.31 

144.97 

135.70 

113.55 

68.060 

210 

202.46 

195.06 

191.74 

175.78 

152.40 

142.70 

119.52 

71.908 

220 

212.11 

205.43 

200.93 

184.24 

159.83 

149.69 

125.48 

75.758 

230 

221.77 

214.79 

210.10 

192.71 

167.24 

156.69 

131.39 

79.603 

240 

231.42 

224.16 

219.27 

201.17 

174.68 

163.68 

137.41 

83.456 

250 

241.08 

233.57 

228.45 

209.64 

182.19 

170.68 

143.39 

87.300 

260 

250.73 

242.89 

237.62 

218.10 

189.53 

177.67 

149.35 

91.150 

270 

260.38 

252.26 

246.79 

226.57 

196.96 

184.67 

155.32 

95.000 

280 

270.04 

261.62 

255.94 

235.03 

204.39 

191.66 

161.29 

98.860 

300 

289.34 

280.35 

264.30 

251.95 

219.24 

205.56 

173.22 

106.550 







































53f) Expansion Table IT. for Double Cylinder Expansion Engines. 


Mean Pressure f during the Expansion. 


Ptvs. 

P. 

i i 

l 

3 

& 

2 


1 2 

I 3 

i 

i 

30 

28-549 

24 

23-50 

20-79 

17-60 

16-31 

13-86 

8-9097 

35 

33-308 

28 

27-41 

24-25 

20-54 

19-02 

16-17 

10-394 

40 

38-066 

32 

31-83 

27-72 

23-47 

21-73 

18-48 

11-879 

45 

42-824 

36 

35-25 

31-18 

26-40 

24-46 

20-79 

13-364 

50 

47-5S2 

40 

39-16 

34-65 

29-33 

27-16 

23-10 

14-849 

55 

52-340 

44 

43-08 

38-11 

32-24 

30-17 

25-41 

16-334 

60 

57-098 

48 

47-00 

41-58 

35-20 

32-62 

27-72 

17-819 

65 

61-853 

52 

50-91 

45-04 

38-14 

35-33 

30-03 

19-303 

70 

66-616 

56 

54-83 

48-51 

41-07 

38-04 

32-34 

20-78S 

75 

71-371 

60 

58-75 

51-90 

44-00 

40-75 

34-65 

22-263 

80 

76-128 

64 

62-66 

55*44 

46-94 

43-47 

36-96 

23-758 

85 

80-885 

68 

66-18 

58-90 

49-87 

46-19 

39-27 

25-243 

90 

86-448 

72 

70-50 

62-37 

52-80 

4S-93 

41-58 

26-729 

95 

90-391 

76 

74-41 

65-73 

55-73 

51-62 

43-89 

28-213 

100 

95-166 

80 

78-33 

69-30 

58-66 

54-33 

46-20 

29-699 

105 

99-910 

84 

82-24 

72-76 

61-57 

57-33 

48-51 

31*183 

110 

104-6S 

88 

86-16 

76-23 

64-48 

60-35 

50-82 

32-669 

115 

109-40 

92 

90-08 

79-69 

67-44 

62-79 

53-13 

34 153 

125 

118-95 

100 

97-91 

97-02 

73-34 

67-95 

57-75 

37-122 

140 

133-23 

112 

109-6 

97*02 

82-14 

76-08 

64-68 

41-576 

150 

142-74 

120 

117-5 

103.9 

88-00 

81-50 

69-30 

44-548 

200 

190-32 

160 

156-6 

138.6 

117-3 

108-6 

92-4 0 

59-398 

250 

237-07 

200 

195-7 

173.2 

146-6 

135.8 

115-5 

74-247 

300 

28S-16 

240 

235-0 

207.9 

176-0 

163.1 - 

" 13S-6 

89-097 


Table III. Economy of Expansion and high Steam. 
Fuel saved or effect gained per cent. 


Pres.; 

P -J 

01 t I 

i 

£ 1 
8 1 

i 

1 . § 

1 

i 

30 

0 

12 

29-5 

32 

41 

49-3 

52 

58 

67-5 

35 

1-6 

13-6 

31 

33-6 

42-6 

51 

53-6 

59-6 

69-1 

40 

2-5 

14-5 

32 

34-5 

43-5 

51-8 

54-5 

60-5 

70 

45 

3-4 

15-4 

33 

35-4 

44-4 

52-7 

55-4 

61-4 

71 

50 

4.3 

16 3 

33-8 

363 

45-3 

53-6 

56-3 

62-3 

71-8 

55 

5-2 

17-2 

34-7 

37-2 

46-2 

54-5 

57-2 

63-2 

72-7 

(50 

6 

18 

35-7 

38 

47- 

55-3 

58 

64 

73-5 

(55 

6-7 

18-7 

36.2 

38-7 

47-7 

56 

58-7 

64-7 

74-2 

70 

7-3 

19.3 

36.8 

39-3 

48-3 

56-6 

59-3 

65-3 

74-8 

75 

7-8 

19-8 

37-3 

39-8 

48-8 

57-1 

59-8 

65-8 

753 

80 

8-5 

20-5 

38 

40*5 

49-5 

57-8 

60-5 

665 

76 

85 

9 

21 

38-5 

41 

50 

58-3 

61 

67 

76-5 

001 

9-5 

21-5 

39 

41-5 

50-5 

58-8 

61-5 

67*5 

77 

95 

10 

22 

39-5 

42 

51 

59-3 

62 

68 

77-5 

100! 

10-4 

22-4 

40 

*42.4 

51.4 

59-7 

62-4 

68-4 

78 

105 

! 10*7 

22-7 

40-2 

42-7 

51-7 

60- 

62-7 

68-7 

78-2 

115| 

I 11 

23 

40*5 

43 

52 

60-3 

63 

69 

78-5 

1251 

11-7 

23-7 

41-2 

43-7 

52-7 

61 

63-7 

69-7 

79-2 

150 

14 

26 

43-5 

46 

55 

63-3 

66 

72 

81-5 

200 

16 

28 

455 

48 

57 

65-3 

68 

74 

83-5 

250 

17*7 

29-7 

46-2 

49-7 

58-7 

67 

69-7 

75-7 

85-2 

300 

19 

31 

48-5 

51 

60 

1 

68-3 

71 

77 

86.5 




































































CONSrMPTION OF FUEI, 537 






Table IV. 





Consmnp tion 

of Coal in pounds per horse power per hoar. 




Grade of Expansion. 




Pres, 


1 l 

1 

, 3 

l 

5 

2 

3 

1 X 

P. 

0 


3 

8 

Z 


3 


1 * 

lbs. 

lbs, 

lb 8, 

lbs . 

lbs. 

lbs, 

lbs, 

lbs, 

lbs, 

lbs. 

30 

5-6 

4-93 

3-95 

3-81 

3-30 

2-84 

2-69 

2-35 

1-82 

35 

5-51 

4-84 

3-86 

3 72 

3-21 

2-74 

2-60 

2-26 

1-73 

40 

5-46 

4-79 

3-81 

3-67 

3'16 

2-70 

2-55 

2-21 

1-68 

45 

5-41 

4-73 

3-75 

3-62 

3-11 

2-65 

2-50 

2-16 

1-62 

50 

5*36 

4-68 

3-71 

3-57 

3-06 

2-60 

2-45 

2-11 

1-58 

55 

5-31 

4 63 

3-66 

3-51 

3-01 

2-55 

2-40 

2-06 

1-53 

60 

5-26 

4-59 

3-60 

3-47 

2-97 

2-50 

2-35 

2-02 

1 '49 

65 

5-20 

4-55 

3-57 

3-43 

2-93 

2-46 

2-31 

1*98 

1*45 

70 

5-19 

4*52 

3-54 

3'40 

2-90 

2-43 

2-28 

1-94 

1*41 

75 

5*16 

4.49 

3-51 

3-37 

2-87 

2-40 

2-25 

1*91 

1-39 

80 

5-12 

4-45 

3-47 

3-33 

2-83 

2-36 

2*21 

1-88 

1 *35 

85 

5-09 

4-42 

3-44 

3-30 

2-80 

2-33 

2-18 

1-85 

1-32 

90 

507 

4-39 

3-41 

3-28 

2-77 

2-31 

2-16 

1-82 

1-29 

95 

5*04 

4-37 

3 39 

3-25 

2-74 

2-28 

2-13 

1'79 

1-26 

100 

5*01 

4-34 

3-36 

3’23 

2-72 

2-26 

2 10 

1*77 

1*23 

105 

5-00 

4-32 

3-35 

3'21 

2-70 

2-24 

2-09 

1-75 

1-22 

115 

4-98 

4-31 

3-33 

3-19 

2-69 

2-22 

2-07 

1-73 

1-20 

125 

4-94 

4-27 

3-29 

3-15 

2-65 

2-19 

2-03 

1-70 

1-17 

150 

4*81 

414 

3-16 

3-02 

2-52 

2-05 

1-90 

1-57 

1-04 

200 

4-70 

403 

3-05 

2-91 

2 41 

1-94 

1-79 

1-46 

0-92 

250 

4-60 

3 93 

3-01 

2-81 

2-31 

1-85 

1-70 

1-36 

0-83 

300 

4 54 

3-87 

2-89 

2-75 

2-24 

1-78 

1 62 

1-29 

0-75 





Table V. 






Consumption of Coal in tons per 100 horses in 24 hours. 

Pres. 


4 . 

1 

3 

1 

& 

2 

a 

1 

P. 

0 

t 

3 

8 

2 

8 

3 

t 

8 

lbs. 1 

tons 

tons, 

tons, 

tons, 

tons, 

tons, 

tons. 

tons, 

tons, 

30 

6-001 5-29 

4-23 

4-09 

3-54 

3-04 

2-88 

2-52 

1-95^ 

35 

5-90 

5-19 

4-13 

3-99 

3*44 

2-94 

2-79 

2-42 

1-86 

40 

5-85 

513 

408 

3-93 

3-39 

2-90 

2 73 

2 37 

1-80 

45 

5-80 

5-07 

4-02 

3-88 

3-34 

2-84 

2-68 

2*31 

1*73 

50 

5-75 

5 01 

3-97 

3-83 

3-28 

2-79 

2-63 

2-26 

1-69 

55| 

5-70 

4-96 

3-92 

3-77 

3-22 

2-73 

2-57 

2*21 

1-64 

60 

5-64 

4-92 

3-87 

3*72 

3-18 

2-68 

2-52 

2-17 

1-60 

05' 

5-58 

4-88 

3S2 

3-68 

314 

2-63 

2-48 

2-12 

1 -55 

70, 

5-561 

4-84 

3-79 

3-64 

3-11 

2-60 

2-44 

2-08 

1*51 

75j 

5-53 

4 81 

3-76 

3-61 

3-07 

2 57 

2-41 

2-05 

1-49 

80 

5-49 

4-77 

3-72 

3*57 

3-03 

2*53 

2-37 

2-01 

1-44 

85 

5-46 

4-74 

3 69 

3-54 

3-00 

2'50 

2-33 

1-98 

1-41 

90 

5-43 

4-70 

3-66 

3-51 

3-97 

2-47 

2-31 

1-95 

1-38 

95! 

5-40 

4-68 

3-63 

3*48 

2-94 

2-44 

2-28 

1-92 

1-35 

100, 

5-37 

4-65 

3-60 

3-46 

2-91 

2-42 

2-26 

1-90 

1 32 

105| 

5-36 

4 63 

3-59 

3*44 

2-89 

2-40 

2-24 

1-88 

131 

11 5| 

5-34 

4-61 

3-57 

3-42 

2-88 

2-38 

2-22 

1-85 

1-29 

125 

5-30 

4-58 

3-53 

3-38 

2 84 

2-34 

218 

1-82 

1-25 

150 

5-16 

4-44 

3.39 

3'34 

2-81 

2-30 

2*04 

1-68 

111 

200 

5*04 

4*32 

3-27 

312 

2'59 

219 

1-92 

1-56 

0 99 

250 

4-93 

4-21 

3-22 

3-01 

2-47 

2-09 

1-82 

1-46 

0-89 

300 

4-87 

4*15 

310 

2 95 

2'40 

2-01 

174 

1 38 

0-83 















































































538 


Expansion of Steam. 


EXPANSION OP STEAM. 

In order to save steam, or more correctly to employ its effect to a higher 
degree, the admittance of steam to the cylinder is shut off when the piston 
has moved a part of the stroke ; from the cut-off point the steam acts ex¬ 
pansively with a decreasing pressure on the piston, as represented by the 
accompanying figure. 

Let the steam be cut off at i of the stroke, and 
Aa represent the total pressure, say 20 pounds 
per square inch which will continue to the point 
E where the admittance of steam is shut off at 
one-third the stroke S. The steam Aa eE, is now 
acting expansively on the piston, and the pres¬ 
sure decreases as the volume increases, when the 
piston has attained Cc or two-thirds of S, the 
pressure C'c=10 pounds, only half the pressure 
Aa 20 because the volume Aa eE is only half of 
Aa cC, and so on until the piston has attained B b 
the pressure = X2d=6 66 pounds. 

The mean pressure, or the effectual pressure, 
throughout the stroke, will be about 13 33 pounds 
per square inch, or 66 per cent., but the quantity 
of steam used is only 33 per cent., hence 33 per 
cent, is gained by using the steam expansively. 

I = part of the stroke S in feet, at which the steam is cut off. 

P= pressure per square inch under full admittance of steam. 

F= mean pressure per square inch throughout the stroke S. 

/=mean pressure per square inch during the expansion, which in 
double expansion cylinder engines will be the average pressure per 
square inch on the large piston A. 
p — end pressure per square inch after expansion. 

S= stroke of the cylinder Piston in feet. 

!■=£[«(*. P = p s '. 

The following Tables are calculated from these formulas. 

On page 594 el svq. will lie found a rational and practical method of com¬ 
puting the most economic number of expansions for steam-engines of any 
type, which is abstracted from The Relative Proportions of the Steam-Engine , by 
Professor Marks. 

Example 3. Required the mean pressure /=! for an initial pressure 
P=43 lbs. under y A expansion! 

For P = 40 lbs. /= 18-48 
P = 30 or 3 lbs. /= 1-38 



Table II. 


P — 43 lbs. f= 19 86 the answer. 

The effect gained or fuel saved by expansion and high steam is calculated 
from the following formulae, in which it is supposed as a unit the work 
of an engine with P = 30 pounds per square inch, or an indicated pressure 
of 15 lbs. without expansion. 

c = per cent or 100, of effect gained or fuel saved. 


For expansion c = 100 (1- 


IP 

SF 


, . . , . 26490, 

). For high steam c = 100 (1— ^ ). 


The preceeding Table III. is calculated from these formulae, in which 
the first line from 30 contains the economy per cent, from expansion 
alone, and the column o contains the economy per cent, from high steam 
above P = 30lbs. The balance of the table contains the conjoined economy 
of expansion and high steam. Required tlie conjoined economy ofP=90 
lbs. under £ expansion? 50'5 per cent, the answer. 





























Inertia of Reciprocating Masses. 


539 


Inertia of Reciprocating Masses. 



When bodies are moved rapidly forward and backward like the reciprocating 
parts of a steam-engine, the force of inertia plays an important part in the ope¬ 
ration. The reciprocating parts must be started and stopped at each end of the 
cylinder. The accompanying illustration shows the inertia diagram a b drawn in 
the cylinder; the ordinates drawn from the centre line represent the force of in¬ 
ertia, which is greatest at the ends and vanishes to nothing near the middle of the 
stroke. For a definite length of connecting-rod the inertia is greater at the back 
than at the front end of the cylinder; but if the connecting-rod was infinitely 
long (like in slot motion), the inertia would be alike at both ends of the 
stroke. 

F — force in pounds consumed in starting the reciprocating parts at the 
beginning of the stroke, which force is restored by bringing the 
moving masses to rest at the end of the stroke. 

IF = weight in pounds of all the reciprocating parts. 

R = radius of the crank in feet. 

L = length of connecting-rod in feet. 
n = revolutions of crank per minute. 

„ IFRn 2 

lorceof inert,a: F= 2mM .1. 


Examples 1 and 2. The reciprocating parts of an engine weigh TF = 1000 
pounds. Radius of crank R = 1 foot, n = 200 revolutions per minute. Required 
the force of inertia at the back and front ends of the stroke? 


F- 


1000 XIX 200 2 

2933.54 


= 13635 pounds. 


d = distance in fraction of a foot from the centre of the stroke to where the 
inertia diagram crosses t he centre line. 


d = L- -(- R^ — L. . . . . .3. 

The piston and the other reciprocating parts move fastest when the crank 
and connecting-rod are at right angles. 

The velocity of the piston when the crank passes at d is to the velocity of 
the piston when the crank passes at c as Ce is to Cf. Where tiie direction 
of the connecting-rod crosses the line CO is the measure of the velocity of 
the piston. 

For a connecting-rod of infinite length the formula for inertia is 

^ IF R n- 

F — --—» 

2933.54 

l 

which is the same as the formula for centrifugal force. 





















540 


Force ahd Air Pumps. 



Force or Feed Pumps. 

Approximate Formulae. 

^ — ptroke^ } of thc f° rce *P ura Pj sin ?' e acting. 

i-V)k« teF } of the ste:im -cylinder piston, in inches, double acting, 

F= volume of steam given in the table at the given pressure. 

The stroke of the steam-piston is only that under which steam is fully admitted 
to the cylinder. 


d — 2D A / 

AM 


4,5. 


-« - * -ol • • 

Vs Vd7 

Slip-water included in the formulas. 

Example. Required, the diameter of a force-pump having the same stroke as the 
cylinder piston s = 38 inches, diameter of cylinder Z> = 30 inches. The steam is cut 
off at £ the stroke, and the steam pressure + 50 pounds per square inch. Here V 
= 437, and S= 19 inches, because steam is cut off at g- the stroke. 


d= 2X30. 




19 


= 2.03 inches. 


\ 437 X 3S 

To liml tlie Quantity of Condensing Water. 

q = condensing water of temp, t in cubic feet._1.4 Q 990 + T — 1'\ 

Q — steam of temperature T in cubic feet. 2 .. '>* 

(' = temperature in the condenser. ^ • 


6 . 


Dimensions of tlie Air-Pump. 


d = diameter) of the air-pump, 
s = stroke J single acting. 

D = diameter (of the steam cylinder, 
S= stroke j double acting. 


d = 2.3 £> 


M 


ls( 990 + 7’— t') 


V s{t'—t ) 


Assume V = 100°, and t — 50°, we shall have— 
Single acting air-pumps. 


rf = 0.326O A ®«+JQ, 

\ Vs 

£(940 + T) 


s = 0.106 Z)2 


Vd? 


8 . 


9. 


Double acting air-pumps. 
d = 0.23 

s = 0.053 Z)2 


St 940 + T) 
Vs 

8(940 + r) 
Vd* ’ 


10 . 


11 . 


Example. A single acting air-pump is to be constructed for an engine D = 
3S inches, S= 45 inches stroke of the cylinder; the stroke of the air-pump can be 
32 inches, and tho exhaust steam is 2G1°. Required, the diameter of the air- 
pump? F=767. 


d -- 0.326 X 38 


(45(940 + 261). = 18 .38 inches. 
\ 767 X 32 


JQfjp* Slip-water included. Tand V must be taken for the exhaust steam, as the 
steam may have worked expansively; the area of the foot valve must be calcu¬ 
lated from the following formulas. 

Foot Valve in tlie Air-Pump. 

To enable an air-pump to work well and with the greatest advantage, it is neces¬ 
sary to pay particular attention to the following formulas. The force by which 
the water is driven from the condenser through the foot valve into the air-pump 
is limited by the pressure in the condenser; this absolute pressure is the vacuum 
subtracted from 14.7 pounds; it is noted in the third column, where the tempera¬ 
ture in the condenser is opposite, in the first column. Every pound of this pres¬ 
sure per square inch balances a column of water 27 inches high, which is the head 
that presses the water from the condenser. 























Air-Pump. 


541 

—i 


Foot-Valves In Air-Pumps. 

= area of the air-pump piston. a = D S n (890-f 1). 

a = area of the foot-valve, or bucket-valve. 23000 m V V V 

33 = diameter of the air-pump-piston. »» = 0-6 to 0‘8 

tl = diameter of the foot-valve, when round. 

<S = stroke of air-pump piston, in feet. 

33 = pressure in the condenser at the temperature T. 
n = number of strokes of the air-pump piston per minute. 


& S n 

12, 

^3 >/ n 

15, 

" loo v 'W 

104/33 ’ 

^ ioo.VTj 

n& ’ 

13, 

100b VI 

n 30 a ’ 

16, 

n _100.lv/33 
&S * 

14, 

n _ 100bV$ 

33 a a ’ 

17. 


Example. The area of an air-pump-piston is 2-35 square feet, stroke of 
piston ^ = 36 feet, to make n = 40 strokes per minute, and the pressure to be 
It} = 3*2 pounds. Required the area of the foot-valve. 


a 


2-3 5X3-6X40 
100 sf 3-2~ 


l - 85 square feet. 


To Find the Velocity and Quantity of the Injection Water 
through the Adjutage into the Condenser* 

Letters denote. 

v — velocity in feet per second. 

h = head of the press water; wnen above, and — below the adjustage. 

F = vacuum, noted — or negative in the last column, but is positive in the 
formulas. 

q == quantity of water discharged in cubic feet, per second. 
a = area of all the holes in the adjustage in square feet. 

jj— ^e injection pipe, in feet. 

n = double strokes of cylinder-piston, or revolutions per minute. 

A, D, and S, dimensions of the steam cylinder, in feet. 

T = temperature, and v e= volume coefficient of the exhaust steam. 


q — 5a 2F+h, . 21 


5 V '2 F±h' 


18, 


v =• 8 <y2F±h 


19, 


d - 


0-35 



L q * 

2 T±ti 


n S DH9-10+T) 0A 

q -55T ’ 20, 


n S 2)’(940 + T) 
a ~ 275Vy/ ZF+JT* 


The above are approximate formula; only.—W. D. M. 


22 , 

23 , 






























542 


Steam. 


Example. Required tlie diameter of an injection-pipe L— 10 feet long, 
which shall supply q = 1.3 cubic feet of water per second into a vacuum ol 12 
pounds per square inch, the head of [tress water h = 2 feet? 


5 /1( 

d = 0.35-W— 


10 x 1.32 
2 X 12“ 


= 0.3211 feet = 3.84 inches. 


Area of Steam Passages. 

7i = number of revolutions per minute, 
a = area of the steam pipe, sq. in, 

A — area of the cylinder piston, sq. in. 
d = diameter of the pipe, in inches. 

D = diameter, 8 = stroke of cylinder, in inches. 


a - 


A S n 
40000 


d = 


D l/S 


n 


200 


24, 25. 


Example. Required the diameter of a steam pipe for a cylinder D = 40 
inches. Stroke of piston S = 48 inches, and n = 38 revolutions per minute? 


d = 


40 ]/48 X 38 
200 


= 8.54 inches. 


c . , ASn 

Steam port, a - 


Steam Ports to tlie Cylinder. 

A S n 


Exhaust port, 


a = 


27000* 


26. 


Safety Valve. 

Three-fourths of the fire grate in square feet is a good proportion for the safety 
valve in square inches. Make area of safety valve 3 g l Si j of boiler heating surface. 
Weisbach, Yol. II. 

Notation of Letters corresponds with Figure 3, Plate V. 


a = area of safety valve in square inches. 

P — pressure per square inch in the boiler 
W= weight on the safety valve lever y in pounds. 
Q = weight of the safety valve and lever 
l — lever for 
e 
x 


— weigni. oi mesaieiy vaivt 
= lever for IF 1 
= “ a P > in inches. 

“ Q ) 


} 


Balance tlie lever over a sharp edge, and the centre of gravity Q is found ; 
measure the distance x from the fulcrum C. 


si Pe = Wl + Qx. 
Wl + Qx 


P = 


si e 


27. 

28. 


W = 


l = 


SI Pe — Qx 
l ' 

a Pe — Qx 

lv~ 


29. 


30 . 


Example. Area of the safety valve a = 9 square inches, e = 4^ inches, 
IF = 50 pounds, weight of tlie lever and safety valve Q = 15 pounds, and x 
- 17 inches. Required at what distances l, l\ and l" will the weight IF indi¬ 

cate pressures of P = 30, / y = 40, and P' 50 pounds ? 


I = 


9 X 30 X 4.5 — 15 X 17 
50 


= 19.2 inches, 


from the fulcrum C the weight IF will indicate 30 pounds. 

V = 27.3 inches, when P’ ~ 40 pounds. 

I" — 35.4 “ “ P"= 50 “ 

and thus the lever can be graduated. 























Condenser and Incrustation. 


543 



The fresh-water or surface-condenser is now considered indispensable on 
sea-going steamers, as it not only saves 15 to 20 per cent, in the consumption , 
of fuel, hut saves the boiler from dangerous incrustation. It is also advisable | 
to use surface-condensers on rivers with muddy water, and on lakes with 
hard water, which is very injurious and treacherous to steam-boilers. 

The condensing surface in a fresh-water condenser should be about five- 
eighths (f) of the heating surface in the steam-boiler, or about two square 
feet per indicated horse-power. 

The jet-condenser cannot be used for steam of 35 pounds to the square 
inch above atmospheric pressure, because when sea-water is raised to the 
temperature of 280° Fahr. its insoluble salts, principally sulphate of lime, 
wholly precipitate and form scale upon the hot surfaces in the boiler. This 
scale sticks fast to the surface of the iron and cannot be dissolved. 

Scale formed of carbonate of lime and salts of soda and magnesia has a 
soft or plastic consistence, which can to a great extent be blown out from the 
boiler when in port. 

For the surface-condenser any steam-pressure can be used, but it is gen¬ 
erally expanded down to a low pressure before it reaches the condenser. 

The fresh-water condenser is more expensive, occupying more room, and 
requires an extra circulating-pump. 

The air-pump need not be so large for a surface-condenser as for the jet- 
condenser. The capacity of the circulating-pump should be about 0.6 of that 
of the air-pump. The velocity of the cold water through the valves of the 
circulating-pump should not be over 450 feet per minute. 


HORSE-POWER, IN STEAM-ENGINES, 

Horse-power in machinery is assumed to be about the effect a horse is able to 
produce, and has been estimated and established by Mr. Watt to be 33,000 lbs. 
raised one foot per minute for one horse, which will be the same as 550 lbs. raised 
one foot per second. Mr. Watt adopted a standard steam-pressure of 7 lbs. per 
square inch, established a simple rule for the nominal horse-power of engines, 
which is, “ The square of the diameter of the cylinder in inches multiplied by the 
cube root of the stroke in feet, and divided by the constant number 47, is the nominal 
horse-power. This rule agreed very near to the actual performance of engines in 
those days, but as improvements were made the steam-pressure and piston-spoed 
were increased. The rule is not now applicable. 

Nominal Ilorse-Povver. 

Assume a standard steam-pressure of 30 lbs. per square inch expanded two- 
thirds, the velocity of the steam-piston to be 200^/$ feet and revolutions per 

minute n = - - — > we will arrive at a formula of nominal horse-power. 

//== ^ I ^ for condensing engines, which will agree very near with the actual 

performance of our present condensing engines. The following tables are calcu¬ 
lated from this formula. 

For high-pressure engines I will assume the steam-pressure to be 80 lbs. per square 
inch expanded one-haif, which will give the nominal horse-power— 

it — - - -, high-pressure engines. 

4 

The horse-power in the accompanying table, divided by 0.4. gives the nominal 
power of high-pressure engines. The diameters /> are contained in the first col¬ 
umn in inches, and the stroke S in feet and inches on the top lino. 






















Nominal Horsepower op Condensing Engines. 


544 


! Diam 


Stroke of Cylinder Piston S in feet. 


13 

V 

.1' 3" 

1' 6" 1' 9’ 

1 2* 

2' 3" 

|2 / 6"2 / 9" 3' 

3' 6" 4' 

4' 6 "| S' 

in. 

; h 

II 

1 11 

11 

II 

1 H 

11 

1 11 

H 

H 

II 

1 11 

w 

6 

3-6 

3*88 

4*12 

4-33 

453 

4-72 

4*88 

5-0^ 

511 

5-47 

5-71 

L 5-94 

6-16 

7 

4-9 

5-27 

5-61 

5-90 

6-17 

6*43 

6-65 

6-8f 

7-07 

7-44 

7-7* 

1 8*1 C 

8-3-8 

8 

6-4 

6-90 

7-32 

7-71 

8-06 

8*39 

S-6S 

8-9f 

9-23 

9-72 

10-1 

110-0 

11-0 

9 I 

8-1 

8*72 

9-27 

9-75 

10-2 

10-6 

11-0 

11-3 

11-7 

12-3 

12-9 

1 13-4 

13-9 

10 1 

10 

10-8 

11*4 

12-0 

12-6 

13*1 

13*6 

14-0 

14-4 

15-2 

15-9 

16*5 

171 

111 

12-1 

13-0 

13-9 

14*6 

15-2 

15*8 

16-4 

16-9 

17-4 

18-3 

19-2 

20-0 

20-7 

12 

14-4 

15-5 

16-5 

17-4 

18-1 

18-9 

19-5 

20-2 

20 8 

21-9 

22-9 

23-8 

24-6 

13 

16-9 

18-2 

19*3 

20-3 

21*3 

22*1 

22-9 

, 23'7 

24-4 

25-6 

268 

27-9 

2S-9 

14 

19-6 

211 

22-4 

23-6 

24-7 

25*7 

26 6 

j 27-4 

28-3 

29-7 

31-1 

32 4 

33’5 

15 

2*25 

24-2 

25-8 

27-1 

28-3 

29*5 

30-5 

31-5 

32-4 

34-1 

35-7 

37-1 

38-5 

16 

256 

27-4 

29-3 

30-8 

32-2 

33*5 

34-7 

35-8 

37-0 

38-9 

40-6 

42-2 

43-8 

17 

28-9 

311 

33-1 

34-8 

36-4 

37-9 

39-2 

40-5 

41-7 

43-9 

45-9 

47-7 

49-4 

18 

32-4 

34-9 

37*1 

39-0 

40*8 

42*5 

44-0 

45-4 

46-8 

49-2 

51 4 

53-5 

55*4 

19 

36-1 

38-9 

41 3 

43*5 

45-5 

47*3 

49-0 

50-6 

52-1 

54-8 

57-3 

59-6 

61-7 

20 

40-0 

43-1 

45-S 

48-2 

50-4 

52-4 

54*3 

56-0 

57-7 

60-7 

63-5 

66-0 

68-4 

21 

44-1 

47-5 

505 

53-1 

55-6 

578 

598 

61-7 

63-6 

67-0 

70-0 

72-8 

75.4 

22 

48-4 

52-1 

554 

5S*3 

610 

63*4 

65-6 

64-S 

69-S 

73-5 

76*8 

SO 0 

82-8 

23 

52-9 

57-0 

60-5 

63*7 

66-7 

69-3 

71-8 

74-1 

76*3 

80-3 

84-0 

87-4 

90-5 

21 

57-6 

62-0 

65-9 

69-4 

72*6 

75-5 

78-1 

80-7 

83-1 

87*4 

91-5 

95-2 

98-6 

25 

62-5 

67-3 

71-5 

75-3 

78-7 

81-9 

84-8 

87-5 

90-2 

94-8 

99-2 

103 

107 

26! 

67-6 

72-8 

77-3 

81-5 

85-2 

SS-6 

91-7 

94-7 

97-5 

102 

107 

111 

116 

27 

72-9 

78-5 

83-5 

87-S 

91-9 

95-6 

990 

102 

105 

111 

116 

120 

125 

28 

78-4 

84-4 

89-8 

94*5 

9S-8 

102 

106 

110 

113 

119 

124 

129 

134 

29 

84.1 

90-5 

96-2 

101 

106 

110 

114 

118 

121 

128 

133 

139 

144 

30 

90-0 

96-9 

103 

108 

113 

118 

122 

126 

130 

137 

143 

149 

154 

31 

96-1 

103 

110 

116 

121 

126 

130 

134 

139 

146 

153 

159 

164 

32 

102 

110 

117 

123 

129 

134 

138 

143 

148 

155 

163 

170 

175 

33 

109 

117 

124 

131 

137 

142 

147 

152 

157 

165 

173 

ISO 

186 

34 

115 

124 

132 

139 

145 

151 

157 

162 

167 

175 

183 

190 

198 

35 

122 

132 

140 

148 

154 

160 

166 

172 

177 

186 

194 

202 

210 

36 

129 

140 

148 

156 

163 

170 

176 

182 

187 

197 

205 

214 

222 

37 

137 

147 

156 

165 

172 

180 

186 

192 

198 

208 

217 

226 

234 

38 

144 

155 

165 

174 

1S2 

190 

196 

202 

209 

218 

229 

238 

247 

39 

152 

164 

174 

183 

192 

200 

206 

213 

220! 

231 

241 

251 

260 

40 

160 

172 

183 

193 

202 

210 

217 

224 

231 

243 

254 

264 

274 

42 

170 

190 

202 

212 

222 

231 

240 

347 

254 

268 

280 

291 

302 

44 

193 

208 

221 

233 

244 

254 

263 

271 

280 

294 

307 

320 

331 

46 

211 

228 

242 

255 

266 

277 

287 

297 

306 

321 

336 

350 

362 

48 

2.40 

248 

264 

277 

290 

302 

313 

323 

332 

350 

366 

3S0 

394 

50 

250 

269 

286 

301 

315 

328 

339 

350 

300 

380 

397 

413 

427 

52 

270 

291 

309 

326 

340 

354 

367 

378 

390 

410 

429 

446 

463 

54 

291 

314 

333 

351 

367 

3S2 

396 

408 

420 

443 

463 

481 

500 

60 

360 

388 

412 

433 

453 

47 2j 

488 

504 

519 

547 

571 

594 

616 

66 

435 

468 

498 

525 

548 

571 

591 

610 

628 

661 

690 

718 

744 

72 

518 

558 

593 

626 

653 

679 

704 

726 

748 

787 

822 

856 

8S6 

78 

60S 

655 

696 

734 

766 

784 

825 

852 

877 

924 

964 

1003 

1039 

84 

705 

759 

807 

851 

88S 

924 

957 

989 

1015 

1071 

1116 

1166 

1206 

90 

810 

872 

927, 

975 

1020 

1062 

1100 

1134 

1168 

1229 

1284 

1336 

1385 

96 

921 

991 

1053,1110 

1160 

1206 

124911291 

1327 

1400 

1460 

1505 

1575 






































































Nominal Horsepower of Condensing Engines, 


545 


Stroke of Cylinder Piston S in feet. 


D 

<V 

T 

tt' 

9' 

1 10' 

IV 

12' 

1 13' 

14' 

15' 

16' 

[ 18' 

| 20' 

ta. 

H 

11 

11 

11 

H 

H 

H 

Ii 

H 

H 

H 

1 H 

f H 

30 

163 

172 

180 

187 

194 

4 200 

200 

> 211 

2V 

r 22 

2 22 

> 23 ( 

>| 24* 

32 

186 

196 

204 

213 

220 

227 

259 

241 

241 

3 251 

3 25? 

3 26? 

3 278 

34 

210 

221 

231 

24C 

249 

257 

264 

1 272 

271 

3 28, 

> 291 

30? 

313 

36 

235 

248 

259 

269 

27? 

288 

290 

3oe 

311 

2 314 

) 32f 

> 334 

351 

38 

262 

276 

289 

299 

311 

321 

330 

341 

34? 

350 

> 36? 

37? 

391 

40 

290 

306 

320 

33? 

344 

355 

360 

377 

38' 

39? 

40? 

41£ 

434 

4 2 

320 

336 

352 

365 

38( 

392 

404 

416 

.42' 

V 43' 

444 

462 

478 

44 

352 

371 

387 

402 

411 

430 

453 

461 

460 

> 471 

494 

507 

525 

46 

384 

405 

423 

440 

460 

470 

484 

497 

5 U 

1 522 

5 533 

554 

587 

48 

418 

441 

461 

479 

490 

512 

527 

541 

55S 

>| 562 

580 

603 

625 

50 

55-1 

478 

500 

520 

538 

555 

572 

588 

602 

’1 617 

630 

655 

677 

52 

491 

518 

541 

562 

582 

601 

619 

635 

651 

667 

681 

708 

733 

. 54 

529 

558 

583 

606 

628 

648 

667 

685 

700 

719 

734 

764 

790 

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35 








































































































Friction Dynamometer. 


546 


Prony’s Friction Dynamometer. 

This dynamometer consists of a friction 
brake, as shown by the illustration. It is 
keyed on t lie shaft A, which transmits tire 
power and work to be measured. 

The lever of the brake should be balanced 
at R before the weight IF is put on the scale; 
ai*d if it is not. balanced, the weight of the 
lever and scale should be weighed at the 
scale and added to the weight IF. 

The weight IF on the scale is the force 
acting on the lever or radius R. 

It is supposed that all the power and 
work transmitted by the shaft is con¬ 
sumed hv the friction in the brake. When the shaft is running with its 
average speed of n revolutions per minute, the strap is tightened up with the 
screws, so that the lever will barely lift the weight IF, which is also adjusted 
to suit the motion. When the weight and friction are well balanced, count 
the revolutions per minute of the shaft. 

The power transmitted through the shaft is equal to the weight IF multi¬ 
plied by the velocity oi the circumference of a wheel of radius li , making the 
same revolutions as the shaft. 

The velocity in feet per second is 

2 it Rn ' 



ty/W/////////////'//'/////''/'>/////V/f'//////////fr* 


V = 


60 


Power P --= 
which divided by 550 give the 


Tr 2 v R n W . _ , 

IF V= ———-in effects, 


Horse-power IP = 


60 


2nRnW WRn 


60 X 650 5252.2 

The work K in foot-pounds consumed by the friction of the brake in the 
time T in seconds will he 


Work K = 


2nRnWT 
60 


W R n T 

9.55 


All this work consumed by the friction is converted into heat, which makes 
the brake so hot that a constant stream of water must run on it to absorb 
the heat whilst the experiment is made, otherwise the wood in the brake 
would take fire. 

When convenient, it. is best to make the lever R = 10.5 feet., or 10 feet 6 
inches, which will make the circumference 66 feet; in which case, the horse¬ 
power will be 

jp = 66 n IF 2 n W 


550 X 60 


1000 


That is to say, the product of the revolutions per minute and weight IF, 
multiplied by 2 and point off three places, will be the horse-power of the 
experiment. 

A lever of R = 5 feet 3 inches will make the circumference 33 feet, and the 
horse-power 

ZP = ^. 

1000 























Horse Power. 547 

ACTUAL HO.RSE POWER. ' 


One actual horse power is 33000 lbs. raised one foot in one minute. This 
applied to steam engines will be the mean steam pressure on cylinder 
piston in pounds, multiplied by the velocity of piston in feet per minute, 
divided by 33,000, is the horse power imparted by the steam. From this 
we shall deduct 25 per cent, in condensing engines, and 13*1 per cent, 
in high pressure engines, for working friction and pumps, the balance 
to be termed the actual horse power. 

Example 1. Fig. and formulae 318. Area ot steam cylinder A =1809 square 
inches, stroke of piston 5 = 4 feet, indicated pressure of steam 30 lbs. to 
which add the atmospheric pressure 15 lbs. or P = 45 lbs. expanded jf, the 
mean pressure will be F=31-459 lbs. (see Expansion Table I.), vacuum 
v = \2 lbs. the engine making n=45 revolutions or double stroke per 
minute. Required the actual horse power, H=1 IF=31 459+ 12-14*7= 
23-759 lbs. ^1809X4X28-769X45 = 425 . 6 

22000 


In this example the actual horse power is 11*6 per cent, more than the 
nominal power from the table. 

Example 2. Fig. 318. A high pressure engine of cylinder piston A =314 
square inches, stroke .5=3 feet, steam pressure 80 lbs, per square inch, to 
which add 15 lbs. P —95 lbs. expanded |, the engine making n—56 revo- 
lutions per minute. Required the actual horse power! From the ex¬ 
pansion table we have the mean pressure F 80-412 lbs., from which sub¬ 
tract the atmospheric pressure 14-7 lbs. W= 65-712 lbs. 

rr 314X3X65-12X56 

H=* ---—= 180-8 horses. 


Annular Expansion Double Cylinder, Fig. 319. 

These kind of engines are noAV sometimes made in Europe with a view 
to economise fuel, and to extend the expansion of steam. The outer 
cylinder A, A, is annular, similar to that made by Mouslay, but in this 
case it is employed only for expansion, the inner cylinder a is used for high 
pressure only. It is so arranged by steam valves that the high steam is 
acting the whole stroke on the small piston a, after which it is conducted 
to the annular cylinder where it acts expansively on the large piston A, A. 
The tw-o pistons being connected by rods to one common crosshead as 
shown by Fig. 319. This arrangement has been successfully carried out 
by Mr. Jagerfelt in Nykoping, Sweden. The inner cylinder can be con¬ 
sidered an ordinary high pressure engine where the utilized steam is set 
free into the air at each stroke ; but in this case the exhaust steam ac¬ 
complishes a second engagement in the annular cylinder, which according 
to the grade of expansion may greatly exceed the original effect im¬ 
parted in the small cylinder during the first engagement. 

Example 3. Fig. 319. Area of the high pressure cylinder piston 
a=254-4 square inches, the annular expansive piston A = 763-2 square 
inches, stroke of pistons 5=3 feet, the high steam pressure P--60 lbs. 
vacuum v=12 lbs., making n=65 revolutions per minute. Required the 
actual horse power of the engine H=1 The grade of expansion will be 

763*2 

1- -= §, for which the mean pressure on the annular piston will be 

254 4 

f— 32-62 lbs. See Expansion Table II. The effective pressure on the two 
pistons will be F=763 2 (32-62-|-12—14-7) -f 254-4 (60—32-62) = 30337 lbs. 

H = 29a»X3X66_ 269 horses. 

22000 

Example 4. Now we will reject the annular expansion cylinder, and 
take the effect of the steam without expansion, when the effectual pres¬ 
sure will be 60—14-7=45-3 lbs. and the actual power, 


H = 25£4X3X45 J 3X65 =118hofsc8i 
19000 























548 


Horse Power. 


If we jonsider the last result as unit we shall have 269 — 118 = 151 horses 
or nearly 128 per cent, gained by the Expansion, omiting the loss of steam 
in the steam passages. 

In the first case about 11 per cent, was gained by vacuum, but that ad¬ 
vantage is rather in favour of the utility of expansion, because the high 
steam cannot so well be introduced into the condenser. 

The economy will be in the same proportion when the same grade of 
expansion is used in one cylinder. 

I do not mean to maintain that this high per centage of economy is al¬ 
ways fully realized in practice, as I am well aware of cases where expan¬ 
sion is of little use, caused by misconception and carelessness in its em¬ 
ployment. (Very little confidence can be placed in computations of economy 
of expansion when the theory is incomplete, as in the above examples. On 
page 594 el. seq. will be found a natural and practical theory of expansion.— 
W. D. M.) 

Half Trunk Expansion Engines. Fig. 320. 

This kind of engines has been introduced by Mr. Carlsund, and are ex* 
tensively used in Sweden, they are well suited for Gunboats where the 
machinery is required to be below the water line. The high steam is em¬ 
ployed throughout the stroke in the annular space around the trunk, 
after which it is conducted to act expansively on the large piston A 
Fig. 320. 

Example 5. Fig. 320. Area of the annular piston a= 662 square inches, 
and A ^2248 square inches, stroke of piston S=4 feet, steam pressure 
P=90 lbs., making n= 68 revolutions per minute. Required the actual 
horse power] 

662 

Grade of expansion = 1 —2248 = 

From the Expansion Table II. we have/=41-58 lbs. mean pressure on A. 

The effectual pressure will be K=2248 (41‘68—14*7) -[-662 (90,—41-68) = 
87639 lbs., high pressure 87639X4X68 

H — - - = 627-3 horses. 


38000 

Double Cylinder Expansion Engines, 


Fig. 321. 


This kind of engines are now made in England and are said to be very 
economical. The small cylinder is used for high pressure, from which 
the steam is conveyed to expand in the large cylinder. In the figure it 
is arranged so that the pistons follow one another in one direction, when 
the steam must be conveyed from the top of the small cylinder to the 
bottom of the large one, and vice-versa ; but it is sometimes arranged so 
that the pistons move in opposite direction, when the steam is conveyed 
direct at the same ends from the small cylinder to the large one, which 
has the advantage of making the steam passages shorter, but is more 
complicated in concentrating the motion. 

Example 6. 

High pressure cylinder, { «= ^quare inches. 

Expansion cylinder, { j. = f^ e s J uare inche8 ‘ 

Steam pressure in the small cylinder P=40 lbs., vacuum v=12 lbs., 
making ?i=21 revolutions per minute. Required the actual horse 

power, H—1 962X5 

Grade of expansion =1—--- - = 

3848X10 

From the Expansion Table II. we have/= 11.879 lbs., mean pressure on A. 

U>=-384SX10 (11.879 + 12-14:7)+962X5 (40-11.879) = 488438 lbs. 


H = 


366767X21 


= 406 horses. 


22000 


l 















Horse Power of Engines. 


549 



318. One double acting Cylinder . 
r jy A S W n 

Actual \ ll = —’ cond. engs. 

horse •( 



W=F-\-v —14-7 for cond. engines. 
W—F- —14-7 forliigli pressure engines. 
319. Annular expansion double Cylinder. 
F=? ' “[2-3 (log.A—log.a)^ 1], 

A 

„ FA—Pa V=A(f-\-v —14*7^) 

J— A—a ’ +<?—/)■ 

, VSn 

Actual j H= 22000 ’ con(L en S ines * 
horse < 

power. ( high pr. engs. 


320. Half trunk expansion Cy Under. 

Pa 

P=—- [2-3 (log.A — log.a)- j-1], 

A 


a- 


-Pa V=A(f+v— 14-7; 

-\-a(P — f). 


' A—a 

( rx VSn 

Actual H =~ cond. engines. 
horse 


44000 


power. I jj_ 


VSn 


38000 


■’ high pr. engs. 


■ 

Aq.-, ' / 321. Double Cylinder expansion. 

! I f==4 „ Pas 





t 

AS 

1-!!.■■- ^ 



F== ^~ l *Wo9'AS-lo 9 .as)-\-l] 

. FA—Pa w=AS(f-\-v —14’7 ) 

f= ~~A^r y -f-a s(P f). 

( tt w n cond. engines. 
Actual lL — ^ 0fti0 

horse ■< 
power. I ]j. 


‘22000 
w n 


‘19000 


high pr. engine®. 
















































































































650 


Indicator Diagrams. 


1 


INDICATOR DIAGRAMS. 

Page 553. 

Fig. 1 represents an ordinary diagram taken from a condensing-engine. 
The line a, b i, e , B, B is drawn by the indicator; the space U is added for 
clearance between ihe piston and cylinder-head and volume of steam-ports. 
In ordinary engines this clearance is between 6 and 12 per cent., to be added 
to the stroke S for proper analysis of the diagram, 
c = volume of clearance in cubic inches. 

A = area of steam-piston in square inches. 

Clearance, £7 = 


The line 6, t,/, which indicates the steam-pressure during the expansion, 
is an equilateral hyperbola which Rankin has given the name of isothermal 
curve, or a curve of constant temp rature. The lines Va and Vd are the 
asymptotes and m the axis of the hyperbola. (See Conic Sections, page 181, 
Figs. 202 ami 203.) The axis m forms an angle of 45° with the asymptotes, and 
the hyperbola passes m at right angles. 

The diagram shows that the steam is cut. off when the piston is at b, and 
the hyperbola should then be the line 6, q, h. t, c ; but for the sake of a better 
explanation 1 have purposely selected a case in which the steam leaked into 
the cylinder during the expansion ; so that the actual line of pressure is 
b i, f, e , which shows that the steam was wire-drawn. When the steam was 
cut off at b, the end-pressure should have been d e; but the steam which 
leaked in during the expansion raised the end-pressure to df. The work done 
by the leakage is represented by the area bounded by b, i, Je t h q b. Had 
there been no leakage, but the same amount of steam admitted from be¬ 
ginning of the stroke and cut-off at c, the end-pressure would have still been 
5,/, with the additional work represented by the area bounded by b,c,g,f,ib, 
or that much work was lost by the leakage. 

A e is the atmospheric line which should always be drawn by the indicator. 

Vd is the perfect vacuum line, and the vertical height, from Vd to A e 
represents the atmospheric pressure, about 14.7 pounds to the square inch. 

u represents the pressure in the condenser, of the deficiency of vacuum. 

v = vacuum in the condenser. 

p = steam-pressure in pounds per square inch above atmospheric pressure 
at that part of the stroke. 

Prepresents the absolute initial steam-pressure in pounds per square inch 
in the cylinder. 

The work done for each single stroke of the piston is represented by the 
area bounded by the diagram a., b, i x f,e, B, B, a, of which the vertical heights 
are expressed in pounds per square inch and the stroke £ in feet. 

The variable steam-pressure must be reduced to a mean pressure before it 
can be applied for calculating the power of the engine. When the point of 
cut-off is correctly known, the expansion tables give the mean pressure, from 
which deduct the deficiency of vacuum u. 

F — mean effective steam-pressure in pounds per square inch on the 
cylinder piston. 

A == area of piston in square inches, double acting. 

V— velocity of piston in feet per minute. 


Indicated horse-power, 


IP = 


AFV 
33000 ’ 


r 




If the velocity is given in feet per second, then divide by 550 instead of 
33000. 

To Construct the Indicator Diagram. 

Having given the point of cut-off b, stroke S, pressure P, and clearance 17, 
draw the rectangle a, v, d. From b draw b, k parallel with av. Divide the 
rectangle into 10 equal parts and number them as shown. From v draw the 
dotted lines to the top of the ordinates, as vop'\ from odraw the line oq; 
then q is a point in tlie diagram, and the other points are obtained in the 
same way, and so the hyperbola is obtained. 












Indicator Diagrams. 


551 


To Find tlie Point of Cut-off when the Final Pressure is Given. 

Measure the end-pressure by any convenient scale (a diagonal decimal 
scale of inches is the nest, such as that on plate I., page 368), say de — 0.35 
of an inch ; then divide 0.35 by the number of each ordinate, and the quotient 
is the ordinate pressure; for instance, 0.35:0.8 = 0.4375 of an inch is the 
height of the ordinate 8 1\ 0.35 :0.4 = 0.875 of an inch, the height of the ordi¬ 
nate 4g; and thus the curve can he plotted out until it reaches the cut-off 
point a b. The curve c, g, /is obtained in the same way. The indicator curve 
b, i is continued to/ to get the end-pressure d /. 

To Find (lie Final Pressure when the Point of Cut-off is Given. 

I = length or part of stroke with full steam. 

/ = end-pressure at end of stroke. 

S — stroke and P = initial steam-pressure. 

P:{U+S)=f-.l 

End-pressure, f— Cut-off at 

To Find the Effective Mean Pressure by Measurement. 

Fig. 2 represents a fairly good card taken from a condensing-engine. The 
stroke is divided into 10 equal parts, and each part measured in the middle as 
indicated by the dotted lines and arrows. Use a decimal diagonal scale of inches 
like that on plate I., page 368. Add all the dotted ordinates together, and point 
off one decimal from the sum, which gives the mean length of all the ordinates. 
Multiply this mean length by the scale of the indicator-spring, and the product 
will be the effective mean pressure. 

Scale of the Indicator-Spring. 

The scale of the spring means the number of pounds of steam-pressure per 
square inch corresponding with one inch on the diagram, which scale is, or 
should be, marked on the spring. 

When the vacuum and atmospheric Hops are known on the diagram, then 
the scale of the spring can be determined, as there is 14.7 pounds pressure 
between these two lines.- 

The vacuum line cannot be drawn by the indicator, because there is never 
a perfect vacuum iu the condenser, but the atmospheric line is always given 
by the indicator. 

AMSLER’S PLANIMETER. 

Area and Mean Pressure of Diagrams. 

The best method of finding the area and mean pressure of indicator cards is by 
using the Amsler Planimeter, which instrument not only gives the area correctly, 
but also the mean ordinate in inches, from which the mean pressure is calculated 
by the scale of the indicator-spring. 

Divide the area of the diagram in square inches by its length in inches; the re¬ 
sult is the mean ordinate. 

Remarks on Card, Fig. 2. 

An indicator card as represented by Fig. 2 is considered to be a good card. The 
inclination of the part a, b shows that the steam has been slightly throttled or 
wire-drawn, or that the steam-port was not large enough for the speed of the 
engine. The steam appears to have been cut off at b, from which the nearly 
regular hyperbola extends to c, where the exhaust opens and rounds off the 
diagram at the end of the stroke. On the return stroke the diagram runs 
nearly parallel with the vacuum line v cl, and rounds off the corner at C, 
which is called compression. The compression is caused by the exhaust being 
closed before the piston reaches the end of the stroke, and also by lead of the 
main valve. 













552 


Indicator Diagrams. 


Diagram Fig. 3. 

The diagram Fig. S is taken from a high-pressure engine and from both 
ends of the cylinder, and is considered a had diagram. The rise c of the 
exhaust is caused by the exhaust-opening being too small, or when two 
cylinders are connected at right angles the exhaust of one cylinder opens 
when the piston of the other cylinder is at half stroke and has its greatest 
velocity, which may interfere with the free exhaust, and thus raise the back¬ 
pressure. The vacuum line is not shown in the high-pressure diagram, and 
the horse-power cannot be calculated without knowing the pressure or scale 
of the indicator-spring. 

Diagram Fig. 4. 

Fig. 4 represents a diagram taken from a compound engine, of which the 
upper part a, b, c,f, g, a is from the high-pressure cylinder, and e, h , V, i, e 
from the low-pressure cylinder. After the steam has worked through the 
high-pressure cylinder, it is expanded into the low-pressure cylinder. 

When the high- and low-pressure cylinders are connected at right angles, 
there must be a steam space between the cylinders for the high-pressure 
exhaust, and it is generally so arranged that the steam entering the low- 
pressure cylinder is cut off at the moment the exhaust of the high steam is 
opened, for otherwise there will be a loss of power. The steam is gen¬ 
erally expanded some in the high-pressure cylinder before it enters the 
other. 

When the high- and low-pressure cylinders are connected to opposite 
cranks, then the steam can expand directly from one to the other without 
more space between the cylinders than that of the conducting ports. 

Compound engines are generally so proportioned that the power of the 
high-pressure cylinder is nearly equal to that of the low-pressure cylinder, 
and on account of the high grade of expansion of steam used in them, and 
the reduction of initial condensation, they sometimes prove very eco¬ 
nomical. 


Nominal Horse-Power of Compound Engines. 

The English have adopted a rule for designating the sii&e of compound 
engines in nominal horse-power, as follows: 

1) = diameter of low-pressure cylinder in inches. 
d — diameter of high-pressure cylinder. 

S = stroke of piston in iuclies. 

Nominal IP = ^ y'' S. 

1UU 

The diameter of the low-pressure cylinder is made about double that of the 
high-pressure cylinder. 


Diagram Fig. 5. 

The diagram Fig. 5 is taken from a locomotive engine, and is a fair 
average of the steam-engineering existing in our day’s locomotive practice. 
About fifty per cent, of the power and fuel is uselessly wasted in the 
locomotive. 

The defect of the valve-gear is clearly illustrated by the diagram Fig. 5; 
namely, that there is no sharp cut-off, the steam is wire-drawn, the exhaust 
is choked and compressed by cutting off the steam with the main valve and 
link-motion. There is plenty of ingenuity among us to devise a locomotive 
valve-gear that would distribute the steam as correctly as required, but how 
to make such a gear simple and substantial in a problem not easily solved. 

To compound the two cylinders of a non-condensing locomotive, when 
properly done, would result in a great economy of steam. There is no prac¬ 
tical difficulty in attaining far better results than have as yet been reached. 
—W. D. M. 









Indicator Cards. 


553 































































554 


Scientific and Technical Terms. 


SCIENTIFIC AND TECHNICAL TERMS. 

Curious names are often given to principles supposed to be newly dis¬ 
covered, but which in reality are old and have proper technical names. One 
vagary of scientists has been the giving individual names 10 physical con¬ 
ceptions which have technical names when those principles are properly 
understood. This annoyance exists more in the electrical profession than in 
anv other branch of science, the result of which is sometimes ridiculous. 

The Electrical Congress, which met in Paris in 1881, occupied much time 
in discussing which individual names should be adopted for electrical con¬ 
ceptions. Many superficial talkers and writers have much bothered earnest 
students by this form of solemn nonsense. 


Acceleration, the increment of 
velocity per second of a moving body. 
The acceleration of gravity is gen¬ 
erally denoted by the letter g and 
amounts to about 32.17 feet, or D.81 
metres, per second, the velocity at¬ 
tained at the end of one second for a 
body free to fall. 

Adiabatic curve, a curve rep¬ 
resenting volume and pressure of a 
gas or vapor without transmission of 
heat. 

Ampere, the unit of measurement 
of electrical intensity. 

Amplitude, the deviation from 
east or west toward north or south in 
the horizon. 

Azimuth, the deviation from the 
meridian east or west. 

Battery of steam-boilers is 
applied to a number of boilers work¬ 
ing together. 

Binnacle, a case in which a mar¬ 
iner’s compass is set on board a 
vessel. 

Binary means doubling or halv¬ 
ing. A binary system of numbers is 
that whose base can be divided by two 
repeatedly without leaving fractions. 
The metric or decimal system is not 
binary, because the base 10 can be di¬ 
vided only once by 2 without frac¬ 
tions. 

Coloinb, unit of quantity of elec¬ 
tricity which is equivalent to work, 
and can be expressed in foot-pounds 
or kilogrammeters. 

Dyna mic quant ity, some quan¬ 
tity containing both force and motion. 

Dynamic effect, used for ex¬ 
pressing either power or work. 

Dyne, a unit of electromotive force 
established by the British Association 
for the Advancement of Science. 

Electro-dynamics, the science 
of e'ectricity producing power and 
work. 

Electrolysis, the science of ana¬ 
lyzing substances by electricity. 

Electrolyte, any substance that 


can be decomposed and analyzed by 
electricity. 

Energy means power, but is most 
frequently and erroneously applied to 
work, and even to force. Scientific 
writers have a great, many kinds of 
energies, distinguished as potential, 
actual , equality of , intrinsic, mechanical, 
kinetic, etc., which appellations have 
no definite meaning. 

Ergf, a unit of electric work equiv¬ 
alent to one dyne lifted one centi¬ 
metre ; established by the British As¬ 
sociation. 

Farad, the unit of electric ca¬ 
pacity, which capacity is equal to one 
colomb divided bv one volt. 

Field, magnetic field, is the space 
bet ween the poles of the magnets in 
a dynamo or electric motor. 

Field magnets, the stationary 
magnets in a dynamo or electric motor. 

Galvanic current, a current of 
electricity direct from a battery. 

Galvanometer, a magnetic nee¬ 
dle acted upon by an electric current 
for measuring the strength and de¬ 
termining the direction of electric 
flow. 

Isothermal line, a curve repre¬ 
senting volume and pressure of a gas 
or vapor whilst the temperature re¬ 
mains constant. 

Kinetic. In mechanics kinetic 
means motion or the science of cause 
of motion. It is superfluous in me¬ 
chanics. 

Mechanical effect means power 
or work. 

Micro-farad, a unit of electric 
capacity, or a one-millionth part of a 
farad. 

Moment of activity means sim¬ 
ply power. 

Moment of inertia means the 
moment of the momentum. 

Moment um, Mv, is the intensity 
of that constant force which will in 
one second give to a body at rest, of 
mass M , the final velocity v. 





Scientific and Technical Terms. 


555 


Olim, the unit of resistance of a 
conductor to the flow of electricity. 
Resistance is the loss of electro-motive 
force per ampere. The latest unit 
ohm established by the Electrical 
Congress is the resistance of a col¬ 
umn of mercury 1.06 metres long by 
1 square millimetre section. 

Origin is the point where the rect¬ 
angular co-ordinates of a curve meet, 
and from which the ordinates and ab¬ 
scissas are measured. 

Parameter is the ordinate which 
passes through the focus of a curve. 

Potential. In electricity, poten¬ 
tial means the available electro-motive 
force, including the combined action 
of both positive and negative elec¬ 
tricity, and called electrical potential. 

Quantity of action means sim¬ 
ply power. 

Quantity of moving force 

means motive force. 


Quantity of motion is often 
used for momentum. 

Hate of work means simply 
power. 

Rheostat, an instrument contain¬ 
ing a number of resistance coils for 
comparing electrical resistances of 
conductors. Each coil has a known 
resistance, marked in ohms. The 
rheostat is analogous to the friction 
dynamometer in mechanics. 

Vis-viva means living force said 
to be stored in a moving body. An old 
term meaning twice the work stored 
in a moving body. 

Volt, unit of electro-motive force 
corresponding nearly with a force or 
weight of one milligramme. 

W aft, unit of electric power estab¬ 
lished by the British Association; it 
is the rate of working one erg per 
second. There are 746 watts per horse¬ 
power. 


Work produced or consumed by combustion of one ounce avoirdupois or 
one gramme of coal, gunpowder, zinc, copper, and hydrogen : 

One ounce coal = 695,000 foot-pounds, 

gunpowder = 100,000 “ 

zinc — 113,000 “ 

copper = 69,000 “ 

hydrogen — 2,925,000 “ 

One gramme coal = 3,390 kilogrammeter. 

“ “ gunpowder— 487 “ “ 

“ “ zinc =■ 

<C << 

u u 


u 

it 

u 


it 

u 

u 

u 


u 

ii 

ti 

u 


copper = 
hydrogen — 


550 

336 

14,225 


U 

u 

u 


u 

u 

« 

















Approximate Horse-Fower. 


556 


Approximate Horse-Power 

3/ —’ 

of small high-pressure engines. II = 0.1 D 2 y S. Steam pressure not less than 80 

pounds to the square inch. 


Diam. 

D 

Inches. 

3 

4 

5 

G 

Strok* 

7 

S of 

8 

pistol 

9 

a in ii 

1 10 

iches. 

13 

14 

15 

1G 


2 

.577 

.634 

.684 

.727 

.765 

.800 

.832 

.862 

.915 

.964 

.985 

1.00 

1.05 

2; 

.900 

.990 

1.07 

1.13 

1.20 

1.25 

1.30 

1.34 

1.42 

1.50 

1.54 

1.57 

1.63 

3 

1.30 

1.43 

1.54 

1.64 

1.72 

1.80 

1.87 

1.94 

2.06 

2.17 

2.22 

2.27 

2.36 

3j 

1.77 

1.94 

2.10 

2.22 

2.34 

2.45 

2.55 

2.64 

2.80 

2.95 

3.00 

3.69 

3.21 


4 

2.31 

2.54 

2.74 

2.90 

3.06 

3.20 

3.33 

3.44 

3.66 

3.85 

3.94 

4.05 

4.19 


4* 

2.92 

3.21 

3.47 

3.68 

3.87 

4.05 

4.42 

4.36 

4.64 

4.88 

5.00 

5.10 

5.30 


5 

3.60 

3.96 

4.27 

4.54 

4.73 

5.00 

5.20 

5.38 

5.72 

6.02 

6.16 

0.30 

6.55 


6 

5.19 

5.70 

>6.15 

6.53 

6.89 

7.20 

7.55 

7.82 

8.31 

8.75 

8.95 

9.15 

9.50 


7 

7.08 

7.78 

8.40 

8.92 

9.40 

9.80 

10.2 

10.6 

11.2 

11.8 

12.1 

12.3 

12.9 


8 

9.25 

10.1 

11.0 

11.6 

12.2 

12.8 

13.3 

13.8 

14.6 

15.4 

15.7 

16.1 

16.8 


9 

11.7 

12.9 

13.9 

14.7 

15.5 

16.2 

16.8 

17.4 

18.5 

19.5 

20.0 

20.4 

21.2 


10 

14.4 

15.9 

17.1 

18.2 

19.1 

20.0 

20.8 

21.5 

22.9 

24.1 

24.6 

25.2 

26.2 


11 

17.5 

19.2 

20.8 

22.0 

23.2 

24.2 

25.2 

26.1 

27.7 

29.2 

29.9 

30.5 

31.6 


12 

20.8 

22.9 

24.7 

26.2 

27.6 

28.8 

30.0 

31.0 

33.0 

34.8 

oO.O 

36.3 

37.8 



The horse-power of small engines, as counted by the English, is only 0.4 of 
that in this table for the same size cylinders. 

To Approximate tlie Size of Steam-Engines. 

Example 1. It is required to build a river steamer of displacement 7 T = 1000 
tons to run M = 16 nautical miles per hour. Required, the size of the cylinder 
for an ordinary overbeam engine? From the table of steamship performance will 
be found the required actual power H = 179S IP. 

From the table of Nominal horse-power select the approximate size of cylinder, 
which may be D =88 inches, diameter of cylinder by S = 14 feet stroke, which 
answers to H = 1866 horses nominal. In this case the nominal horse-power can 
be considered the same as the actual. 

Example 2. A propeller steamer is to run M = 10 nautical miles per hour, with 
a displacement T = 0400 tons. Required, the size of the cylinders? 

From table of steamship performance H — G92 horses, to be divided into two 
cylinders of 406 each. Select from table of Nominal horse-power D = 60 inches 
diameter of cylinders and <S = 2' 19" stroke of piston, which answers to II = 504, 
or 504 X 2 = 1008 horses of the two cylinders. After these approximations are 
made, make a careful calculation from the original formulas. 

Example 3. Suppose the propeller for the steamer in the preceding Example 2 
makes n = 60 revolutions per minute. Required, the diameter of the propeller- 
shaft? See Table, page 418, for wrought-iron shafts, for 1000 horses and 60 revolu¬ 
tions, the shaft should be 12.8 inches. 

Example 4. A steamer of T = 2500 tons is to run M = 9 nautical miles per 
hour with an indicated steam-pressure of 20 lbs., or P = 35 lbs. per square inch, 
expanded £. Required, the consumption of fuel in tons per 24 hours? 

Table of steamship performance II — 585 IP. 

Table V., page 405, consumption of fuel, 3.44 tons. 

The required consumption will be 5.85 X 3.44 = 20.124 tons per 24 hours’ 
steaming. 

[The resistance of vessels or the work to be done is to be calculated and the 
proper size of engine, with margin for frictional loss, computed from indi¬ 
cated horse-power. W. D. M.] 











































Prices and Weights op Machinery. 


557 


PROPORTIONATE PRICES OF MACHINERY. 

Machines or engines made of different sizes, but of uniform proportion, 
generally vary in prices as the square root of the cube of any linear dimen¬ 
sion. Suppose two engines, one of exactly double the linear dimensions of 
the other; then the proportionate prices will be as 1 : }/2 a = 1 : 2.828. 

For steam-engines the volume of the cylinder—that is, the displacement 
of the steam-piston—is a good representation of the cube of any linear di¬ 
mension of the engine. 

C— volume of the steam-cylinder in cubic inches. 

X = a coefficient to be determined by the manufacturer. 

$ = price of the engine in dollars. 

$ = A"1 / C. 

For a double engine it will not answer to add the volumes of the two cyl¬ 
inders, but the price to be calculated for one engine, and then doubled. 

For ordinary stationary engines the coefficient AT is between 20 and 30; for 
highly-finished high-speed engines it runs up as high as A^=50; and for 
donkey-pumps as low as Z= 15. 

Example 1. What will be the price of an ordinary stationary engine of 
D = 30 inches diameter of the cylinder, by S— 48 inches stroke? 

Assume the coefficient to be X— 25. 

Area of cylinder piston, 706.86 sq . in ._ Volume, C — 33929.28 cub. in. 

Price, $ = 25 |/33929.28 = 4605 dollars. 

Example 2. What will be the price of a similar engine to that in F.xample 
1, hut D — 15 inches diameter of the cylinder and S = 24 inches stroke ? 

Area of cylinder piston, 176.71 sq. in. Volume, C = 4241.04 cub. in. 

For a similar engine use the sam e coeffi cient, X— 25. 

Price, $ = 25 ]/4241.04 = 1628.125 dollars. 

Thus a regular scale of prices can be made for different-sized engines. 

Example 3. What, will be the price of a donkey-pump of D — 6 inches 
diameter of cylinder and = 9 inches stroke of piston? 

Assume the coefficient X — 16. 

Volume of cylinder, 28.27 X 9 = 254.43 cub. inches. 

Price, $ = 16 ]/254.43 — 255.20 dollars. 

Example 4. What will be the price of a two-cylinder donkey-pump con¬ 
nected so that they work one another valve motion, when D = 9 inches 
diameter of cylinders and S = 15 inches stroke of pistons? 

Volume of one cylinder , 63.6 17 X 15 = 954.255 cub. inches. 

Price for one, $ = 16 |/954.255 = 494.24 dollars. 

Price for the double engine, 494.24 X 2 = 998.48 dollars. 

For compound engines calculate the price for each cylinder separately; 
then add the two prices, and the sum is the price of the compound engine. 

When the price of one engine is determined, the coefficient X will be 



Then fix the prices of other sizes, but similar engines, by the same coefficient. 

Example 5. An engine of D = 12 inches diameter of cylinder by S — 24 inches 
stroke costs $ = 1500 dollars. Required the coefficient AT? 

The volume of the cylinder is C— 2714.16 cub. inches. 

Coefficient X — ——i=£E=- = 28.8 ; say 29. 

1/2714.16 

The weight of engines of different sizes should be as the cube of any linear 
dimension, but the smaller engines are generally made heavier in proportion 
to the larger ones; so that in ordinary practice the weight varies nearly as 
the price. 

]V= weight of the engine in pounds. _ 

W = 2.5 X \/ C. 










Sl.je valves. 


558 


SLIDE VALVES. 

The slide valve motion is one of the most important features in the steam 
engine. Plate TV. is a drawing of the Gonzenbach valve motion as used in 
Europe. (See Zeuner’s Treatise on Valve Gears.) 


Main Valve. 


It will be best to assume a certain size cylinder, and at the same time give 
the proportions for any size. 

D = *34 inches, diameter of the cylinder. 

S = 18 inches, stroke of pisiou.* 
n = 56 double strokes per minute. 

We have the area of the steamports m, from Formula 26, page 542. 


342 x 0-785 X 18 X 56 
35000 


= 26 square inches, nearly. 


m 


D + S 
26 


34 + 18 
26 


= 2 inches, 


the width of the steamport; if the quotient gives a fraction take the nearest 
quarter or eighth. 


— = — = 13 inches, breadth of steamport. 
m 2 ’ * 


r = 1 m about = 1 inch, the exhaust port o = 2m — ir = 3$ inches, and 
/ = o 4- 2r = 5* inches, h = f — inches, k = lfm = 3 inches, and 

i = h-\-2k — 11^ inches, t — m = 2 inches. 

* The stroke and diameter is here rather out of proportion, but we wiii maintain 
them in the calculations as they suit the drawing, which is purposely made to 
show the slide valves on a large scale. The rules will however suit any propor¬ 
tions of diameter and stroke. 


To Find the Stroke off the Eccentric* 

s =r stroke of the eccentric in inches. 
s — i — / — = 5i inches. 

The lap L — J(i — /— 2 to) = } inches. 

The lead of the valve, or opening of the steamport when the crank pin stands 
on the centre should be about 

, mi/n 2i/56 ... , 

l = -Jj- = -g- = i inches, nearly. 


Having finished the main valve and ascertained the stroke of the eccentric, 
it is now required to find the position of the centre b , (Plate V.,) of the eccentric, 
to the crank-pin. Suppose the crank pin of the engine stands atva on the centre 
nearest to the cylinder, and the eccentric rods are attached direct to the valve 
rods; draw the line dd, at right-angle to the centre-line au" of the engine, 
then 

the angle, sin.lF= 2 ^^±^ = * = °' 409 > or w = 240 lO*. 

See Plates IV. and V. 

To Find the position of tl»e Crank«Pin at the moment the 

Main Valve opens. 

y-'rsi \w = S5xo°9i23 - °' 9 i “ ches - ne ‘ r, 7> 

from the centre line. 



























■■I 

■■■ 


W'Wmm 




wm* 


'yyyyyyyy// 




WM 


KM\ 






;/ .. : 


- ''■''' \\ 
Vv\.V. 


S/u/r l?//irs. 


Jl! J i i/i'vm 










































































































































/v/vy/Z/y/.y. 

Hntr / 




































Slide Valves. 


559 


To Find the position of the Crank at the moment the 

Exliaust opens. 

* = |( sin - ,F - 7 (/-*)) = 7(o-409 - ^(5'5-5-25)\ = 3-27 inches 
from the centre line. 

To Find the position of the Crank Pin when the Main 

Valve cuts off the Steam. 


2S L _2X18XJ ,. 797 . , 

- = —— 5q27 inches. 

s 0 0 


To Find at what part of the Stroke the Main Valve Cuts 

off tile Steam, 

/ 9 V' ^ 

Will cut off at = 1-= 1 — ( •— 3 * * * * 8 | = 0-899 of the stroke. 

s a \ 5-5 / 

The greater the lap is, the sooner will the main-valve cut off, hut if the lap is 
increased the stroke of the eccentric must also be equally increased. It does 
not work well to cut off much by the main-valve, especially when the engine 
works fast; for very slow motion it may answer to cut off at $ the stroke. 

It will be noticed that the centre of the eccentric is always ahead of the crank 
pin with an angle 90 o -fu>. Hence when the engine is to be reversed, the centre 
b must have the same position on the opposite side of the centre-line, or the 
eccentric must be moved forwards aD angle of 90° — 2 w. 

C«t“Olf Valve. 

The width of the cut off ports should be about d = j}m = 1J inch, and their 


breadth 


a 30 
2d ~ 2 XU 


=» 12 inches, when two ports are used. 


Proportions of the Valve. 

a — b — c — d , a-f d — b-\-c, and a = 2d, and the stroke of the cut-off valve 
eccentric s = 2b, we shall have a = 2 $, b = 2 £, c — 1 £, c = l£, and 
s — 4J inches. 

Let us assume the steam to be cut off at f = l of the stroke S, the position of 
the crank-pin a' will then be sin.w — 21 — 0-666, or n = 70° StF; at the same 
time the position of the centre c’ of the cut off tccentric will be 


sin .3 


d+c li+ 1 * 
s U 


= 0-612, or z = 37° 50', 


and V — u — z = 70° 30' — 37° 50' = 32° 40', the position of the centre c when 
the crank-pin a is on the centre. This Table will show the positions of the 
centre a and c, at different cut offs. Letters correspond with Figure 1, Plate Y T I. 


Cut off 

at 1. 

A 

a 

i 

# 

* 

2 . 

3 _ 


V 

sin.a 

strolce of 

z 

u 

F. 

V . 



eccen. s. 



22 0 10 ' 

0-377 

2b 

37° 50' 

60° 

0-5880 

0-250 

32° 40' 

0-539 

2b 

37° 50' 

70° 30' 

0-6914 

0-333 

31° 55' 

0-527 

c+a, 

43° 35' 

75° 30' 

0-7332 

0375 

42° 35' 

0-675 

b+c 

47° 25' 

90° 

0 8350 

0-500 

46° 30' 

0-7193 

a+ 6 — c 

58° 

104° 30' 

0-910 

0-625 

50° 30' 

0-7933 

a+ 6 — c 

58° sty 

109° 30' 

0-985 

0-666 


It will now be observed that the effectual pressure F in this Table is less 

than in the Table on page 403, owing to the valve not cutting off the steam 

instantly, but gradually, so that the density of the steam in the cylinder is 
already diminished at the cut off point. The valve will cut off quicker the less 
the angle z is. 

See Figure 2, Plate VIII. The actual pressure will not form a sharp corner at 
e, or follow the line e,e,e, as would be due when cut off at j the stroke, but the 
line //'//will be the true diagram. Including the steam in the ports and 
steamchest, the density at the end of the stroke will correspond nearly with the 

Table. 



































560 


Steam-Boilers. 


STEAM-BOILERS. 


The accompanying proportions are averages of a great number of good 
marine boilers. 

Letters Denote. 

D = diameter of the steam-cylimler in inches. 

S= stroke of piston under which steam is fully admitted in inches. 
n = number of double strokes or revolutions per minute. 
w — pounds of water evaporated per pound of coal per hour. 

V— volume coefficient from the steam table. 

S = lire grate in square feet for each cylinder and with natural draft. 

To Find the Area of Fire Grate. 

„ U 1 S n 4.6G w V a 


4.66 wV' 


n- 


V 1 S 


. 1 , 2 . 


Example 1. A steam-engine of D = 54 inches diameter of the cylinder, and 
stroke of piston 96 inches, cut off at £ £=48 inches, is to make 22 revolutions 
per minute. Anthracite coal to be used, that evaporates tc = 7 pounds of 
water per pound of coal, and to carry 27 pounds ot steam per square inch. 
V= 649. Required the area of fire grate E = ? in square feet. 


a _ _54 2 x 48 x 22 


145.34 square feet. 


4.G6 x7 x 649 

Example 2. A steam-boiler of E = 128 square feet is to be used for an 
engine of D = 86 inches diameter and 64 inches stroke, cut off the steam at 
|, then £ = 42.66 inches. Steam-pressure to be kept at 25 pounds per square 
inch F=679, w = 6.5. Required for how many revolutions per minute can 
the steam be kept at 26 pounds? 


71 = 


4.66x6.5x679x128 


36 2 x 42.66 


= 47.6 revolutions. 


Horse-Power of the Fire Grate. 

UP = horse-power of the fire grate. 

P = pressure in the boiler in pounds per square inch excluding the 
atmosphere. 

p = vacuum in the condenser in pounds per square inch. 


B = . 


IP X 


IP = 


e Vw (P+0.8 p) 


Fic(P+0.8p)’ x 

' 1 the stroke, x = 27700, saves 55 

Cut off the 
steam at 


3, 4. 


1 

8 

u 

* = 31400, “ 

49 

1 

T 

it 

x = 38400, “ 

38 

0 

3 

(i 

* = 45500, “ 

26 

.1 

U 

* = 49100, “ 

20 


per cent. 
y of fuel. 


Steam admitted throughout the stroke x — 61700, saves 0 per cent. 

Example 3. Steam-boilers are to be constructed for an engine of 650 horses, 
the steam to he cut off at £ the stroke; and P=36 pounds per square inch, 
V= 544, w = 7.5 pounds of water evaporated per pound of coal. Required the 
tire grate in the boilers E = ? in square feet. 

b = _ 650x 3840 0_— 134 square feet 

554x7.5(36 + 0.8x11) 16 sqnare leet * 
























Steam-Boilers. 


561 


Example 4. Required the horse-power of a fire grate B = 112 square feet 
to carry 18 pounds steam, and cut off at | the stroke? F=810, m> = 7 pounds. 


LP 


112x18x810x7 

45500 


= 251.2 horses. 


Consumption of Coal. 

C— coal consumed in pounds per hour. 


< 7 = 


3 D*Sn 
w V ’ 


( 7 = 


14 ipx 


V w(P+0.8 p)* 


. 5, 6. 


Example 5. A steam-engine of Z> = 42 inches diameter and 48 inches stroke, 
cut off the steam at j, S= 16 inches is to make n = 6o revolutions per minute 
with a pressure of 34 pounds per square inch, F=564 and w; = 6 pounds. 
Required the consumption of coal in pounds per hour C=1 


c 


3 x 42 2 x 16 x 65 
6 x 564 


1625 pounds per hour. 


Examples. A pair of steam-engines of ZP = 260 horses are to be worked 
with P = 28 pounds per square inch, cut off at | the stroke, F=635, the coal 
to evaporate w — 6.5 pounds of water per pound of coal. Required the con¬ 
sumption of coal in pounds per hour C=? 


( 7 = 


14x260x31400 
630x6.5(28 + 0.8x10) 


= 775 pounds per hour. 


It will be observed in the Formulas 4 and 6, that the higher steam used the 
less fuel and fire-grate is required for the same power; the proportion of fuel 
will be nearly as the square root of the steam-pressure, and still more fuel 
is saved by cutting off the steam at an early part of the stroke. 

Heating Surface O Compared with Grate. 


In common stationary boilers, . 
Returning flue boilers, 

Tubular boilers (marine), . 
With vertical tubes (Martin), . 


0 = 20 B. 
0=25 B. 
0 = 30 B. 
0 = 35 B. 


Cross-Area of Flues (Calorimeter). 

In the common single returning flue boilers the cross-section area of the 

first row should be,.0.18 B, 

Returning row, flues or tubes,.0.13 B. 

Cross-section area of chimney at the top, A — . . . 0.16 B. 

Height of Chimney. 


rn 

h = -p—- 2 , h = 

4 E 2 

H>= 1.45 AV h y A = 


ip* 


2.1 A’ 

ip 


C=2 sVh+2, 


B =; 


c 


of 


1-45 \/h’ “ 2/4 + 2 

Example. Area of fire-grate E = 140 square feet to consume C= 2100 pounds 
coal per hour. Required the height h of the chimney ? 

2100 2 


h = 


4x 140 2 


= 56.3 feet, the answer. 


36 
















562 


Horse-Power of Boilers. 


HORSE-POWER OF STEAM-BOILERS BY EVAPORATION. 

IP = horse-power of evaporation. 

P= steam-pressure in pounds per square inch above vacuum. 

W— cubic feet of feed-water evaporated per hour from 32° F. 

V= steam volume compared with that of water at 32°. 


IP = W 


144 P(V —1) 
1980000 


1 


This formula gives the natural effect of the evaporation without expand¬ 
ing the steam. 


With expansion IP 


WP ( V — 1)(1 -f hyp.log. X) 
13748.4 


-3T=grade of expansion. 


13748.4 IP 
w ~ P ( V — 1)' 


2 . 


Natural Effect of Evaporation without Expanding 

the Steam. 


Steam- 

pressure 

above 

Water evaporated per hour per horse¬ 
power. 

Horse-power 
per cubic 
foot. 

Equivalent 
work per 
unit of 

vacuum. 

Cubic feet. 

Cubic in. 

Pounds. 

heat. 

P 

W 

w 

lbs. 

IP 

J 

5 

0.0024 

1041.0 

29.852 

1.6600 

46.584 

10 

0.5796 

1002.0 

28.723 

1.7253 

48.032 

14.7 

0.5701 

985.2 

28.252 

1.7540 

48.583 

20 

0.5641 

974.7 

27.954 

1.7727 

48.902 

25 

0.5593 

966.5 

27.717 

1.7879 

49.040 

30 

0.5553 

959.6 

27.518 

1.8008 

49.403 

35 

0.5516 

953.2 

27.337 

1.8130 

49.665 

40 

0.5483 

947.4 

27.170 

1.8238 

49.832 

45 

0.5451 

941.9 

27.012 

1.8345 

50.150 

50 

0.5420 

936.6 

26.861 

1.8540 

50.244 

55 

0.5391 

931.5 

26.715 

1.8549 

50.440 

60 

0.5362 

926.6 

26.573 

1.8049 

50.651 

65 

0.5334 

921.6 

26.429 

1.8747 

50.861 

70 

0.5305 

917.1 

20.300 

1.8850 

51.000 

75 

0.5280 

912.5 

26.168 

1.8936 

51.265 

80 

0.5254 

907.9 

20.038 

1.9033 

51.470 

85 

0.5228 

903.5 

25.910 

1.9127 

51.670 

90 

0.5203 

899.1 

25.783 

1.9219 

51.865 

95 

0.5178 

894.7 

25.660 

1.9312 

52.077 

100 

0.5153 

890.5 

25.537 

1.9406 

52.264 

105 

0.5129 

880.2 

25.415 

1.9497 

52.513 

110 

0.5104 

882.0 

25.295 

1.9592 

52.722 

115 

0.5081 

877.9 

25.177 

1.9681 

53.053 

120 

0.5057 

873.8 

25.060 

1.9774 

53.137 

125 

0.5034 

809.8 

24.945 

1.9805 

53.351 

130 

0.5008 

865.3 

24.815 

1.9968 

53.572 

135 

0.4988 

861.9 

24.71S 

2.0048 

53.788 

140 

0.4965 

858.0 

24.006 

2.0140 

54.000 

145 

0.4943 

854.1 

24.494 

2.0230 

54.206 

150 

0.4921 

850.4 

24.387 

2.0321 

54.427 

























Horse-Power of Boilers. 


563 


Legal Horse-Power of Steam-Boilers. 

The legal horse-power of a steam-boiler fired with a given kind or quality 
of fuel should be the power passing from the boiler into the steamer-pipe 
with pressure above that of the atmosphere, because the boilermaker is not 
responsible for how the steam-user employs that steam. The only difference 
in the formulas for horse-power and evaporation will then be in taking the 
steam-pressure above that of the atmosphere. 

ro _TFP(F-l) > 13748.4 IP 

13748.4 P ( V — 1) 

The last column (J) in the tables gives the equivalent work in foot-pounds 
per unit of heat as realized in steam without expansion. 

For further information on this subject, see Nystrom’s S:eam Engineering. 


Reduction for Temperature of Feed-Water. 


Temp. /. 

Seduction E. 

Logarithm. 

Temp. t. 

deduction R. 

Logarithm. 

40 

0.9932 

9.9970367 

130 

0.9105 

9.9592620 

50 

0.9851 

9.9934803 

140 

0.9000 

9.9546693 

60 

0.9761 

9.9895039 

150 

0.8912 

9.9499637 

70 

0.9671 

9.9854516 

160 

0.8815 

9.9451979 

80 

0.9577 

9.9812455 

170 

0.8719 

9.9404765 

90 

0.9486 

9.9770612 

180 

0.8625 

9.9357359 

100 

0.9392 

9.9727643 

190 

0.8529 

9.9308916 

110 

0.9296 

9.9683116 

200 

0.8432 

9.9259440 

120 

0.9199 

9.9637468 

212 

0.8317 

9.9199515 

Legal Horse-Power of Steam 

-Boilers per Rate of Evanora- 


tiou of Water to Steam without Expansion. 

Steam- 

pressure 

Water evaporated per hour per horse¬ 
power. 

Horse-power 

nor pnhip. 

Work, 
ft.-lbs. per 

above at- 




foot. 


unit of 

mosphere. 

Cubic feet. 

Cubic in. 

Pounds. 


heat. 

P 

W 

w 

lbs. 

IP 

J 

5 

2.2562 

3898.8 

140.76 

0.4433 

12.225 

10 

1.3983 

2416.2 

87.235 

0.7150 

19.616 

15 

1.1106 

1919.0 

69.284 

0.9005 

24.701 

20 

0.9654 

1668.1 

60.226 

1.0358 

28.380 

25 

0.9770 

1515.4 

54.711 

1.1403 

31.145 

80 

0.8176 

1411.9 

51.010 

1.2231 

33.433 

35 

0.7743 

1338.1 

48.308 

1.2914 

35.171 

40 

0.7412 

1280.9 

46.244 

1.3490 

36.683 

45 

0.7150 

1235.5 

44.605 

1.3986 

37.988 

50 

0.6935 

1198.3 

43.264 

1.4420 

39.124 

55 

0.6755 

1167.2 

42.140 

1.4S04 

40.118 

60 

0.6600 

1140.6 

41.1*80 

1.5150 

41.012 

65 

0.6467 

1117.5 

40.345 

1.5463 

41.819 

70 

0.6349 

1097.1 

39.607 

1.5750 

42.551 

75 

0.6243 

1078.9 

38.951 

1.6016 

43.221 

80 

0.6149 

1062.5 

38.360 

1.6263 

43.854 

85 

0.6062 

1046.6 

37.822 

1.6495 

44.425 

90 

0.5983 

1033.9 

37.328 

1.6713 

45.011 

95 

0.5910 

1021.3 

36.873 

1.6919 

45.533 

100 

0.5847 

1009.6 

36.451 

1.7115 

46.027 

105 

0.5779 

998.67 

36.056 

1.7303 

46.495 

110 

0.5720 

988.44 

35.686 

1.7482 

46.949 

115 

0.5664 

978.75 

35.337 

1.7655 

47.390 

120 

0.5611 

969.62 

35.007- 

1.7822 

47.812 

125 

0.5561 

960.96 

34.964 

1.7982 

48.213 

130 

0.5513 

952.65 

34.394 

1.7083 

48.604 

135 

0.5468 

945.00 

34.111 

1.8288 

49.737 






































504 


Hokse-Power of Boilers. 


The actual quantity of feed-water of temperature t°, multiplied by the re¬ 
duction in the table, gives the quantity of wafer that would have been 
evaporated when heated from temperature 32° F. 

Example 11. A steam-boiler evaporates IF = 125 cubic feet of water per 
hour under a pressure of P= 75 pounds to the square inch above vacuum, or 
60 pounds above the atmosphere, the temperature of the feed-water being 
— 110°. liequired the natural etlect or horse-power of the evaporation? 


Formula 11. 


IP = 


125 X 75(348.15 — 1) 
13748.4 


236.73 horses. 


That is, 0.528 cubic feet of water evaporated per hour per horse-power, or 
1.893 horse-power per cubic foot of water evaporated per hour. 

Making correction for the temperature of the feed-water 110° (see table), 
the horse-power wiil be 236.73 X .9392 = 222.4 liorse-power, the natural 
effect of the evaporation. 

Example 12. What quantity of water of temperature f° = 90° must be evap¬ 
orated under a pressure of P = 90 pounds to the square inch in order to 
generate a natural effect of IP = 150 horse-power? 


Formula 12. 1F= = ”8.043 cubic feet. 

This volume corrected for temperature gives 78.043:0.9486 = 82.275 cubic 
feet, the quantity of water required. 

Standard Horse-Power of Steam-Boilers. 

The power of a steam-boiler ought to be graded by the dimensions of the 
areas of the fire grate and heating surface, like that of a steam-engine is 
graded by the diameter and stroke of the steam-piston, without taking into 
consideration the evaporative power of the fuel, expansion of the steam, etc., 
which are independent of the size of t he boiler, as well as that of the engine. 

Let 3 denote the area of the fire grate. 

O *** the area of the heating surface in square feet. 

P = pressure of steam in pounds per square inch above vacuum. 

Then the standard nominal horse-power II of a steam-boiler can be ex¬ 
pressed by 


IP 


=v 


B O 1 / P 


10 


10 7P2 

"ov/p- 

Example. Suppose 3 = 100, O =* 3000, and P = 75. 


• 

. 1. 

10 £P2 

V-*' j • 

3 \/ P 

• 

• 

. 3. 

• 

. 2. 

„ /10 /P 2 \ 

• 

■ 

. 4. 


Then, IP 


=v 


100 x 3000 l/75 
10 


510, the standard nominal horse-power. 


Oi'dinnry Performance of Steam-Boilers. 

Natural draft consumes about 12 to 15 pounds of coal per square foot of 
grate per hour, and generates about 4 to 5 horse-power per square foot, of 
grate. 

The heating surface should be about 4 to 5 square feet per horse-power, and 
evaporate 4 to 5 pounds, or 92.5 to 115.5 cubic inches, of sea water per hour at 
the above-mentioned rate of combustion. 

Good coal evaporates about 6 to 8 pounds of water per pound of coal. 

Each horse-power requires the consumption of about 3 to 4 pounds of coal 
per hour. 

Locomotive boilers with forced draft dispose of 80 to 120 pounds of coal per 
hour per square foot of grate surface, and evaporate less water per pound of 
coni* 















Steam-Boilers. 


565 


Ultimate Strength of Tubes and Flues 

for External Pressure against Collapsing. 
Notation. 

D= diameter of tube or flue in inches. 

L = length of the tube or flue in feet. 
t = thickness of iron in decimals of an inch. 

P— external collapsing pressure in pounds per square inch. 


200.000 £ 2 

D/Z 


and 


t= V PD \/L 

447.2 


Example 1. A flue of D = 15 inches diameter, and L = 12 feet long, thickness 
of iron t = 0.25. Required, the collapsing pressure ? 

„ 200,000x0.25 2 , 

P— -—-= 241 pounds to the square inch. 

15 x ]/12 

Example 2. D = 9, L — 10 and t = 0.2. 

200,000x0.04 


P= 


- 1 — = 282 pounds. 


9 \/10 

Example 3. D = 6, L = G, and t = 0.2. Required the pressure PI 

200,000 x 0.04 


P= 


6 x j/ 6 


— : — = 552 pounds. 


Staying Steam-Boilers. 

d = diameter of good iron stay-bolts in inches. 

P = distance apart in inches in salt water on 
flat surfaces. 


74 ' 


P = 


74 d 


P= pressure of steam in pounds per square inch. P= 



D 2 

The following table is given by Mr. Fairbairn, as exhibiting the strongest form 
and best proportions of rivet joints, as deduced from experiments and actual 
practice: 


Thickness 
of plate. 

Diameter of 
rivet. 

Length of rivet 
from head. 

Distance from 
centre to cent. 

Quantity 
single riveted. 

of iap in 
double riveted. 

in. lGths. 

in. 

Ratio. 

in. 

Ratio. 

in. 

Ratio. 

in. 

Ratio. 

in. 

Ratio. 

0.19 = 3 

0.38 

2 

0.S8 

4.5 

1.25 

6 

1.25 

6 

2.10 

10 

0.25= 4 

0.50 

2 

1.13 

4.5 

1.50 

6 

1.50 

6 

2.50 

10 

0.31= 5 

0.63 

2 

1.38 

4.5 

1.63 

5 

1.88 

6 

3.15 

10 

0.38= 6 

0.75 

2 

1.63 

4.5 

1.75 

5 

2.00 

5.5 

3.33 

9.2 

0.50= 8 

081 

1.5 

2.25 

4.5 

2.00 

4 

2.25 

4.5 

3.75 

7.5 

0.63 = 10 

0.94 

1.5 

2.75 

4.5 

2.50 

4 

2.75 

4.5 

* 4.58 

7.5 

0.75 = 12 

1.13 

1.5 

3.25 

4.5 

3.00 

4 

3.25 

4.5 

5.42 

7.6 



































566 


Strength of Boiler-Shells. 


STRENGTH OF BOILER-SHELLS. 

The steam-pressure per square inch in the boiler, multiplied by the inside 
diameter of the shell in inches, is the strain on the plates per inch of length 
of the shell; and as this strain is borne by two sides of the shell, only oue- 
half of it is borne by each side. 

S = ultimate strength in pounds per square inch of section of the plate. 

t = thickness of the plate in fractions of an inch. 

D — inside diameter of the boiler in inches. 

p = steam-pressure in pounds per square inch above that of the atmo¬ 
sphere. 

Coefficients X Cor Safety Strength of Lap-joints. 


Construction of Shell. 

X 

Per cent, 
of strength. 

Solid plate without joints. 

0.5 

100 

Double-riveted drilled holes. 

0.4 

80 

Double-riveted punched holes. 

0.35 

70 

Single-riveted drilled holes. 

0.3 

60 

Single-riveted punched holes. 

0.25 

50 


Steam-pressure, p = 

Diameter of boiler, D = 

Thickness of plate, l — 

Breaking-strain, S = - 


XtS 

D 

Xt S 
V 

Dp 


X S 

Dp 

Xt 


The safety strength is taken one-quarter (£) of the bursting strength. 

The static condition of riveted joints is that the sheering strain on the 
rivet, is equal and opposite to the tearing strain on the plate, and the strength 
to resist these two strains must therefore be alike for the greatest strength 
of the joint. 

It has been found by experiments that the sheering and tearing strengths 
of wrought iron are nearly alike per section strained, and the slight differ¬ 
ence varies either way according to the particular iron experimented upon, 
but on an average the sheering strength appears to have some advantage 
over that of tearing. 

Assuming these two strengths to be alike, the section of the rivet should 
be equal to the section of the plate between the rivets. 

d = diameter of the rivets. 

5 = distance between centres of rivets. 

t — thickness of plate. 

Areas of sections, 0.7854 d 2 = t (5 — d). 8 = — (0.7854 d + 1). 

t 

The English Board of Trade has adopted a very good and proper rule for 
determining the strength of riveted joints; namely : 

Strength of plate = - 10 ° ^ —— per cent, of solid sheet. 

• . 100 a n 

Strength of rivet = ——— per cent, of solid sheet. 

o t 










































Strength of Boiler-Shells: 


567 


a = area of rivet (cross-section) in square inches. 
n— number of rows of rivets. 

p — safety pressure allowed in the boiler or on safety-valve. 

& = breaking strength per square iuch of the iron. 

S X smallest # -f 2 1 

P ~ iJf 

The factor of safety F varies between 5 and 7 under different conditions 
of the boiler. 

For drilled holes make the distance between the centres of the rivets one- 
eighth (|) of an inch less than for punched holes. 


Proportion of Single-riveted Lap-joints with Punched Holes. 


Thickness 
of plate. 

Itiv 

Diameter. 

ets. 

Length. 

Distance 

between 

cent. 

Lap 

of 

joint. 

Area 

of 

rivet. 

Area 

of 

plate. 

Per 

cent, of 
solid 

t 

d 

1 

8 

inches. 

sq.inch. 

sq. inch. 

plate. 

i/8 

5/16 

1/2 

7/8 

1.1 / 4 

0.0767 

0.07031 

64 

3 /16 

7 /16 

3/4 

1.5 /16 

1.1 /2 

0.1-503 

0.16406 

66 

1/4 

1/2 

1.1 /8 

1.1 / 2 

1.3 / 4 

0.1963 

0.25000 

66 

5 /16 

5/8 

1.3/8 

1.7/8 

2 in. 

0.3067 

0.39062 

66 

3/8 

3/4 

1.11 /16 

2.1 / 4 

2.1/4 

0.4417 

0.56250 

66 

7/16 

13/16 

1.15/16 

2.3/8 

2.3/8 

0.5184 

0.68359 

65 

1/2 

7/8 

2.1 / 4 

-.1/2 

2.1 /2- 

0.6013 

0.75250 

64 

9 /16 

1 in. 

2.1 /2 

2.5/8 

2.5/8 

0.7854 

0.91406 

63 

5/8 

1.1 /16 

2.13 /16 

2.3/4 

2.7/8 

0.8904 

1.05468 

62 

11 /16 

1.1 /8 

3.1 /8 

2.7/8 

3.1/8 

0.9940 

1.03125 

61 

3/4 

1.3/16 

3.5/8 

3 in. 

3.3/8 

1.3603 

1.35937 

60 

13 /16 

1.5/16 

3.11 /16 

3.1 / 4 

3.5/8 

1.3605 

1.57422 

60 

7/8 

1.3/8 

3.15 /16 

3.1 /2 

4 in. 

1.4840 

1.85937 

60 

15 /16 

1.1 /2 

4.1 / 4 

3.3 / 4 

4.1 / 4 

1.767 

2.10937 

60 

1 in. 

1.5/8 

4.1/2 

4 in. 

4.5/8 

2.073 

2.375 

60 


Double-Riveted Lap-Joints. 

Double-riveted joints, if properly proportioned, increase the strength of 
the boiler about 40 per cent, on account of the rivets being spaced farther 
apart, leaving more section of plate between them to resist the strain. The 
rivets are arranged in two rows, zig-zag, over one another, as shown in the 
accompanying illustration. For the greatest strength the distance between 
the rivets in the direction of the joint should be double the distance between 
the centre lines of the two rows, and the rivets will then form a right angle, 
or 90°, with one another. 

The distance between the rivets in the direction of the joint can be made 
42 to 50 per cent, greater than between rivets in single-riveted joints. 

The diagonal distance between centres of rivet should be made equal to the 
distance in the direction of the joints in single riveting. (See page 568.) 

























568 


Double-Riveted Joints. 



2 )— B --.0 

' ?K / \ i X 



Double-riveted joints with punched holes, proportioned according to this 
rule, should he 40 per cent, stronger than single-riveted joiuts, and with 
drilled holes about 60 per cent, stronger. 

A. Proportions of Double-Riveted Lap-Joints with 

Drilled Holes. 


Thickness 
of Plate. 

Riv 

Diameter. 

ets. 

Length. 

Dist.betwe 

Central. 

en Rivets. 

Diagonal. 

Distance 
between 
Cent. Lines. 

Lap of 
Joint. 

t 

1/8 

d 

5/16 

1 

1/2 

6 

1.1 / 4 

7/8 

5/8 

1.5/8 

3 /16 

7/16 

3/4 

1.7/8 

1.5/16 

15 /16 

2.3 /16 

1/4 

1/2 

1.1 /8 

2.1 /8 

1.1 /2 

l.i /16 

2.9 /16 

5/16 

5 /8 

1.3/8 

2.5/8 

1.7/8 

1.., '"le 

3.1 /4 

3/8 

3/4 

1.11 /16 

3.3 /16 

2.1 / 4 

1.3/8 

3.7/16 

7/16 

13 /16 

1.15/16 

3.3/8 

2.3/8 

1.11 /16 

4 inches. 

1/2 

7/8 

2.1 / 4 

3.9/16 

2.1 /2 

1.13 /16 

4.1 /4 

9 /16 

1 inch. 

2.1 /2 

3.3 / 4 

2.5/8 

1.7/8 

4.1 /2 

5/8 

1.1 /16 

2.13/16 

3.7/8 

2.3 / 4 

1.15 /16 

4.7 /16 

11 /16 

1.1 /8 

3.1 /8 

4.1 /16 

2.7/8 

2.1 /16 

5.1 /8 

3/4 

1.3/16 

3.5/8 

4.1 / 4 

3 inches. 

2.1 /8 

5.7/16 

13/16 

1.5/16 

3.11 /16 

4.9 /16 

3.1 / 4 

2.5 /16 

5.7/8 

7/8 

1.3/8 

3.15 /16 

4.15 /16 

3.1 / 2 

2.1 /2 

6.7 /16 

15/16 

1.1 /2 

4.1 / 4 

5.5 /16 

3.3 / 4 

2.11 /16 

6.15 /16 

1 inch. 

1.5/8 

4.1 /2 

5.5/8 

4 inches. 

2.7 /8 

7.1 /2 


B. Proportions of Double-Riveted 

Punched Holes. 


Dap-Joints with 


Thickness 
of Plate. 


1/8 
3 /16 
1/4 
5 /16 
3/8 
7/16 
1/2 
9/16 
5/8 
11 /16 
3/4 
13/16 
7/8 
15/16 
1 inch. 


Rivets. 


Diameter. 


d 

5 /16 
7/16 
1/2 
5/8 
3/4 
13 /16 
7/8 
1 inch. 
1.1 /16 
1.1 /8 
1.3/16 
1.5/16 
1.3/8 
1-1 /2 
1.5/8 


Length. 


I 

1/2 
3/4 
1 . 1/8 
1.3/8 

1.11 /16 

1.15 /16 

2.1 / 4 

2.1 /2 

2.13/16 

3.1 /8 
3.5/8 

3.11 /16 

3.15 /16 

4.1 / 4 

4.1 /2 


Dist. between Rivets. 


Central. 


5 

1.3/8 
2 inches. 

2.1 / 4 
2.13 /16 
3.3/8 
3.9/16 
3.13/16 
4 inches. 

4.1 /8 
4.5/16 

4.1 /2 
4.7/8 
5.1/4 
5.5/8 

6 inches. 


Diagonal. 


Distance 
between 
Cent. Lines. 


1 inch. 
1.7/16 
1.9 /16 
2 inches. 

2.3 /8 

2.1 /2 

2.11 /16 
2.13/16 

2.15 /16 

3.1 /16 
3.3/16 
3.7 /16 

3.11 /16 
3.15/16 
4.3/16 


11 /16 

1 inch. 

1.1 /8 

1.7/16 

1.11 /16 
1.13/16 
1.15/16 

2 inches. 

2.1 /16 

2.3 /16 

2.1 / 4 
2.7 /16 
2.5/8 
2.9 /16 

3 inches. 


Lap of 
Joint. 


1.7/8 

2.1 /8 

2.3/8 

2.3 / 4 
3.3/8 

3.1 / 4 
3.3/4 

4.1 /4 

4.3 / 4 

5.1 /8 
5.3/8 
5.5/8 

6.1 /8 

6.5/8 

7 inclies. 























































Steam-Pressure in Marine Boilers, 


569 


Government Inspector’s Table for 

Steam-Pressure in 

Marine Boilers. 


Thickness 
of Boiler 
Plate. 

Tensile Strength of Iron per sq. in., Stamped on Plate. 

Diameter 

50,000 lbs k 

60,000 lbs. 

70,000 lbs. 

of Boiler. 

Riveted. 

Riveted. 

Riveted. 


Single 

Double. 

Single. 

Double. 

Single. 

Double. 

Inches. 

Inches. 

Pounds. 

Pounds. 

Pounds. 

Pounds. 

Pounds. 

Pounds. 


f 

0.25 

115.74 

138.88 

138.88 

166.65 

162.03 

194.43 

36 


0.29 

134.25 

161.11 

161.11 

193.33 

187.90 

225.48 


0.3125 

144.67 

173.60 

173.6 . 

208.32 

202.5 

243.04 


( 

0.375 

173.61 

208.33 

208.33 

249.99 

243.05 

291.66 


f 

0.25 

109.64 

131.56 

131.57 

157.88 

153.5 

184.2 

38 


0.29 

127.19 

152.62 

152.63 

183.15 

178.06 

213.67 


0.3125 

137.00 

164.46 

164.47 

197.36 

191.88 

230.25 


( 

0.375 

164.73 

197.67 

197.36 

236.83 

230.26 

276.31 


( 

0.25 

104.16 

124.99 

125. 

150. 

145.83 

174.99 

40 


0.29 

120.83 

144.99 

145. 

174. 

169.16 

202.99 


0.3125 

130.2 

156.24 

156.25 

187.45 

182.29 

218.74 


( 

0.375 

156.24 

187.48 

187.5 

225. 

218.74 

262.48 


( 

0.25 

99.2 

119.04 

119.04 

142.84 

138.88 

166.65 



0.29 

115.07 

138.08 

138.09 

165.7 

161.11 

193.33 


0.3125 

124.00 

148.8 

148.74 

178.56 

173.61 

208.23 



0.375 

148.8 

178.56 

178.57 

214.2S 

208.33 

249.99 


0.25 

94.69 

113.62 

113.63 

136.35 

132.56 

159.07 

4 -i 4 


0.29 

109.84 

131.80 

131.S1 

158.17 

153.78 

184.53 


0.3125 

118.36 

142.03 

142.04 

170.44 

165.71 

198.85 


{ 

0.375 

142.04 

170.44 

170.45 

204.54 

198.86 

238.63 


0.25 

90.57 

108.68 

108.69 

130.42 

126.8 

152.16 

40 

i 

0.29 

105.07 

126. 

126.09 

151.3 

147.1 

176.52 


0.3125 

113.21 

135.86 

135.86 

163.03 

158.51 

190.21 

. 

l 

0.375 

135.86 

163.03 

163.04 

195.64 

190.21 

228.25 


r 

0.25 

86.8 

104.16 

104.16 

124.99 

121.52 

145.82 

48 - 


0.29 

100.69 

120.82 

120.83 

144.99 

140.97 

169.16 


0.3125 

108.5 

130.2 

130.21 

156.25 

151.9 

182.28 


L 

0.375 

130.2 

156.24 

156.25 

187.50 

182.29 

218.74 


r 

0.25 

77.16 

92.59 

92.57 

111.10 

108.02 

129.62 

54 - 


0.29 

0.3125 

89.5 

96.44 

107.4 

115.72 

107.41 

115.55 

128.88 

138.66 

125.3 

135.03 

150.36 

162.03 



0.375 

115.74 

138.88 

138.88 

166.65 

162.03 

194.43 


r 

0.25 

69.44 

83.32 

83.33 

99.99 

97.22 

116 66 

GO 


0.29 

80.55 

96.66 

96.66 

115.99 

112.77 

135.32 


0.3125 

86.8 

104.16 

104.18 

124.99 

121.52 

145.82 


0.375 

104.16 

124.99 

125. 

150. 

145.83 

174.99 


r 

0.25 

63.13 

75.75 

75.75 

90.90 

88.37 

106.04 

60 < 


0.29 

73.23 

87.87 

87.87 

105.44 

102.52 

123.02 


0.3125 

78.91 

94.(59 

94.69 

113.62 

110.47 

132.56 

1 


0.375 

94.69 

113.62 

113.62 

136.34 

132.57 

159.08 


0.25 

57.87 

69.87 

69.44 

83.32 

81.01 

97.21 

72< 


0.29 

67.12 

80.54 

80.55 

96.66 

93.98 

112.77 


0.3125 

72.33 

86.8 

86.8 

104.16 

101.27 

121.52 

i 


0.375 

86.8 

104.16 

104.16 

124.99 

121.52 

145.82 



0.25 

53.41 

64.09 

64.4 

76.92 

74.78 

89.73 

T8-| 


0.29 

61.96 

74.35 

74.35 

89.22 

88.75 

104.1 


0.3125 

66.77 

80.12 

80.12 

96.14 

93.48 

112.17 

1 


0.375 

80.12 

96.14 

96.15 

115.38 

112.17 

134.6 

| 


0.25 

49.6 

59.52 

59.52 

71.42 

69.44 

83.32 

84 J 


0.29 

57.53 

69.03 

69.04 

82.84 

80.55 

96.66 


0.3125 

62. 

74.4 

74.4 

89.28 

86.8 

104.16 

i 


0.375 

74.4 

89.28 

89.28 

107.13 

104.16 

124.99 



0.25 

46.29 

55.54 

55.55 

66.66 

64.81 

77.77 

90 J 


0.29 

53.7 

64.44 

64.44 

77.32 

75.18 

90.21 


0.3125 

57.86 

69.43 

69.44 

83.32 

81.01 

97.21 

1 


0.375 

69.44 

83.32 

83.33 

99.99 

97.22 

116.66 

/ 


0.25 

43.4 

52.08 

52.08 

62.49 

60.76 

72.91 

96 J 


0.29 

50.34 

60.4 

60.41 

72.49 

70.48 

84.57 


0.3125 

54.25 

65.1 

65.1 

78.12 

75.95 

91.14 

l 


0.375 

65.1 

78.12 

78.12 i 

93.74 

91.14 

109.6 

























570 


Rules for Boilers. 


LLOYD'S RULES FOR BOILERS (BRITISH), ETC. 

Cylindrical Shells. 

The strength of circular shells to be calculated from the strength of the 
longitudinal joints by the following formula: 

C X T X B 

-^-= working pressure, 

where C = coefficient as per following table ; T= thick ness of plate in inches; 
D = mean diameter of shell in inches; B = percentage of strength of joint 
found as follows (the least percentage to be taken): 

p — d 


For plate at joint B = 
For rivets at joint B = 


B = 


B - 


B = 


P 

n X a 
P X T 
n X a 
PX T 
n X a 
p XT 
n X a 


X 100. 

X 100 with iron rivets in iron plates with 
punched holes. 


iu iron plates with 
X 85 with steel rivets in steel plates. 


X 90 with iron rivets 
drilled holes. 


X 70 with iron rivets in steel plates. 


j»XT 

(In case of rivets being in double shear, 1.75 a is to be used instead of a.) 

where p = pitch of rivets; d = diameter of rivets; a = sectional area of 
rivets; n = number of rows of rivets. 

Mem. —In any case where the strength of the longitudinal joint is satisfac¬ 
torily shown by experiment to be greater than that given by this formula, 
the actual strength may be taken in the calculation. 

Table of Coefficients. 

Iron Boilers. 


Description of Longitudinal Joint. 

For 
Plates 
\ inch 
thick 
and 
under. 

For 
Plates 
% thick 
and 
above 
£ inch. 

For 
Plates 
above 
£ inch 
thick. 

Lap joint, punched holes. 

Lap joint, drilled holes. 

Do’ut le butt strap joint, punched holes. 

D'.tpUe butt strap joint, drilled holes. 

155 

170 

170 

180 

1G5 

180 

ISO 

190 

170 

190 

190 

200 

Steel Boilers. 

Description of Longitudinal Joint. 

For 
Plates 
§ thick 
and 
under. 

For 
Plates 
t !, c thick 
and 

above §. 

For 
Plates 
| thick 
and 

above T e c 

Lap joints. 

200 

215 

215 

230 

230 

250 

Double butt strap joints. 


For 
Plates 
above 
f thick. 


240 

2(50 


Note.— The inside butt strap to be at least. £ the thickness of the plate. 

Note. —For the shell plates of superheaters or steam chests enclosed in 
the uptakes or exposed to the direct action of the llame, the coefficients 
should be § of those given in the above tables. 

Proper deductions are to be made for openings in shell. 

All manholes in circular shells to be stilfened with compensating rings. 

The shell plates under domes in boilers so fitted, to be stayed from the 1 
of the dome or otherwise stiffened. 


top 























































Rules for Boilers. 


571 


Stays. 

The strength of stays supporting flat surfaces is to be calculated from the 
weakest part, of the stay or fastening, and the strain upon them is not to 
exceed the following limits; namely, 

Iron Stays.—For screw stays, and for other stays not exceeding 11 inches 
effective diameter, and for all stays which are welded, 6000 lbs. per square 
inch; for unwelded stays above il inches effective diameter, 7500 lbs. per 
square inch. 

Steel Stays.—For screw stays, and for other stays not exceeding 11 
inches effective diameter, 8000 lbs. per square inch ; for stays above 11 inches 
effective diameter, 9000 lbs. per square inch. No steel stays are to be welded. 

Flat Plates. 

The strength of flat plates supported by stays to be taken from the follow¬ 
ing formula: 

C X T 2 

--= working pressure in lb. per square inch, 

where T — thickness of plate in sixteenths of an inch, 
p = greatest pitch in inches. 

C = 90 for plates T 7 S thick and below fitted with screw stays with 
riveted heads. 

C = 100 for plates above TE fitted with screw stays with riveted heads. 
C = 110 for plates x % thick and under fitted with screw stays and nuts. 
C == 120 for plates above T 7 5 fitted with screw stays and nuts. 

C = 140 for plates fitted with stays with double nuts. 

C = 160 for plates fitted with stays with double nuts, and washers at 
least \ thickness of plates and a diameter of § of the pitch, 
riveted to the plates. 

Note.— In the case of front plates of boilers in the steam space, these num¬ 
bers should be reduced 20 per cent., unless the plates are guarded from the 
direct action of the heat. 

Girders. 

The strength of girders supporting the tops of combustion chambers and 
other flat surfaces to be taken from the following formula: 

C X <P X T 

--iX---— = working pressure in lb. per square inch, 

where L = length of girder; P = pitch of stays; D = distance apart of 
girders; d = depth of girder at. centre; T = thickness of girder at centre. 
All these dimensions to be taken in inches. C — 6000, if there is one stay 
to each girder; 9000, if there are two or three stays to each girder; 10,200, if 
there are four stays to each girder. 

Circular Furnaces. 

The strength of plain furnaces to resist collapsing to be calculated from the 
following formula: 

89600 XT 2 . . ... . , 

— ^ working pressure in lb. per square inch, 

where 89600 = constant. 

T = thickness of plates in inches. 

D = outside diameter of furnace in inches. 

L = length of furnaces in feet. If rings are fitted, the length be¬ 
tween rings to be taken. 

, 8000 XT 

The pressure in no case to exceed-^-. 

The strength of the corrugated furnaces (corrugations 11 inches deep) to 
be calculated from the following formula: 

.1000 XJT ~ ~i — working pressure in lb. per square inch. 

where T = thickness of plate in sixteenths of an inch. 

D = greatest diameter of furnace in inches. 









572 


Calorimeters. 


CALORIMETERS. 


G. A. Hirn of Mulhouse devised a simple method in 1868 for measuring 
the humidity of steam by mixing it with cold water. The apparatus used is 
called calorimeter , consisting of a common wooden barrel of about 8 cubic feet 
capacity, which is filled with water, and a steam-hose bads from the boiler 
through the water to the bottom, where the steam is condensed and its 
humidity determined by the weight and temperature of the water. 


w = pounds of cold water put into the barrel. 

h = units of heat per pound of w when cold and above 82°. 

W= pounds of heated water in the barrel after the completion of the ex¬ 
periment ; that is, including the weight of the condensed steam. 

h' = units of heat per pound of lb above 32°. 

f — pounds of priming or water carried over with the steam into the 
barrel. 

S = pounds of saturated steam blown into the barrel. 

If = units of beat per pound of the steam S. 

H'= units of heat per pound of the priming /. 

p = pounds of steam and priming carried over from the boiler into the 
barrel. 

P= units of heat passed over with the steam and priming into the barrel. 


The weight p must then be equal to the sum of the weights of the steam 
S and priming /, which is evidently the same as the difference between the 
weights IF and w. 


That is, 


p = S + f = W—w. 


The total units of heat P passed over with the steam S and priming /must 
then be: 

P=1IS + H'f=WV — wh .2. 


By solving this formula for the steam S, we have : 




P — H’f 


II 


— W — w —/. 


3. 


S = p—f. 

II (P —/) = P — IP f- 
Up — Hf = P — II'f. 
f l (ii-ir) = ifp — p. 


4. 

5. 

6 . 


From this formula we have the weight of priming carried over with the 
steam from the boiler into the barrel; namely, 


/ = 


Up 


II- 


-P 


But P — Wh' — wh, which, inserted in Formula 7, gives: 
Pounds of priming, 


, Hp + w li — Wh' 

H—ir . 8 - 


The percentage of humidity of the steam will then be: 

/ 




r 

P 


IF—u’' 


i 




Formula 8 is readj%for use of the data obtained by the calorimeter when 
p = IF— w. 


























Humidity of Steam. 


573 


Another form of calorimeter depends upon the specific volume of the 
steam from tables. 

A vessel of accurately measured capacity is inserted into the steam-pipe 
on the way to the engine, it is surrounded by the steam or a steam jacket 
carefully clothed, through which the steam is forced to pass. 


This vessel is opened to the steam from the boiler for a sufficient period to 
become thoroughly warmed to the temperature of the steam inside and out, 
so that condensation is not caused by it. 

It is then closed to the steam from the boiler, and opened to a closed coil 
surface condenser. A steam-gauge is on the vessel and a vacuum-gauge on 
the condenser. 

Let V — volume of vessel in cubic feet. 

*• S — speedfic volume for steam pressure. 

“ aS 1 , = specific volume tor vacuum pressure. 

“ W = weight calculated for each evacuation of vessel. 

“ )Vi = observed weight of each evacuation of vessel. 


Then 


Wi — IF = weight of priming 


w x —w 


= priming in hundredths. 


To Approximate the Horse-Power of Horizontal Tubular 

Steam-Boilers. 

D = diameter, and L = length in feet of the boiler. 

For 3" tubes x *= 5. 3y" tubes x — 6. 4" tubes x — 7. 

x 

To Approximate the Weight, of Horizontal Tubular 

Steam-Boilers. 

T , r 144 D n ~L . . 

IF=—- in pounds. 


To Approximate tlie Heating Surface of Horizontal Tubular 

S tenm-Boilers. 

„ 15 D-L . . . 

D —- m square feet. 








574 


IIOKSE-POWER OF CHIMNEYS. 




Horse-Power of Chimneys. 

For safety this table gives the horse-power about 25 per cent, less than may 

be attained in practice . 


Height 

chim - 


Area of chimney in square feet at the top. 


ney . 

0.5 

1 

3 

4 

G 

10 

15 

30 

30 

| 40 

Feet . 

IP 

IP 

IP 

IP 

IP 

IP 

IP 

IP 

IP 

IP 

20 


6.7 

13.4 

26.8 

40.2 

67 

100.5 

134 

201 

268 ^ 

25 

3.7 

7.4 

14.8 

29.6 

44.4 

74 

111.0 

148 

222 

296 

30 

4.0 

8.0 

16.0 

32.0 

48.0 

80 

120.0 

160 

240 

320 

35 

4.25 

8.5 

17.0 

34.0 

51.0 

85 

127.5 

170 

255 

340 

40 

4.5 

9.0 

18.0 

36.0 

54.0 

90 

135.0 

180 

270 

360 

45 

4.75 

9.5 

19.0 

3S.0 

57.0 

95 

142.5 

190 

285 

380 

50 

5.0 

10.0 

20.0 

40.0 

60.0 

100 

150.0 

200 

300 

400 

55 

5.2 

10.4 

20.8 

41.6 

62.4 

104 

156.0 

208 

312 

416 

60 

5.4 

10.S 

21.6 

43.2 

64.8 

108 

162.0 

216 

324 

432 

65 

5.6 

11.2 

22.4 

44.8 

67.2 

112 

168.0 

224 

336 

448 

70 

5.8 

11.6 

23.2 

46.4 

69.6 

116 

174.0 

232 

348 

464 

75 

6.0 

12.0 

24.0 

48.0 

72.0 

120 

180.0 

240 

360 

480 

80 

6.15 

12.3 

24.6 

49.2 

73.8 

123 

184.5 

246 

369 

492 

85 

6.35 

12.7 

25.4 

50.8 

76.2 

127 

190.5 

254 

381 

508 

90 

6.5 

13.0 

26.0 

52.0 

78.0 

130 

195.0 

260 

390 

520 

95 

6.65 

13.3 

26.6 

53.2 

79.S 

133 

199.5 

266 

399 

532 

100 

6.8 

13.6 

27.2 

54.4 

82.8 

136 

204.0 

272 

414 

544 

no 

7.1 

14.2 

28.4 

56.8 

85.2 

142 

213.0 

284 

426 

568 

120 

7.4 

14.8 

29.6 

59.2 

88.8 

148 

222.0 

296 

444 

592 

130 

7.65 

15.3 

30.6 

61.2 

91.8 

153 

229.5 

306 

459 

612 

140 

7.9 

15.8 

31.6 

63.2 

94.8 

158 

237.0 

316 

474 

632 

150 

8.15 

16.3 

32.8 

65.2 

97.8 

163 

244 . 5 . 

326 

489 

652 

160 

S.4 

16.8 

33.6 

67.2 

100.8 

168 

252.0 

336 

504 

672 

170 

8.65 

17.3 

34.6 

69.2 

103.8 

173 

259.5 

346 

519 

692 ' 

180 

8.9 

17.8 

35.6 

71.2 

106.8 

178 

267.0 

356 

534 

712 

190 

9.2 

18.2 

36.4 

72.8 

109.2 

182 

273.0 

364 

546 

728 

200 

9.3 

18.6 

37.2 

74.4 

111.6 

186 

279.0 

372 

558 

744 

210 

9.5 

19.0 

38.0 

76.0 

114.0 

190 

2S5.0 

380 

570 

760 

220 

9.7 

19.4 

38.8 

77.6 

116.4 

194 

291.0 

388 

582 

776 

230 

9.9 

19.8 

39.6 

79.2 

118.8 

198 

297.0 

396 

594 

792 

210 

10.1 

20.2 

40.4 

SO.8 

121.2 

202 

303.0 

404 

606 

808 

250 

10.3 

20.6 

41.2 

82.4 

123.6 

206 

309.0 

412 

618 

824 

260 

10.5 

21.0 

42.0 

84.0 

126.0 

210 

315.0 

420 

630 

840 

270 

10.65 

21.3 

42.6 

85.2 

127.8 

213 

319.5 

426 

639 

852 

2S0 

10.8 

21.6 

43.2 

86.4 

129.6 

216 

324.0 

432 

648 

864 

290 

11.0 

22.0 

44.0 

88.0 

132.0 

220 

330.0 

440 

660 

880 

300 

11.15 

22.3 

44.6 

89.2 

133.8 

223 

334.5 

446 

669 

892 

310 

11.35 

22.7 

45.4 

90.8 

136.2 

227 

340.5 

454 

681 

908 

320 

11.5 

23.0 

46.0 

92.0 

138.0 

230 

345.0 

460 

690 

920 

330 

11.65 

23.3 

46.6 

93.2 

139.8 

233 

349.5 

466 

699 

932 

340 

11.8 

23.6 

47.2 

94.4 

141.6 

236 

354.0 

472 

708 

944 

350 

12.0 

24.0 

48.0 

96.0 

144.0 

240 

360.0 

480 

720 

960 

360 

12.15 

24.3 

48.6 

97.2 

115.8 

243 

364.5 

486 

729 

972 

370 

12.3 

24.6 

49.2 

98.4 

147.6 

246 

369.0 

492 

738 

984 

380 

12.45 

24.9 

49.8 

99.6 

149.4 

249 

373.5 

498 

747 

996 

390 

12.6 

25.2 

50.4 

100.8 

151.2 

252 

387.0 

504 

756 

1008 

400 

12.75 

25.5 

51.0 

102.0 

153.0 

255 

382.5 

510 

765 

1020 























































































































































































■ 
























. 
























7 
















To Reduce Actual Evaporation-. 


575 


To Reduce Actual Evaporation to the Standard at and from 

^l:i° Falir. 

The quantity of heat required for evaporating one pound of water under 
atmospheric pressure at and from 212° Fabr. is 965.66 units. 

IF== actual evaporation in pounds of water per unit of time. 

w = standard evaporation at and from 212° Falir. 

H— units of heat per pound of the steam actually evaporated, and to be 
found in the steam table. 

h = units of heat per pound of the feed-water, to be found in the water 
tables. 

W (If — h) 
w = ---—. 

965.66 

Example .—A steam-boiler evaporated W — 36000 pounds of water per hour 
under a pressure of 50 pounds to the square inch indicated by gauge; the 
temperature of feed-water, 180°. Required the equivalent standard evapora¬ 
tion at and from 212°? 

From steam table, If = 1172.8 

From water table, h = 148.5 


tv = 


1024.3 

36000 (1172.8— 148.5) 


= 38186 pounds. 


965.66 

The actual evaporation multiplied by the tabular number is the standard 
evaporation. 


Temp. 



Steam-pressure in 

boiler above atmosphere. 

\ 


feed- 














water. 

30 

40 

50 

60 

40 

SO 

90 

IOO 

110 

130 

130 

140 

150 

32° 

1.207 

1.211 

1.214 

1.217 

1.220 

1.223 

1.225 

1.227 

1.229 

1.231 

1.233 

1.234 

1.236 

40° 

1.199 

1.203 

1.206 

1.209 

1.212 

1.214 

1.217 

1.219 

1.221 

1.223 

1.224 

1.226 

1.228 

50° 

1.188 

1.192 

1.196 

1.199 

1.201 

1.204 

1.206 

1.208 

1.210 

1.212 

1.214 

1.216 

1.217 

60° 

1.178 

1.182 

1.185 

1.188 

1.191 

1.194 

1.196 

1.198 

1.200 

1.202 

1.204 

1.205 

1.207 

70° 

1.167 

1.171 

1.175 

1.178 

1.181' 

1.183 

1.185 

1.188 

1.190 

1.191 

1.193 

1.195 

1.196 

80° 

1.157 

1.161 

1.165 

1.168 

1.170 

1.173 

1.175 

1.177 

1.179 

1.181 

1.183 

1.185 

1.186 

90° 

1.147 

1.151 

1.154 

1.157 

1.160 

1.162 

1.165 

1.167 

1.169 

1.171 

1.172 

1.174 

1.176 

100° 

1.136 

1.140 

1.144 

1.147 

1.150 

1.152 

1.154 

1.156 

1.158 

1.160 

1.162 

1.164 

1.165 

110° 

1.126 

1.130 

1.133 

1.136 

1.139 

1.142 

1.144 

1.146 

1.148 

1.150 

1.152 

1.153 

1.155 

120° 

1.116 

1.120 

1.123 

1.126 

1.129 

1.131 

1.134 

1.136 

1.138 

1.140 

1.141 

1.143 

1.145 

130° 

1.105 

1.109 

1.113 

1.116 

1.118 

1.121 

1.123 

1.125 

1.127 

1.129 

1.131 

1.133 

1.134 

140° 

1.095 

1.099 

1.102 

1.105 

1.108 

1.110 

1.113 

1.115 

1.117 

1.119 

1.120 

1.122 

1.124 

150° 

1.085 

1.088 

1.092 

1.095 

1.098 

1.100 

1.102 

1.104 

1.106 

1.108 

1.110 

1.112 

1.113 

160° 

1.074 

1.078 

1.081 

1.084 

1.087 

1.090 

1.092 

1.094 

1.096 

1.098 

1.100 

1.101 

1.103 

170° 

1.064 

1.067 

1.071 

1.074 

1.077 

1.079 

1.081 

1.084 

1.086 

1.0S7 

1.089 

1.091 

1.092 

180° 

1.053 

1.057 

1.060 

1.064 

1.066 

1.069 

1.071 

1.073 

1.075 

1.077 

1.079 

1.080 

1.082 

190° 

1.043 

1.047 

1.050 

1.053 

1.056 

1.058 

1.061 

1.063 

1.065 

1.067 

1.068 

1.070 

1.072 

200° 

1.032 

1.036 

1.040 

1.043 

1.045 

1.048 

1.050 

1.052 

1.054 

1.056 

1.058 

1.059 

1.061 

210° 

1.022 

1.026 

1.030 

1.032 

1.035 

1.037 

1.040 

1.042 

1.044 

1.046 

1.047 

1.049 

1.051 

















































576 


Fuel and Timber. 


"1 


WOOD FOR COMBUSTION. 


A cord of wood is 8 feet wide by 4 feet high and 4 feet deep, or the wood is 4 feet 
long. The cord contains 8X^X4 = 128 cubic feet, of which only 74 cubic feet 
is solid wood and 54 cubic feet of space. 

Two cords of wood evaporate about the same quantity of water as one ton of 
anthracite coal. 

The best pine wood evaporates 5 pounds of water per pound of wood consumed 
in a steam-boiler furnace. One cord of wood can be consumed per hour on 60 square 
feet of grate. 




Weight in Pounds per Cord of Different Woods. 




Woods, Seasoned. 

lbs. 

Woods, Seasoned. 

lbs. 

Woods, Seasoned. 

Shell-bark Hickory. 

4469 

Hard Maple . . . 

2878 

Cedar . 

White Oak.... 

3821 

Beech. 

2S75 

Yellow Pine . . . 

Ked-heart Hickory . 

3705 

Hazel. 

2870 

White Pine . . . 

Southern Pine . . 

3375 

Virginia Pine . . 

2689 

Spruce . 

Red Oak .... 

3254 

New Jersey Pine 

2137 

Hemlock .... 


lbs. 


1910 

1904 

1868 

1685 

1240 


Wood requires 32 per cent, more fire grate than mineral coal, for equal genera¬ 
tion of steam. The furnace should be 60 per cent, of cubical space more for wood 
than for coal, or about 4.5 cubic feet per square foot of grate. 


Properties of Fuel. 


' 


Kind of Fuel. 

Units of heat 
per pound of 
fuel. 

Pounds of water 
evaporated per 
lb. of coal. 

Per cent, of 
carbon. 

Cubic feet of air 
requ. for one lb. 
of coal. 

Weight per 
cubic foot. 

Cubic feet to 

stow a ton. 

Bituminous coal 

• 

11600 

7 to 9 

80 

265 

50 

44 

Anthracite coal . 

• 

13340 

8 to 10 

92 

282 

54 

40 

Coke • • • • 

• 

12420 

8 to 10 

86 

245 

31 

72 

Coke, nat. Virginia . 

• 

11600 

8 to 9 

80 

260 

48 

48 

Coke, Cumberland 

• 

11600 

8 to 10 

80 

250 

32 

70 

Charcoal . . . 

• 

13920 

5 to 6 

96 

265 

24 

104 

Dry wood . 

• 

6380 

4 to 5 

44 

147 

20 

100 

Wood with 20 per ct. w'ater 

4930 

4 

34 

115 

25 

100 

Turf, dry . 

• 

7395 

6 

51 

165 

28 

80 

Turf, 20 per ct. water 

• 

5800 

6 

40 

132 

30 

75 

Oil, Wax, Tallow 

• 

11165 

14 

77 

200 

69 

37 

Alcohol (from market) 

• 

8410 

9.56 

58 

154 

52 

42 


Chemically, one pound of carbon burnt 
153 cubic feet of 


to carbonic acid 
atmospheric air. 




requires the oxygen of 


Timber, Green and Seasoned. 


Timber. 

Green. 

Seasoned. 

American Pine . 

44.75 

30.69 

Ash * t • • • 

58.19 

50.00 

Beech .... 

60.00 

53.37 

Cedar. 

32.00 

28.25 

English Oak . 

71.62 

43.50 

Riga Fir . 

48.75 

35.50 


Comparative weight per cu¬ 
bic foot in pounds of green 
and seasoned timber. 


Board Measure. 

Multiply together the three dimensions, width and thickness in inches and the 
length of the lumber in feet; divide the product by 12, and the quotient will be 
the board measure. 


_ 





















































Combustion - and Effect of Fuee. 


577 


Combustion and Effect of Fuel. 

Combustion is the rapid chemical combination of substances with oxygen. 
Carbon C and hydrogen H, are the substances most generally employed for 
generating heat. Carbon is fully consumed when combined with oxygen 0, 
to form carbonic acid gas C0 2 , and partly consumed when in the form of car¬ 
bonic oxide gas CO or smoke, h — units of heat generated by one pound of 
fuel. The heat necessary to raise one pound of water one degree Fah. is one 
unit of heat, w = pounds of water at 212° evaporated per pound of fuel. 
A — volume in cubic feet and a = weight in pounds of atmospheric air re¬ 
quired for the perfec t combustion of one pound of fuel. 

From Mr. Johnson’s Experiments for the U. S. Government it would appear 
that 24 pounds of air is required for each pound of fuel burned in order to 
get complete combustion with chimney draught, and 18 pounds with forced 
draught. That is, with chimney allow 800 cubic feet of air per pound of 
fuel, and with forced draught allow 225 cubic feet. 

The following theoretical table is taken from Rankin’s The Steam Engine. 

One unit of heat = 772 foot pounds, if generated per second will be II— l - 4 
horses, of which we in present practice utilize about one-twentieth. The 
following table will show how important it is to fully consume the com¬ 
bustibles to acid. One pound of carbon consumed to oxide will generate 
only 1*72 horses, instead of 5*66 when consumed to acid. 


Properties of Combustion, per Hour. 


c 

CO 

CO ., 

O 

a 

A 

k 

w 

H 

lbs. 

lbs. 

lbs. 

lbs. 

lbs. 

cub. ft. 

heat. 

lbs. 

bosses. 

1 


3-666 

2-666 

12 

149 

14500 

15 

5-660 

1 

2*666 


1*333 

6 

74*50 

4400 

4-55 

1-720 ; 

0*433 

1 


0-566 

2*550 

31*65 

1650 

5-633 

1*200 ! 

0-272 


1 

0*727 

3-275 

40*56 

3960 

4*100 

1*545 


1-730 

1.375 

1 

3-500 

43*33 

5440 

5*633 

2-125 


0*445 

0*392 

0*222 

1 

12*38 

1210 

1*250 

0*472 i 


*0358 

•0246 

*0261 

•0808 

1 

97*3 

0*100 

0-038 ! 


0*584 

0244 

0170 

0*800 

9920 

966 

1 

0*378 ; 


1*550 

0-651 

0*470 

2*120 

.26-30 

2558 

2-645 

1 


t. 




37 




























578 


Radiation ok Heat from Steam-Pipes. 


RADIATION OF. HEAT FROM STEAM-PIPES, 

Boilers or Steam Cylinders. 

Notation. 

D = outside diameter of steam-pipe, without casing and limited to not more 
than 12 inches. 

T = temperature of the steam, Fahr. degrees. 
t — temperature of the external air. 

h = heat units radiated per square foot per hour, on uncovered pipe. 

A — outside area in square feet of steam-pipe. 


H = horse-power lost by radiation of heat. 



Wind. 

Exp. n. 

n = exponent of the wind, which varies with the cur- 

Calm. 

1.20 

rent of air or draft about the steam-pipe, as in 

Gentle. 

1.22 

the following table: 

Brisk. 

1.24 


Storm. 

1.26 

The loss of heat will then be per hour (empirical formula)- 



h = 0.001122 [450 + (12 — £>) 2 ] (T—lf 

• • 

. . 1. 

Horse-power lost II — A .!. 1 ... 

2564 

• • • 

. 2. 

One horse-power consumes or generates 2564 heat units per hour. 

By logarithms the Formula 1 is reduced to— 

> 

log. h = log. & -f n log. (T — t ). . 

• • 

. . 3. 

The log. 7c is contained in the second column of the accompanying table for 
different diameters of pipes. 

For any uncovered plane or cylindrical surface above 12 inches in diameter the 
radiation in units of heat per square foot per hour will be— 

/< = 0.505 (T— tf. 

• • • 

. 4. 


The effect of thickness of metal is inappreciable for practical purposes. 

Example 1. The California S. N. Co.’s steamer Julia has a steam-pipe 40 feet 
long by D — 9 inches in diameter, and two branch-pipes 12 feet long by D = 4 
inches each, all uncovered. Pressure of steam, 100 lbs. T = 337°. Temperature 
of the external air t = 70°. Required, the loss of heat and power by radiation? 

In calm wind n =■ 1.2. See table fur n. 

h = 0.001122 [450 + (12 — 9) 2 ] (.337° — 70 0 ) 1 - 2 = 420.34, 
the units of heat lost per square foot per hour. 

Area of pipe, A = 0.75 X 3.14 X 40 = 94.24 square feet. 

Power lost, 11 = X 420 34 = 15.5 horses. 

2564 

The branch-pipes lose 4.6 horses. 

The total loss of power 20.1 horses. 

The same pipes covered with 2-inch-thick felt would gain 20.1 X 0.93 == 18.7 
horsepower. 

Example 2. In the factory of Bellavista, Peru, are 150 feet of uncovered steam- 
pipes 1) = 3 inches in diameter. Steam-pressure, 45 lbs. T= 292° Fahr. 
Temperature of external air t — 68, and wind gentle, n = 1.22. Required, the 
horse-power and fuel lost? 

Formula 3. log. h = 0.77408 — 1 +1.22 log. (292 —68) = 2.50708, 
or 178 units of heat lost per hour. 

Area of steam-pipes, A = 0.785 X T50 = 117.8 square feet. 

117 8 X 173 

Formula 2. H = —- = 8 horse-power nearly, which is lost by radiation. 

2564 























Radiation of Heat from Steam-Pipes, 


579 


The same pipes covered •with one-inch felt will gain 8 X 0-89 (see table) = 7.12 
horse-power. The steam-engine in Bellavista works without expansion, and con¬ 
sumes about 10 lbs. of coal per horse-power per hour = 7.12 X 10 = 71.2 pounds, 
and for 8 hours’ working = 569.6 lbs. ol' coal lost per day. 

The radiation of heat from steam-pipes causes a condensation of steam to water, 
and the weight in pounds of water 6o condensed is equal to the units of heat 
radiated, divided by the latent heat of the steam in the pipe. The Formula 1 will 
also answer for calculating the quantity of heat radiated from steam or water- 
pipes for heating rooms. 

| Many experiments have been made proving no practical gain with a thickness 
of covering exceeding one inch.—W. I). M.] 

Percentage of Heat or Power Gained 


by covering steam-pipes with felt and canvas outside. 


Diam 

D 

Logarithm 

k. 

1 

¥ 

3 

8 

Thick 

1 

S 

.ness o 

3 

¥ 

f felt 

1 

joverii 

n 

lg in i 
2 

nches. 

3 

4 

6 

1 

0.80663—1 

65 

76 

81 

86 

92 

94 

96 

98 

99 

100 

2 

0.79561—1 

63 

74 

80 

85 

90 

93 

95 

97 

98 

99 

3 

0.77408—1 

61 

72 

79 

84 

89 

92 

95 

96 

98 

99 

4 

0.76096—1 

59 

71 

77 

83 

88 

92 

94 

96 

97 

99 

5 

0.74809—1 

57 

69 

76 

82 

87 

91 

94 

96 

97 

99 

6 

0.73670—1 

54 

67 

74 

81 

86 

91 

91 

95 

97 

99 

7 

0.72668—1 

52 

66 

73 

81 

85 

90 

93 

95 

97 

99 

8 

0.71838—1 

50 

64 

71 

80 

85 

90 

93 

95 

97 

99 

9 

0.71179—1 

47 

62 

70 

79 

85 

89 

93 

95 

97 

99 

10 

0.70705—1 

45 

61 

69 

78 

84 

89 

92 

95 

96 

99 

11 

0.70417—1 

42 

59 

67 

78 

83 

88 

92 

94 

96 

98 

12 

0.70321—1 

40 

58 

66 

77 

83 

88 

92 

94 

96 

98 


Lap-welded American Cliarcoal Iron Poiler Tulbes. 

Pascal Iron Works Tasker Iron Works, 

Philadelphia. New Castle, Del. 


Diameter of 

Heating surface 


Length of tube 

Area of cross- 

! Weight 

the tube. 

per foot of length. 

of metal. 

per square loot. 

section. 

per foot of 

Outside 

Inside. 

Outside 

Inside. 

Outside. 

Inside. 

Outside 

Inside. 

length. 

Inches. 

Inches. 

Sq. ft. 

Sq. ft. 

Wg. inches. 

Feet. 

Feet. 

Sq. in. 

Sq.in. 

Pounds. 

1 

0.856 

0.2618 

0.2241 

15 0.072 

3.819 

4.460 

0.785 

0.575 

0.708 

1.25 

1.106 

0.3272 

0.2895 

15 0.072 

3 056 

3.455 

1.227 

0.960 

09 

1.50 

1.334 

0.3926 

0.3492 

14 0.083 

2 547 

2.863 

1.767 

1.396 

1250 

1.75 

1.560 

0.4580 

0.40S4 

13 0.095 

2.183 

2.448 

2.405 

1.911 

1.665 

2 

1.804 

0.5236 

0.4723 

13 0 098 

1.909 

2.118 

3 142 

2.556 

1.981 

2.25 

2.054 

0.5890 

0.5377 

13 0.098 

1.698 

1.850 

3.976 

3.314 

2 238 

2 50 

2.283 

0.6545 

0.5977 

12 0.109 

1.528 

1 673 

4.909 

4.094 

2.755 

2.75 

2.533 

0.7200 

0.6631 

12 0.109 

1.390 

1.508 

5.940 

5.039 

3.045 

3 

2.783 

0.7853 

0 7285 

12 0.109 

1.273 

1.373 

7.069 

6 083 

3.333 

3 25 

3.012 

0.8508 

0 7885 

11 0.119 

1 175 

1.268 

8.296 

7 125 

3.958 

3.50 

3.262 

0.9163 

0 8430 

11 0.119 

1.091 

1.171 

9.621 

8.357 

4.272 

3.75 

3.512 

0.9S17 

0.9194 

11 0.119 

1.018 

1.088 

11.045 

9.687 

4.590 

4 

3.711 

1.0472 

0.9794 

10 0.130 

0.955 

1.023 

12 566 

10.992 

5.320 

4.50 

4.241 

1.1781 

1.1105 

10 0.130 

0.849 

0 901 

15.904 

14126 

6.010 

5 

4.720 

1.3680 

1.2357 

9.5 0.140 

0.764 

0.809 

19.635 

17 497 

7.226 

6 

5.699 

1.5708 

1.4920 

9 0.151 

0.637 

0.6:0 

28.274 

25.509 

9.046 

7 

6.657 

1.8326 

1.7428 

7.5 0172 

0.545 

0.574 

38.484 

34.8i'5 

12.435 

8 

7636 

2.0944 

1.9991 

7 0.182 

0.478 

0.500 

50.265 

45.795 

15.109 

9 

8.615 

2.3562 

2.2553 

6.5 0.193 

0.424 

0.444 

63.617 

58.291 

18.002 

10 

9.573 

2.5347 

2.5022 

5.5 0.214 

0.382 

0 399 

78.540 

71.975 

22.19 


The length of tube and thickness of metal can be varied to suit orders. 

The heating surface of a boiler tube is that exposed to the fire. 

Safe ends of thicker metal welded on the ends of tubes as may be required. 


























































580 


Blqwing OFF. INCRUSTATION. 


BLOWING OFF. SALT WATER. INCRUSTATION. 

Sea water contains about 0.03 its weight of salt. When salt water boils, fresh 
water evaporates and the salt remaihs in tlie boiler; consequently the proportion 
of salt increases as the water evaporates, until it has reached 0.36 weight to the 
water; the salt will then commence to crystallize in the boiler, and the water in so¬ 
lution will hold 0.36 weight of salt to 1 of water. 

To prevent deposit in the boiler, it is necessary to keep the salt below this pro¬ 
portion, which is overcome by withdrawing (blowing off) part of the supersalted 
water, while less salted (feed) water is replaced. It is found in practice that when 
the proportions are kept 0.12 of salt to 1 weight of water, the deposit will bo very 
slight. To obtain this it will be necessary to blow off— 


(L03 

0.12 


= 0.25 parts of the feed water, or, 


if a brine-pump is used, it should be at least 0.25 of the feed-pump. 

IF= cubic feet of supersalted water to be blown off per minute. TV = 
D, S, n and F, as before, we shall have— 


1 )2 Sn 
3000 V‘ 


Example. D = 30 inches, stroke of piston 36 inches, cut off at half stroke 8= 
18, making 14 revolutions per minute, with a pressure of 30 pounds per square 
inch, F=610. How much water must be blown off per minute? 

W = 31^- X l gX 1 1=0.124 cubic feet. 

3000 X 610 

Heat "Wasted l»y Blowing Off. 

Letters denote, 

w = water evaporated ) . , . , . ... * ... 

and IF= water blown off } m cub,c feet P er unit of tlIue - 

t = temperature of the feed water. 

T= “ “ blowing off. 

11= heat wasted, per cent. 

jj _ W(T-t) 

to (990 + T-T) 

Example. Let the quantity of water blown off be X / A of the food water, we have 
TF= 1, and m> = 2; the boiling-point of the water will then bo T= 215.5°; let the 
feed water taken from the hot-well be t = 100°. Required, the quantity of heat 
lost? 

XT— 1(215.5°- 100) n ni>i< r> I* 

11 — ■-= 0.066 or 6.6 per cent. 

2(990 + 215.5 —100) 

This is a very trifling quantity of heat lost. 


Heat AVasted by Incrustation. 

The conducting power of iron for heat is about 30 times that of scale; hence a 
considerable portion of heat is lost when the scale becomes thick in a boiler. 

t = thickness of the scale in 16ths of an inch. 

II = per cent, of heat wasted."" 


11 = 


?2 


32 +£2 

Example. The 6cale in a boiler is 5-sixteonths of an inch thick. IIow much 
heat is lost by it? 

5 2 

0.438, or 44 per cent., nearly, 


11= 


32 + 53 

which goes out through the chimney. 

This is merely to show that the heat lost by blowing off is but triflln* compared 
with the heat lost by scale formation, which additionally iryuros the boiler bv soft¬ 
ening and fracturing the iron. J 

When boilers are taken good care of by cleaning and blowing off at short inter¬ 
vals, the scales need not exceed 1-sixteenth of an inch. 




























Bell Signals. 


581 


Proportions of Salt in Water! 

its boiling-point and weight per cubic foot. 


Salt 
in 100 
Weights. 

Boiling 

temp. 

Fuhr. 

Weight 

per 

cub. ft. 

Spe¬ 

cific 

grav. 

Salt 
in 100 
weights. 

Boiling 

temp. 

Fa hr. 

Weight 

per 

cub. ft. 

Spe¬ 

cific 

gniv. 

0 

212° 

pounds. 

59.837 

1.00 

21 

218.304 

pounds. 

72.224 

1.21 

1 

212.205 

60.431 

1.01 

22 

218.690 

72.728 

1 22 

2 

212.422 

61.024 

1.02 

23 

219.082 

73.395 

1.23 

3 

212.619 

61.617 

1.03 

24 

219.183 

73.980 

1.24 

4 

212.887 

62.209 

1.04 

26 

219TSS7 

74.565 

1.25 

5 

2131136 

62 801 

1.05 

26 

220.296 

75.148 

1.26 

6 

213.394 

63.393 

1.06 

27 

220.713 

75.732 

1.27 

7 

213.664 

63.984 

1.07 

28 

221.131 

76316 

1.28 

8 

213.912 

61.575 

1.08 

29 

221.558 

76.899 

1.29 

9 

214.229 

65.166 

1.09 

30 

221.984 

77.482 

1.30 

10 

214.526 

65.756 

1.10 

31 

222.119 

78.064 

1.31 

11 

214.801 

66.346 

1.11 

32 

222.857 

78.646 

1.32 

12 

215.115 

66.935 

1.12 

33 

223 302 

79.228 

1.33 

13 

215.446 

67.524 

1.13 

34 

223.733 

79.810 

134 

14 

215.797 

68.113 

1.14 

35 

221.208 

80.390 

1.35 

15 

216.132 

68.701 

1.15 

36 

221.668 

80.970 

1.36 

16 

216.177 

69.289 

1.16 

37 

225 139 

81.550 

1.37 

17 

216.826 

69.877 

1.17 

38 

225.611 

82.130 

1.38 

18 

217.186 

70464 

1.18 

39 

226.087 

82.709 

1.39 

19 

217.550 

71051 

1.19 

40 

226.572 

83.288 

1.40 

20 

217.924 

71.377 

1.20 

Saturates with 40 parts of salt. 


"Water does not increase in volume by addition of the above proportions of salt. 


Code of Signals for Engine-Room. 


Go ahead, . . 

• • • 

one stroke. 

Back, 999 

ii • • 

. two strokes. 

StOp, 9 9 9 

i • • • 

one stroke. 

Slowly, . 

Jl. • • 

. two short. 

Full speed, 

HI . . 

three short. 

Go ahead slowly, 

JJ.i. . 

. one long, two short. 

Back slowly, 

JJ..U . . 

two long, two short. 

Go ahead, full speed, 

1 “1 . 

■ 0 - - 0 - 0 - 0 - • 

. one long, three short. 

Back fast, ♦ . 

JLJU1 • 

two long, three short. 

Hurry, . 


. three short repeated. 


It is also customary to have two bells in the engine-room—a large bell for the 
long strokes, and a smaller for the short strokes. 




























582 


Driving a Nail. 


Fig. 1. 


\V 


M 

\ 

V 

Tv*. 

«|s/:_y 




N 

1 

• 

i 


Fig . 2 . 


On Driving a Nail into a Piece of Wood. 

The illustration represents a nail driven 
through a piece of wood by a weight TF rest¬ 
ing on the head of the nail. It is supposed that 
the resistance to tlie nail in the wood is equal to 
the weight W, so that the slightest additional 
force would cause the weight to drive the nail 
down to its head, as shown by the dotted lines. 

In driving a nail into a piece of wood the re¬ 
sistance is not uniform, for the deeper the nail 
is driven in the greater is the resistance; but 
the mean force of resistance will always he as 
the following Formula.2. 

Lets denote the space which the nail is driven 
into the wood by the weight. 

Let the same nail be driven into the same 
piece of wood by the aid of a lever, as repre¬ 
sented by Fig. 2. The force F, acting on the 
long lever L, presses on the nail equally to the 
weight IF. The force of resistance F' to t he nail 
in the wood, which is equal to t he weight IF, Fig. 

1, acts on the short lever l. The fulcrum of the 
lever is at c. 

The force F with the lever L is adjusted so 
that it balances the resistance F’ acting on the 
lever l. 

From the well-known law of levers we 
have 

F:F' = l: L, 

That is to say, the weight F is as much smaller than the weight IF as the 
lever l is smaller than L. 

Let S represent the space which the weight falls in pressing down the nail 
in the wood, and s = the space the nail was driven in, which is the same as 
the space s. 

It is well known in geometry that 

s : S = l : L, 
and, as F\F'±l:L ,’ 
we have F: F' = s : S. and FS = F's. 

{ . ' ft ♦ S • • 

~ F's 







F' = 


F's 
S ’ 
FS 


2 . 


F • 
FS 
F> * 




r 

V ** 4 ‘ 


r * 


VI* 


1 , " * 

























Circular Elbow. 


583 


To Construct a Circular Elbow. 



i _^rfrn 





n 



* is LI ft M $ 0 <f t, 


3 



Having given the radii R and r of the circular elbow, divide the arc into as 
many sections as desired, say three, as shown on the illustration. Each sec¬ 
tion is cut out of a rectangular plate A BCD , of which the length A B is 
equal to the circumference of the pipe, and the breadth B C is equal to the 
outer side (a + 2 b) of each section. The curve c d e is the same as the wave¬ 
line and constructed in the same way, as will he understood by the illus¬ 
tration. 

The wave-line can be laid but by ordinates, for which divide half the 
length A B into 16 equal parts; and number 
them as shown. The adjoining table gives 
the length of each ordinate when the height 
b is the unit. Multiply each tabular ordi¬ 
nate with the actual height b, and the prod¬ 
uct is the actual ordinate to be laid down for 
the curve. 

It is not necessary to use all the 16 ordi¬ 
nates, for half of them, or those with even 
numbers, may be sufficiently correct. 

For a rectangular elbow pipe the height b 
is equal to the diameter of the pipe, and the 
lengt h A B is equal to the circumference of 
the pipe. 


Ab. 

Ordinates. 

Ab. 

Ordinates. 

1 

.009607 

9 

.59755 

2 

.03806 

10 

.69134 

3 

.08426 

11 

.77778 

4 

.14644 

12 

.85355 

5 

.22221 

13 

.91573 

6 

.30866 

14 

.96194 

7 

.40245 

15 

.99039 

8 

.60000 

10 

1.0000 



























































584 


Weight or Boilehs and Engines. 


To Approximate the Weight of Steam Boilers. 

The area of fire grate gives a nearer approximation to the weight ef 
Marine boilers, than the heating surface. 

Notation. 

prj = total fire grate in square feet. 

}V = weight of the boiler in pounds, including fire bars, doors, smoke 
pipe, fire tools and appendages, but without water. W=800 
Examvle. Required the weight W=1 of a steam boiler of E3=2o0 
square feet, grate surface. 

W —800X250 = 200,000 lbs. 

Weight of the water is about three-fourths of W or of the total weight 
of boilers. 

Weight of rivets, braces or stays, doors and fire bars, is about one 
quarter of W or of the total weight of boilers. 

To Approximate the Weight of Engines. 

Notation. 

*S = stroke ^ } of c y linder in inches. 

TE= weight of engine in pounds, including engine room tools, oil and 
tallow tanks, wheels, propeller and shafts. 

coefficient k. 

Trunk and oscillating engines, -------4 

Direct action paddle wheel engines, ------ 4-25 

Horizontal direct action propeller engine, - ... 4-5 

Geared propeller engines, - -- -- -- -6' 

American overhead beam engines, ------ 6-5 

Side lever engines, - - ---'6* 

Horizontal direct action high pressure, ----- 3-5 

IV = k Ivy'S. 

Example. Require the weight W=1 of a pair of Horizontal direct ao- 
tion propeller engines of D=72, S'—36 inches, k= 4-5. 

W= 4 , 5X'72V36 — 139968 lbs. for one cylinder, multiplied by 2=279936 
lbs. the weight required. 

For trunk engines must be taken the largest diameter. 


Practical Thickness in Decimal© of an Inch of Good 
Plate Iron in Steam-boilers, Single Riveted. 


P = steam, pressure in pounds per square inch above atmospherre. 


Press. 





Diameter 

of Boiler in 

Inches. 

• 




P. 

10 

15 

20 

25 

30 

35 

40 

50 

GO 

70 

80 

90 

100 

130 

150 

300 

10 

.10 

.10 

.11 

.11 

.12 

.12 

.13 

.13 

.14 

.14 

.15 

.15 

.15 

.16 

.17 

.20 

15 

.10 

.10 

.11 

.12 

.13 

.13 

.13 

.14 

.15 

.15 

.16 

.18 

.19 

.19 

.22 

.25 

20 

.11 

.11 

.12 

.12 

.13 

.14 

.14 

.15 

.16 

.17 

.18 

.20 

.20 

.22 

.26 

.30 

25 

.11 

.12 

.12 

.13 

.14 

.15 

.15 

.16 

.18 

.19 

.20 

.22 

.23 

.25 

.30 

.35 

30 

.12 

.13 

.13 

.14 

.14 

.15 

.16 

.18 

.19 

.20 

.22 

.24 

.25 

.28 

.33 

.40 

40 

.12 

.13 

.14 

.15 

.16 

.16 

.18 

.20 

.22 

.24 

.26 

.28 

.30 

.34 

.40 

.50 

60 

.13 

.14 

.15 

.16 

.18 

.18 

.20 

.22 

.25 

.28 

.30 

.33 

.35 

.40 

.47 

.60 

60 

.14 

.14 

.16 

.17 

.19 

.20 

.22 

.25 

.28 

.32 

.34 

.37 

.40 

.46 

.55 

.70 

70 

.14 

.15 

.17 

IS 

.20 

.22 

.24 

.28 

.31 

.35 

.38 

.42 

.45 

.52 

.60 

.80 

80 

.15 

.16 

.18 

.20 

.22 

.23 

.26 

.30 

.34 

.38 

.42 

.46 

.50 

.58 

.70 

.90 

90 

.15 

.17 

.19 

.21 

.22 

.24 

.25 

.28 

.32 

.37 

.42 

.46 

.50 


.60 

.77 

1.0 

100 

.15 

.18 

.20 

.25 

.27 

.30 

.35 

.40 

.45 

.50 

.55 

.60 

.70 

.85 

1.1 

120 

.16 

.19 

.22 

.25 

.28 

.31 

.34 

.40 

.46 

.52 

.58 

.60 

.70 

.80 

1.0. 

1.3 

150 

.17 

.22 

.26 

.30 

.33 

.36 

.40 

.47 

.55 

.60 

.70 

.77 

.85 

1.0 

1.2 

1.6 

200 

.20 

.25 

.30 

.35 

.40 

.45 

.50 

.60 

.70 

.80 

.90 

1.0 

1.1 

1 1-3 

1.6 

2.1 




















































Punching and Sheering. 


5S5 


Punching Iron Plates. 

To punch iron plates of from j to 1 inch thick requires 24 tons per 
square inch of metal cut; that is, the circumference of the hole multi¬ 
plied by the thickness of the plate is the area cut through. 

Notation. 

d = diameter of the punch o^ hole. 

D = diameter of the hole in the die. 

t — thickness of the iron plate. 

All dimensions in 16ths of an inch. 

W= weight or force in pounds required to punch the hole. 

W = 660 / d. D =d-\-0'2 t. 

Example 1. An iron plate of t —6 sixteenths of an inch thick, and the 
I hole to be d=12 sixteenths in diameter. Required the force W—1 
W= 660X6X12=47520 lbs., the answer. 

Example 2. Under the same conditions require the diameter D—l of 
the die. 

U—12+0'2X6=13‘2 sixteenths. 

Example 3. Required the diameter of piston for a direction action 
steam punch, for the plate and hole as in example 1, pressure of steam to 
be 60 lbs. per square inch. 

Force 47520=^X50 of which area of piston will be A = ^~—= 950'4 
square inches, which answers to a diameter of 34-8, say 36 inches. 


Shearing Iron Plates. 


It requires the same force per section cut, for shearing as for punching, 
namely, 20 to 24 tons per square inch. If the shears are good, sharp, and 
well adjusted, 16 tons may be sufficient. 

When the cutters in the shears are inclined to one another, the area 
cut, will be the square of the thickness of the plate multiplied by half 
the cotagent for the angle of the shears. Let u=angle of the shears, IV 
and t same as for punching. 

IF=88 IP cot.v. 

Example^. What force is required to cut a half inch plate/=8 sixteenths 
with a pair of shears forming an angle of v^12°1 Cot.l2°=4-7. 

lF=88X8'X4-7=26470 lbs. 


Atmospheric Columns. 

Wafer=33-95 feet. 2-3 feet for 1 lbs. per square inch. 
Seawater=33-05 ft. 2-23 “ “ 

Mercury at 60°=30 inches. 2-05 inches, “ 

Atm. air=28l83 feet. 1912 feet, “ “ 

Atmospheric air Required for each. 


Blacksmith’s forge. 
Charcoal forge, - 
Finery forge, - - 

Charcoal furnace. 
Anthracite furnace 


100 to 200 
- 400 to 500 




800 to 1000 
1000 to 3000 
2000 to 5000 


>- 




Cubic feet per minute. 


Cupola. 

In a cupola of 3 feet 4 inches diameter, and 10 feet high, can be melted 
down 1000 lbs. of cast iron, 200 lbs. of bitumninous coal per hour, with a 
blowing machine of 4-5 horses making 1700 cubic feet of air per minute 
into a pressure of 2-25 inches of mercury at which the temperature of 
the blast will be about 70° Fah. 












588 


Steam-boiler Explosions. 


Steam-boiler Explosions. 

The steam-boiler is a reservoir of work. Each unit of heat in the steam and 
watef is equivalent to a work of 772 footpounds. 

The steam-table gives the units of heat per cubic foot, or per pound, in the 
steam and water at different temperatures and pressures. Work is the product 
of the elements force F and space, or K= F V T, when the force of the work 
K 

will be F = ——. When the pressure in any part of a steam-boiler is suddenly 

removed, the entire work in the steam and water is at the same time started with 
a velocity proportionate to the removed pressure. The steam and water, in the 
form of a foam, strike the sides of the boiler, by which the work is suddenly 
arrested. If the time of arresting the work is infinitely small, we see from the 
above formula that the force of the blow will be infinitely great, and thus the 
boiler explodes. 

Steam-boiler explosions are caused in various ways, namely: 

1st. By long use boilers become corroded and, from neglect, give way in some 
unexpected place. 

2d. The general construction with staying and bracing of steam-boilers is 
often very carelessly executed and results in explosion. This kind of explosion 
is often indicated, long before the accident occurs, by leakage of the boiler; 
when the engineer, not suspecting the approaching danger, limits the remedy 
generally to efforts to stop the leak. The leakage from bad calking, or pack¬ 
ing, is easily distinguished from that of bad or insufficient bracing. In the 
latter case the fire ought to be hauled out, the steam blown off, and the boiler 
secured with proper bracing. 

3d. Explosion is sometimes caused from low water in the boiler, but more 
rarely than generally supposed. When the fire-crown and tubes, subjected to a 
strong heat and not covered with water, the steam does not absorb the heat 
fast enough to prevent the iron from becoming so hot that it cannot withstand 
the pressure, but collapses from weakness. 

4th. Steam-boilers often burst by strain in uneven expansion or shrinkage, 
occasioned by the fire being too quickly lighted or extinguished. 

5th. It is a very bad practice to make boiler-ends of cast-iron, composed of a 
flat disc of from two to three inches thick, with a flange of from one to two 
inches thick, with cast rivet-holes. The first shrinkage in the cooling of such a 
plate causes a great strain, which is increased by riveting the boiler to it. Any 
sudden change of temperature, therefore, either in starting or putting out the 
fire, might crack the plate and thus occasion an explosion. 

6th. In cold weather, when the boilers have been at rest for some time, they 
may be frozen full of ice: then, when fire is made in them, some parts are sud¬ 
denly heated and expand, whilst other parts still remain cold,causing an undue 
strain, which may also burst the boilers. Such accident can be avoided by a 
slow and cautious firing. 

7th. Sometimes a great many boilers are joined together by solid connections 
of cast-iron steam-pipes, which expand when heated; whilst the masonry en¬ 
closing the boilers contracts. Should such a steam-pipe burst from expansion 
or shrinkage, explosion will likely follow in all the connected boilers, of which 
numerous examples have occurred. Such accident may be avoided by making 
the connection elastic, or free to expand or contract without moving the boilers. 

Steam-boiler explosion is thus not always caused by the pressure of steam 
[alone,but often by the expansion and contraction of the materials of the boiler. 
A steam-boiler which is perfectly safe with a working pressure of 200 lbs. may 
explode with a pressure of 20 lbs. to the square inch,. 

The bursting of a boiler is a preliminary process to explosion. A boiler may 
burst without exploding. A boiler full of steam may burst, but never explode. 
It is the work in the heated water which makes the explosion. 

It is evident from the results of explosions that a much higher pressure had 
been acting than the normal working pressure. 








Destructive Work of Steam-boiler Explosion. 


587 


Destructive “Work of Steam-boiler Explosion 


When a steam-boiler explosion takes place, the enclosed water is resolved 
into boiling hot water and steam. 

The greatest precautions against explosion are sometimes unavailing. 
Careful attendants, regular cleaning, inspection by the authorized inspectors 
of the government or boiler insurance companies, will do much to prevent 
accidents. Close attention should be given to the safety-valve, which must 
be free and of sufficient area to blow out all the steam if the demands upon 
the boiler suddenly cease. 

Opinions differ very widely as to the theoretical action of the steam and 
water in an explosion, but are of little consequence, as the legal duty of 
EVERY USER IS TO TAKE STEPS WHICH SHALL PREVENT AN EXPLOSION. 


Precautions against Fire on Steamboats 


Each steamer should have three buckets for every 100 tons measurement, 
plus 10 buckets. That is, a steamer of 800 tons should have 8 X 3 + 10 = 34 
buckets. Also one axe for every 5 buckets. 


U. S. Steam-boilers Inspector’s Rule for Strength of Boilers. 

Multiply one-sixth (%)of the lowest tensile strength found stamped on any 
plate in the cylindrical shell, by the thickness expressed in parts of an inch 
of the thinnest plate in the same cylindrical shell; and divide the product by 
the radius or half the diameter of the shell expressed in inches, and the 
quotient will be the steam pressure iii pounds per square inch allowable in 
single riveted boilers, to which add twenty per ceutum for double riveting. 

S = breaking-strain in pounds per square inch stamped on the plate. 

t — thickness of the plate in fraction of an inch. 

Z)= diameter of the boiler in inches. 

P— steam pressure in pounds per square inch. 




3 DP 

S 


t 


V 










683 


Superheating. 


SUPERHEATED STEAM. 

The Author’s experience in superheated steam has been sufficient to 
convince him of its great importance, it appears that in order to utilize 
the maximum effect of steam or at least to attain the maximum result of 
expansion, it is not necessary to overheat it after a pure steam is formed, 
that is, when all the small particles and bubbles of water are evaporated. 
Water which accompanies the steam in such a form has the same tempera¬ 
ture as that due to the surrounding steam pressure, preventing its vaporiza¬ 
tion ; but when it passes through the superheating apparatus more heat is 
added, while the pressure remains the same because it being in connection 
with the steamroom in the boiler allows the water to vaporize and a pure 
steam may be formed. 

if steam with particles of water is admitted into the cylinder part of 
the stroke and then allowed to expand, it is generally found that the end 
pressure does not come up to that of theory, from which it has been said 
that the expansive quality of steam does not follow' that of a perfect gas, 
and that steam has condensed during the stroke; but if we knew the cubic 
contents of all the particles of water and subtracted that from the cubic con¬ 
tents of the steam it might be found that its expansive quality is not so far 
from that of a perfect gas. It appears also that the expansive quality is 
diminished by overheating pure steam. 

The small particles of water contain a great deal more heat per volume 
than the surrounding steam, consequently when admitted into the condenser 
a good vacuum cannot be formed so well as with pure steam. It is therefore 
of great importance to pay particular attention to the superheating of steam, 
otherwise economy by expansion will not be realized to the extent herein 
given by formulas and tables. It is also of great importance that the piston 
and steam valves be perfectly tight. 

SUPERHEATING APPARATUS. 

The accompanying figure represents a 
superheating apparatus such as the Au¬ 
thor has built it in Russia, and is found 
to answer exceedingly well. The figure 
is a section of the forend of an ordinary 
tubular boiler with steamdrum and uj>- 
take. The chimney is made a great deal 
wider in the steamdrum and contracted 
to the usual size at e, of 0T6 times the 
area of the firegrate ; if a strong fan blast 
is applied it may be better to contract it 
to 0T1£3- the inside of the chimney 
are placed a number of copper tubes a, 
a, b, b, with flanges screwed to the side; 
the area of these tubes should be about 
four times that of the steampipe c. In 
the steamdrum is riveted steamtight a 
conical plated, d, so that the steam can¬ 
not pass to the top without passing the 
superheating pipes. This superheating 
apparatus is in successful operation in 
three first class passenger steamers on 
the River Volga in Russia, each of 600 
horse-power, and one in a steamer of 100 
horse power on the Black Sea. 

The steamdrum can be placed around 
the chimney separately from the boiler 
anu the steam led either above or below 
the plate d, d, by pipes from the steam- 
room, as may suit the circumstances. 

This superheating apparatus may also be well suited for locomotives. 












































Giffard Injector, 


589 



The Giflfard Injector 

is marie by many manufacturers 
in different forms and at differ¬ 
ent prices. 


The following table has 
been furnished by William 
Sellers & Co., Philadelphia, 
manufacturers of this injec¬ 
tor. It gives the quantity 
of water injected per hour 
in cubic feet. 

The first column No. is 
the size or diameter of the 
throat in French millime¬ 
ters. The last column is the 
size in 16ths of an inch. 

Capacity and Size of Giffard’s Injector. 



Size 


Pressure of ste; 

irn in pou 

nds i 

>er sq 

uare 

inch 

above atmosphere. 

Size 

No. 

10 

20 

30 

40 

50 

GO 

70 

80 

90 

100 

110 

130 

130 

150 

16ths 

2 

8.3 

9 

9.7 

10.4 

11.1 

11.8 

12.5 

13.2 

13.9 

14.6 

15.3 

16.0 

16.7 

18.1 

1.26 

3 

19.3 

21.0 

22.8 

24.G 

26.3 

28.1 

29.9 

31.6 

33.4 

35.2 

37.0 

38.7 

40.5 

44.1 

1.89 

4 

3G.6 

39.6 

42.7 

45.9 

49.0 

52.1 

55.3 

58.4 

61.6 

64.7 

67.8 

71.0 

74.1 

80.4 

2.5 

5 

57.G 

62.5 

67.4 

72.3 

77.2 

82.2 

87.1 

92.0 

96.9 

102 

107 

112 

116 

126 

3.6 

G 

83.5 

90.6 

97.7 

105 

112 

119 

126 

133 

140 

147 

155 

162 

169 

183 

3.78 

7 

114 

124 

133 

143 

153 

162 

172 

182 

192 

201 

211 

221 

231 

250 

4.41 

8 

149 

162 

174 

187 

200 

213 

226 

239 

251 

264 

277 

290 

303 

328 

5.04 

9 

189 

205 

221 

237 

254 

270. 

286 

302 

318 

334 

351 

367 

383 

415 

5.67 

10 

234 

254 

274 

294 

313 

333 

353 

373 

393 

413 

433 

453 

473 

513 

6.30 

12 

837 

366 

395 

423 

452 

481 

510 

539 

567 

596 

625 

654 

682 

740 

7.56 

14 

451 

491 

531 

571 

611 

651 

691 

731 

771 

811 

851 

891 

931 

1011 

8.80 

16 

600 

651 

703 

784 

805 

857 

908 

959 

1010 

1062 

1113 

1164 

1215 

1318 

10.1 

18 

760 

825 

890 

955 

1020 

10851 

1149 

1214 

1279 

1344 

1409 

1474 

1539 

1669 

11.3 

20 

939 

1019 

1099 

1179 

1259 

1339 

1420 

15001 

1580 

1660 

1740 

1820 

1900 

2061 

12.6 


Method of Working the Injector. 

First. —See that the steam-plug is closed down, and waste-valve stem is raised. 
Second. —Admit steam from boiler to Injector, which should cause the water to 
flow from the waste pipe. 

Third.— Turn up the steam-plug until the waste valve can be closed without 
causing the Injector to cease working. 

F’ocrth.— Turn up the steam-plug to increase the delivery, and down to decrease 
it. When this Injector has to lift its supply water, the steam valve between the 
Injector and boiler must be opened very slowly, until the water flows out of the 
waste pipe. 

N. B.—A failure to work will .always be indicated by an escnpe of steam and 
water from the waste check attached to check valve in water-supply pipe. 





























































































500 


Blowing Engines. 


BLOWING EN.GINES. 

Notation. 

?=JSSfS ffet° ches> } » f ««*■« <*»“><" d '>“ We -“»*■ 

Z = part of the stroke S under which the air compresses from the 
atmospheric density to that in the reservoir. 
jF= mean resistance in pounds per square inch of the air on the 
cylinder piston (average for one stroke). 

P == pressure in pounds per square inch of the blast in the reservoir. 
C— cubic feet of air of atmospheric density, delivered from the blow¬ 
ing cylinder to the reservoir per minute. 

H= actual horse power required to work the blowing engine, includ¬ 
ing 13 per cent, for friction. 
d = diameter of blast pipe in inches, 
n — number of revolutions or double stroke per minute. 

A = area of supply valve to the blowing cylinder in square inches, at 
each end of cylinder. 

p = vacuum in pounds per square inch, on the supply side of the cylinder 
piston, which should not exceed 0-1 lbs. 

V = velocity of the blast through the tuyeres in feet per second. 
v = velocity of the air through the supply valve A, in feet per second, 
which should not exceed 100 feet. 
a = area of the orifice or tuyeres in square inches. 
h = height of mercury in inches, in the gauge on the blast reservoir. 

L = length of the blast pipe in feet from the receiver to the tuyeres. 
k = volume coefficient, see Table. 

t — temperature Fah. of the blast caused by compression or heating. 
Example 1. Formula 8. For an Anthracite blast furnace is required 
4000 cubic feet of air per minute, under a pressure of 6 inches mercury. 
Required the horse power necessary for the blowing machine 1 The ef¬ 
fectual resistance F= 2 365 lbs. see Table. Assume the vacuum to be 
v=0-09 lbs. 

. rr 4000(2-365 + 0-09) „ . . . 

>V e have II =-—- = 49‘6 actual horse-powers. 

198 

Example 2. Fbrmula 10. Suppose the blast cylinder to be T> — 144 inches 
diameter with = 15 feet stroke. Required the number of double strokes 
per minute n — ? 

_ 96 X 4000 
11 1442 x 15 

Example 3. Formula 9. Under the above conditions, require the area 
of the supply valves A=1 when the velocity u=105 feet, per second. 

, 144 2 X15X12-3 . 

A— -- —^-= 911 square inches. 

40X105 

Capacity of Blast Reservoir should not be less than the following 

proportions, but more is better. 

For one single acting cylinder, 201 

For one double acting cylinder, 10 > times the capacity of one cylind’r. 

Two double act. cyl. cranks at 90° 5 ) 

One double acting cylinder, same as two single acting. The more 
cylinders the less capacity required for blast reservoir. 


F= 2 46 /F^TfOaJW t ), 

P=14 7 (fc — 1), 
f —32 


P = 


3365’ 


/ = 32-1-493 (ft—1), 
t = 33-55 P+ 32, 

'ft 












Blowing Machines. 


591 


Approximate Formulas for Blowing Machines. 


1= 


S h 


'30-H ’ 

P=0-49A, 

„ IT S n 


C= 


96 


198 H 


- 1 
- 2 
- 3 


C , =l:83«7iC30-l-AJ),6 , _ v/ (7+10 L „ 

(L __ — i 1 


D'Sn(F+ P ) 
19000 ’ 


H= 


C= 


F+p’ 


ah V 


- - 4 A= 


CJF+p) 

198 

D 1 S n 


r=350 y~p 

D 2 S n 
’ 8 V == 40 A ’ 


26 ’ 


- - 5 


40y 


96 C 

71= - 

IT S 


- 9 


P~i 


D'S'n* 


180000000 A* 


- 12 
- 13 

14 


, A ., 30+A 

10 ^ ^30 ’ " 16 


Table for Blast ahd Blowing Machines. 


Vblume and tcmperat. 

Guage in inches. 

Pressure lbs. sq inch. 

Stroke. 

Velooifcr. 

k 

t 

water. 

h 

P 

F 

1 

V 

1-002 

33 ° 

1 

0-073 

0-036 

0-032 

0-0024 

72 

1*005 

34*5 

2 

0-147 

0-079 

0 063 

0-0049 

102 

1-007 

35-5 

3 

0-220 

0-108 

0-095 

0-0073 

125 

1-010 

37 

4 

0-294 

0-144 

0-128 

0-0097 

144 

1-012 

38 

5 

0-368 

0-180 

0159 

0-0121 

161 

1-015 

39-5 

6 

0-441 

0-216 

0-191 

0-0145 

176 

1-020 

42 

8 

0-588 

0-288 

0-253 

0-0192 

204 

1-025 

44-5 

10 

0-736 

0-360 

0-309 

0-0239 

228 

1-030 

47 

12 

0-884 

0-432 

0-379 

0-0287 

249 

1-035 

49-5 

14 

1-030 

0-503 

0-437 

0-0334 

269 

1-043 

53-5 

17 

1-250 

0-612 

0-531 

0-0400 

297 

1-051 

57-5 

20 

1-470 

0-719 

0-623 

0-0467 

322 

1-002 

63 

24 

1-766 

0-863 

0-745 

0-0556 

352 

1-074 

69 

28 

2-060 

1-008 

0*865 

0/0643 

381 

1-082 

73 

31 

2-281 

1-116 

0-955 

0 0706 

401 

1-091 

77.3 

34 

2-501 

1-223 

1-043 

0-0769 

420 

1-100 

82 

37 

2-720 

1-332 

1-130 

0-0833 

438 

1-109 

86-5 

46 

3-000 

1-470 

1-205 

0-0908 

460 

1-116 

90 

47-5 

3-500 

1-715 

1-431 

0-1045 

496 

1132 

98 

54-3 

4-000 

1-961 

1-636 

0-1178 

530 

1-165 

114-5 

67-7 

5-000 

2-450 

2-010 

0-1431 

593 

1-200 

132 

81-4 

6-000 

2-941 

2-365 

0-1667 

650 

1-265 

164-5 

108-5 

8-000 

3-925 

3-088 

0-2105 

751 

1-400 

232 

163 

12-00 

5-900 

4-389 

0 - 2 S 59 

918 

1-500 

282 

203-7 

15-00 

7-375 

6-875 

0-3333 

1077 

1-625 

344*5 

254-6 

18-75 

9-217 

8-831 

0-3846 

1393 

1-750 

407 

305-5 

22-49 

11-06 

10-67 

0-4285 

1590 

1-875 

469-5 

356-4 

26-24 

13-90 

11-64 

0-4666 

1760 

2-000 

532 

407-4 

30-00 

14-75 

12-50 

0-5000 

1955 


































































592 


Fans or Ventilator. 


FAN OR VENTILATOR. 

Fans constructed as the accompanying 
figure have been found by the Author who 
has made several of them, to be the most 
effective. 

The vanes are each one quarter of an 
arithmetical spiral with a pitch twice the 
diameter of the fan, that is, each vane should 
be constructed in an angle of 90° from centre 
to tip. Length of fan to be from * to £ the 
diameter. Inlet to be half the diameter of 
the fan. Number of vanes to be not more 
than six, and not less than four. Six vanes 
work softer and better, but they give no 
better effect than four. 

The housing should be an arithmetical 
spiral with sufficient clearing for the fan at a, and leaving a space at b 
about £ of the diameter. Fans of this construction make no noise. 

Notation. 

1 :-i“ r } ° f fan in inches - 

L = fen' d h\ n* feet'! of blast pipe, to be as straight as possible. 

a = area in sq. in. tuyeres or outlet. 

C= cubic feet of air delivered per minute. 
h — inches of mercury. 
v = velocity in feet per second through a. 
k — volume coefficient., see Table, page 441. 
n = revolutions of fan per minute. 

H— actual horse power required to drive the fan. 



W 


Formulas for Fans. 


h = 


d n° 


60000000 




dl 


25 a-\-dl 


11 = 


dlhn 

24000’ 


h = 


24000 II 
din ' 


V = 244 yh 

A = a\L 
Example 1. 


v = 


n )/ d 4 / 
28*86 \/ 


dl 

2 5ar£Tt 


n = 


24000 II 


dlh 


g_vak 
~ 2-6 ’ 

C= 94 a fcj/ h 

A = - C -. fh. 
94 k\/ h 


8 


10 


A fan of d =36 inches diameter, 1 =12 inches, making n= 725 
revolutions per minute, area of tuyere being a=25 square inches. Re¬ 
quired the density of the blast in inches of mercury h=1 


Formulae 1. h = 




/= 


36X12 


60000000 \/ 25X25+ 36 X 12 


= 0-242 inches. 


Example 2. Under the same conditions require the cubic contents or air 
delivered per minute, C=t fc=l-0l the nearest in the Table. 

Formula: 9. C=94X25Xl'01v / 0-242 = 1167-7 cubic feet. Required the 
horse power H=\ 


Formula: 2. H = 


36X12X0-242X725 


24000 


= 3-16 horses. 


-A 



























iron Furnaces. 


593 


BLAST OR IRON FURNACES. 

It is almost impossible to foresee the many variable circumstances con¬ 
nected with the performances of blast Furnaces. The data herein given are 
deduced as an average from the performances of a great many furnaces both 
in America and Europe. 

The accompanying Tables are so arranged that the numbers in Table 
I., multiplied by the numbers in Table II., gives the corresponding charge 
of Iron ore, lime stone, coal, and the produce of pig iron in pounds per 
24 hours, with the consumption of air in cubic feet per minute. 

Table II. contains the effective capacity of blast furnaces in cubic yards. 

Example. It is required to construct a blowing machine for an Anthra¬ 
cite blast furnace of 12 feet diameter of boshes, height of stack 45 feet, to 
be worked with hot blast. Required the produce of pig iron per 24 hours, 
cubic feet of air per minute and actual horsepower of the blowing engine f 

Produce of pig iron 155 Table I.X123 Table II.=19065 lbs. or 8-5 tons per 
24 hours. 

Consumption of air 20X123=2460 cubic feet per minute. Suppose the 
blast to be blown into the furnace at a pressure of P = 2 - 94 lbs., vacuum 
in the supply side in cylinder to be p=~- 0-07 lbs. we shall have the required 

actual power. 2460 (2 - 3S -f 0*07) . , 

„ , _ 11= --' = 30 2 horses. 

Formula 8, p. 591. 198 

Consult Bauerman, Metallurgy of Iron. 

Table I. Iron or Blast Furnaces. 


The unit being the capacity 
of the Furnace in 
cubic yards. 


Soft charcoal 


( Cold blast, 
t Warm blast, 


, , r Cold blast, 

Hard charcoal } Warm bla5tj 


Goko 

Bituminous 

Anthracite 


f Cold blast, 

(Warm blast, 

fCold blast, 
t Warm blast, 

f Cold blast, 
t Warm blast, 


Charge and produce per 24 hours. 

Air 

Iron 

Fig 

Lime 

Coal 

per 

Ore. 

Iron. 

Stone. 


minute. 

lbs. 

lbs. 

lbs. 

lbs. 

cub, feet. 

535 

215 

196 

400 

24 

700 

292 

256 

350 

19 

670 

270 

245 

400 

24 

875 

365 

320 

350 

19 

268 

108’ 

98 

515 

26 

350 

146 

128 

397 

20 

252 

101 

92 

515 

24 

327 

136 

120 

397 

19 

287 

115 

105 

515 

26 

373 

155 

137 

597 

20 


Table II. Capacity and Dimensions of Iron Furnaces. 


Diameter of 
B Piles in ft. 


■ 8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 


25 


40 

50 

62 

75 

90 

105 

121 

140 

160 

280 

202 


30 


44 

55 

68 

82 

98 

115 

133 

153 

174 

197 

220 


35 

40 

45 

50 

55 

60 

47 

51 

54 

58 

62 

65 

60 

64 

69 

73 

7S 

83 

74 

79 • 

75 

91 

96 

102 

89 

96 

103 

110 

117 

123 

106 

114 

123 

130 

139 

147 

125 

134 

144 

153 

163 

172 

145 

155 

167 

178 

189 

200 

166 

178 

191 

204 

217 

230 

189 

203 

217 

232 

247 

261 

21? 

229 

245 

262 

279 

295 

239 

257 

275 

293 

312 

330 


38 


















































594 


Expansion of Steam. 


THE EXPANSION OF STEAM. 

(Continued from page 538.) 

To find the most economical point of cut-off—that is, its inverse that number 
of expansions which will result in the greatest economy of steam from the boiler 
per horse-power per hour. 

dotation. 

e — the true point of cut-off — the reciprocal of the true number of 
expansions. 

B = the absolute back-pressure during exhaust in pounds per square 
inch. 

P b = the absolute pressure at cut-off. 

5 = the stroke of cylinder in feet. 

d = the diameter of cylinder in feet. 

62-5 


A = 


S 


S = the specific volume of steam at cut-off. 
T b -T e „ 


D = 2 


N 


T\y — the temperature of the steam at cut-off (Fahr.). 

T e = the temperature of the steam during exhaust. 

N — the number of strokes per minute — twice the revolutions of 
crank. 

C = the constant of condensation = ’018 pounds of steam for about 82 
pounds gauge-pressure. 


l)d 


P , /I , '194\ _ . . 

P b + ( s + d I Ad + D nat log * 


Example: 

Let P b — 100 pounds absolute. 
“ B = 15 “ “ 

“ s = 4 feet. 

“ d = 1'5 “ 

“ N — 150 per minute. 

We have— 


A = *233. 
I) = -0274. 


e = 015 + 

\ 4 1 I'd / '2 


02 74 X 1’5 X 2 3026 
233 X 1*5 + -0274 


com. log. —. 


•0944 1 

e = 0'15 + '3793 log. 


e — *15 + *0952 log. —. 

e 


We must solve this transcendental equation tentatively, trying values 
until the two members balance. 

Assume e = £ of stroke plus clearance. We have . 

•20 = *15 + -066 = *216. 

This error of -016 is closer work than can be realized in practice, and 
we can take 5 expansions as the best number. 

Between £ and $ would have been near enough for all practical pur¬ 
poses. 






















Expansion of Steam. 


To find the proper ratio of stroke to diameter undo' the given conditions, 
assuming 5 expansions and diameter = V/ 2 feet. 

Inverting the above equation, we have 

d 


595 

i 


D 


— 8‘56. 


8 = 


, d +1 


1*5 



- | - 0194. 


(w.XW + l) 


= 6.4 ft., nearly. 


With slow-moving engines it will be found that long stroke is most 
economical, while on the other hand high-speed engines require short 
stroke for greatest economy. If we double the speed of this engine, 
making N — 300, the stroke s — 2‘4 feet, for greatest economy. 

Every detail of the steam-engine will be found discussed in a rational 
and practical manner in The Relative Proportions of the Steam-Engine, by 
Wm. D. Marks (3d Ed., J. B. Lippincott Co.). 

The subject requires more space than can be given in a pocket book. 


ELECTRO-DYJfAMICS. 

{Begun on page 644.) 


These results show that this high efficiency is not extraordinary, but is 
and should be attained by all dynamo-makers building similar types. 

In the case of the Weston (7 M.) dynamo, already specified, the power 
applied was distributed as follows in the first full load test: 

Friction and wind resistance of armature .... '0106 total. 

Electrical energy lost as heat in armature .... '0559 “ 

“ “ “ in creating field. ‘0170 “ 

“ “ “in external circuit .... '8992 “ 

Total of power accounted for. ‘9827 

Electrical energy lost in eddy currents, heat, and 
otherwise unaccounted for._d)173 

Total power as per dynamometer.TOOOO 

* 

This differs from the average already quoted, because slight variations 
of the conditions would cause any of the machines to vary somewhat in 
their percentages. 

The greatest cause of uncertainty in experiments heretofore made 
upon the transformation of mechanical power has been our lack of cer¬ 
tainty of accuracy in the measurements of the mechanical power driv¬ 
ing the machine. The dynamometer must sum up the whole power 
yielded to the dynamo with as great accuracy as is possible for all other 
measurements in part. 

This dynamometer must be capable of being standardized by absolute 
measurement, and, after being standardized, the machines to be tested 
must be able to be attached to it or removed from it without altering the 
centres or adjustments of the dynamometer. It must be of great sensi¬ 
tiveness to small variations of load, while measuring large amounts of 
power with great steadiness. 













596 


Electro-Dynamics. 


All of these conditions were fulfilled by the dynamometer invented by 
Mr. Wm. P. Tatham, President of the Franklin Institute. Its extreme : 
capacity is 100 horse-power, and yet, while making 1040 revolutions per 
minute) carrying a load of 29 horse-power, it was possible to measure 
with certainty the difference of power required by an Edison voltmeter 
requiring two-tenths of a liorse-power. It announced at once the mak¬ 
ing and breaking of the circuit of this voltmeter, measuring the work 
lost in it with accuracy. Still other tests showed its capability to 
promptly register small changes of power while carrying great loads, 
and proved that the slight and rapid jar of the parts, due to a high speed, 
increased its sensitiveness of measurement. 

Finally, this dynamometer was calibrated by the agitation of water, 
heating something over five tons of water through 15’5° C., giving, as the 
mechanical equivalent of heat, 772'81 foot-pounds per British unit of heat. 

While less can be claimed in the way of originality of apparatus or 
methods used in the electrical measurements of these tests, an examina¬ 
tion of the precautions will convince one of the extreme care taken to 
obtain correct results. (See “ Competitive Tests of Dynamo Electric Ma¬ 
chines,” Journal Franklin Institute , Nov., 1885.) 

The dynamo electric machine has grown out of the fact that, if we 
move a dead wire in the field of another fixed wire, through which a 
current is passing, the dead wire will have a current generated in it 
whose electro-motive force is proportional to (1) the intensity of the cur¬ 
rent in the fixed wire, (2) to the velocity of motion of the moving wire, 
and (8) to the acting length of the moving wire. 

If we take a single wire, Fig. 2, and pass a current through it, its field 
will resemble a whirlpool, of which the wire is the centre. 



If we take two wires and place them a short distance apart, Fig. 3, and 
pass a current in the same direction through both, their fields will com¬ 
bine to form an oval field, and any number of adjacent wires, with cur¬ 
rents in the same direction, will do the same thing, forming a field of an 
intensity proportional to the number of wires and the intensity of the 
current in each. The field would, however, be of loose texture, so to 
say, and the lines of force far apart if the lines of force formed them¬ 
selves around the wires in the air. 

Iron, because of its great permeability to the lines of magnetic force, 
enables us to concentrate this field, and to place it, so to speak, where we i 
desire to use it. 

Pure, soft wrought iron may be said to be 20,000 times more permeable i 
than air. 

We will see from Fig. 4 what is meant. The wrought-iron cores C 
afford the easiest path for the lines of force, and they therefore follow 
them until they reach the armature-space A, between N and .S’, where 
they take their airy path across, because the lines of force m ust always close. 

We see that we have thus managed to concentrate the lines of the field 
of a large number of coils in a small space A. In this space the wires 
of the armature are revolved so as to generate a current which is either 
alternating or approximately continuous. j 

The details of armatures and winding of them, as well as of the com- 



P. Thompson. 


A. 



















Electro-Dynamics. 


597 


There is nothing: written better calculated to give the novice a clear 
idea of the principles involved in a continuous-current machine than 
Pacinotti’s own description of his machine, 
which can be found in the translation of 
Schellen by Dr. Keith, on page 209. 

Taking matters as they are, the most 
economical engine used for the purpose 
of driving dynamos at the Electrical Ex¬ 
hibition of the Franklin Institute required 
about 30 pounds of steam at 90 to 100 pounds 
pressure, and the most economical boiler 
evaporated about 8 pounds of water per 
pound of anthracite coal at the same pres¬ 
sures. That is, an indicated horse-power 
required 3% pounds of average anthracite. 

It can be assumed, with close approx¬ 
imation to average correctness, that 15 per 
cent, of the indicated horse-power is lost 
in the most direct method of transmission 
of power from engine to dynamo. 

So we can say that one utilizable elec¬ 
trical horse-power per hour may, in good 
practice, be obtained from 

3-75 100 



85 


X -qq- = 4'-&% pounds of coal 


(such as is sold in the open market as chestnut anthracite), and neglect 
the loss of electrical energy in the conductors. 



Assuming the mechanical equivalent of one British unit as 774T foot¬ 
pounds, we have very nearly 2558 British units for one horse-power per 
hour. Dividing the last by the first, we find that nearly 4 per cent, of the 
power latent in the coal appears as electrical power in the circuit: 96 per 
cent, of our potential energy is lost, principally in the steam-engine. 

These facts, taken from the labors of many impartial and skilful work¬ 
ers in scientific research, do not correspond'with the alluring statements 
frequently set before us, but are reliable and practical. 

The broad lesson to be drawn from them is that we do not obtain of 
the power in coal in the form of electricity, and that remain to he 
obtained by the discoverer of an economical method of direct conversion 
of heat into electricity. 

When the direct method of conversion of heat into electrical energy 
yields a larger percentage of the power in coal than the indirect method 
which has just been described, at the same cost, then will the dynamo 
supplant the steam-engine. Until then it must remain what it is—a dis¬ 
tributor of power for the steam-engine or other mechanical motor. 

The Conversion of Electrical Power into Light. 

There are at present in use two methods of converting electrical power 
into light. The first and apparently the most economical is by means of 
the voltaic arc between carbon points; the second, by means of the 
incandescence of a carbon filament in a vacuum. 

The Arc Light. 

The first method is open to severe criticism, save on the point of econ¬ 
omy, and for lighting large spaces. 

The briefest look at the intense spot of light formed by the arc between 
the points of carbon causes a painful and persistent image on the eye. 
The light has a vicious way of hissing, which becomes unendurable to 
sensitive nerves, and it varies the monotony of this noise by sudden 
jumps and flickers. Its ghastly effects are due to its bluish color and the 
deep, sharplv-delined shadows. 

In some cases the arc has a way of rotating around the axis of the car¬ 
bons. which also causes variations of the intensity of the light in differ¬ 
ent directions. 

















598 


Electro-Dynamics. 


Opal glass globes, which cut off something more than one-half the 
light, are required to make the light tolerable; and as for the lamp itself, 
the greatest skill and taste of designers has not yet rendered it orna¬ 
mental when not lighted. 

As a rule, the arc light is most intense when viewed at an angle of 45° 
from the vertical, and for this reason it is usually used for lighting open 
spaces from a considerable height. Its power is assumed as an average 
of the illumination at 30°, 45°, and GO 0 from the vertical. 

From the report on Electric Lamps of the Franklin Institute, June, 
1885, the following data are taken: 


Machine. 


Arago Disc. 

Ball. 

Brush (1200 c.-p.) . . . 
Brush (2000 C.-p.) . . . 

Diehl. 

Richter. 

Van Depoele, 20 lights 
Van Depoele, GO lights 
Western Electric . . . 
_Average 



Angle 

with 

Vertical. 

Candles. 

Caudles 

per 

El. H. P. 

Average 

Candles 

per 

El. H. P. 

30° 

645 

783 

) 

45 

583 

708 

V 685 

GO 

465 

565 

f 

30 

182 

421 

) 

45 

485 

1123 

y 9i6 

60 

520 

1204 

f 

30 

355 

762 

) 

45 

613 

1316 

> 1076 

60 

537 

1152 

) 

30 

1200 

1529 


45 

1373 

1750 

l 1553 

60 

1082 

1379 

) 

30 

887 

1176 


45 

830 

1101 

y 1079 

GO 

725 

961 

) 

30 

603 

743 

) 

45 

894 

1101 

y 1009 

GO 

960 

1183 


30 

670 

780 

) 

45 

1377 

1604 

y 1206 

GO 

1060 

1235 

) 

30 

500 

612 

) 

45 

1162 

1423 

y 1045 

60 

900 

1101 

) 

30 

75 

121 

) 

45 

266 

431 

y 37G 

60 

355 

575 

f 


994 


The average candles per electrical horse-power obtained from measure¬ 
ments upon the Arago disc, Ball, Brush, Diehl, Richter, Van Depoele, 
and Western electric machines was 994. 

The efficiency of these arc-light machines was not obtained, but we 
are justified in assuming that 70 per cent, of the absorbed power should 
reappear as electrical power in the circuit, neglecting its losses. 

That is, 7 ^^:^ 0 = pounds of ordinary anthracite coal per elec¬ 


trical horse-power per hour. 


994 

= 158 candles. 


If we divide the candles per electrical horse-power by the weight of 
coal required to produce them, we find in the arc system that we obtain 
lj>s caudles per pound of coal for the naked light,' and something less 
than /o candles if ground glass or opal globes are used and the light 
seen from the most favorable position. 


I 



























Electro-Dynamics. 


599 


The Incandescent Electric Light. 

Very different from the arc light is the incandescent. Its light is so 
soft that we do not realize its brilliancy until we submit it to measure¬ 
ment. It gives out no products of combustion to poison our air; it shows 
colors truly. A delicate hair of carbon, sealed within a vacuum by walls 
of glass, glitters and glows until at almost limpid incandescence it gives 
us a light steady, clear, colorless as daylight. 

If we take a book and hold it from one to two yards away from a 
sixteen-candle light, we find the light sufficiently diffused to read with 
comfort. 

Now, all know that the intensity of illumination varies inversely as 
the square of the distance. Therefore, roughly estimating a shaded arc 
light at 500 candles, the same book would have to be held somewhere 
between five and one-half and eleven yards away from it to be read with 
equal comfort, assuming the light to be steady. We can then say that a 
sixteen-candle incandescent light will illuminate a circle of 1‘2^ square 
yards area, and that a shaded arc light giving 500 candles out of 994 will 
illuminate a circle of 400 square yards area, or 32 times as great. That 
is to say, about 32 sixteen-candle lamps would supply an equal illumina¬ 
tion with a vastly better distribution of light for the use of the eyes. 

We can therefore say that 500 candle-power from incandescent lamps 
will far more than replace 1000 candle-power from the arc light, under 
the conditions of actual usage. 

We can safely say that, for all purposes save that of obtaining light to 
dispel darknes's, the incandescent light is twice as valuable, light for 
light, as the arc light, and therefore should be multiplied by 2 when 
compared with it. 

The objections most vehemently urged against incandescent lamps 
have been their short life and lack of economy; this is not true of them 
in all cases. 

The first public test of the life of incandescent lamps was made by the 
Franklin Institute in the early months of 1885 (Journal of the Franklin 
Institute, Sept., 1885). The record of these tests is given in a pamphlet 
of some 130 pages, and with a detail which renders it impossible, in our 
limited space, to do more than gather from its averages such general 
lessons as we may learn. 

From the efficiency test, which was preliminary to the prolonged-dura¬ 
tion test, we find that 194T spherical candles were realized per electrical 
horse-power. 

Spher. Candles. 


Edison’s 97-volt lamps, per El. II.-P..109'2 

Stanley’s 96 “ “ “ “ 1891 

“ 44 “ “ “ “ . 2161 

Woodhouse & Rawson’s 55-volt lamps, per El. H.-P. . . . 209'0 
“ “ “ 55 “ “ “ « 210"8 

White’s 50-volt lamps, per El. H.-P.182 - 6 

Weston’s 110)4-volt lamps, per El. H.-P. 209'8 

“ 70 “ “ “ “ .166-3 


Average, per El. H.-P 


194-1 


The committee was forced by the different forms of carbon filament 
used to take the illuminating power of the lamps from all points, and to 
call the mean the spherical intensity of illumination. This procedure 
perhaps gives a better idea of the practical value of the incandescent 
lamp, because it is customary to place these lamps in any position that 
convenience may dictate. . 

The incandescent lamp, by reason of its smaller quantity of light and 
better distribution, is worth at least twice as much as the arc light. One 
electrical horse-power costs, with Weston’s incandescent dynamo electric 
machine, about 4'9 pounds of ordinary anthracite. Therefore one pound 
of C oal will give about 40 candles by the incandescent lamp, and this is 
equivalent to 80, and probably many more, candles by the arc light, 
whenever we have to use our eyes for any purpose save guarding our 
footsteps. 














600 


Electro-Dyxamics. 


You will recall that, under assumptions most favorable to the arc light, y 
we probably do not get more than 75 candles per pound of anthracite 1 
from the shaded arc light. Had the Committee on Arc Lights obtained 
the spherical intensity of illumination of these lamps, their showing 
could have been made much less favorable than the one given. The 
present method of arc lighting must ultimately give way before the 
incandescent light, save for large spaces not requiring a close use of the 
eyes. 

The low potential and larger current of the incandescent dynamos 
render necessary a lower resistance in the conductors, and so the cost 
of wiring for incandescent* lamps is much greater, because of the 
increased weight of copper wire demanded to convey the current with¬ 
out too great a loss in the form of heat. This is the pecuniary obstacle, 
and about the only one that prevents the entire disappearance of the arc 
light before the incandescent light. Could an incandescent lamp be 
made of sufficiently high resistance to enable the use of high potentials, 
the last objection to the system would vanish. 

The Edison 97-volt lamp in this test outlived all the others, demanded 
the least weight of conductors, and was 13 per cent, less economical of 
power. It was the only lamp in the test that justified a claim to 1000 
hours of life. 

Out of 20 lamps entered by this company, 19 survived a continuous test 
of 1006 hours. 

The more successful lamps were found to undergo a process of gradual 
degradation which is attributable to two causes—an increase of the 
resistance of the carbon filament, and a deposit of carbon upon the inte¬ 
rior of the glass of the lamp. 

The discoloration of the various lamps was carefully compared after 
their life had ceased, and was remarkably deep in the case of the Wood- 
house & Rawson and the Stanley 44-volt lamps. Indeed, it would seem 
as if this discoloration was in some wise proportional to the economy 
of the lamp, as these tw T o were the most economical of the makes of 
lamps entered. 

A lamp may live a long time and yet be of little value for the purpose 
of giving light, because of this degradation. If you will take a lamp 
which lias been used some time, and lay it upon a white handkerchief, 
the gray coloring-matter on the globe will be brought out quite dis¬ 
tinctly. 

Thus we see that great length of life with little usefulness may be 
attained by lamps. Indeed, the Edison lamps, which outlasted all 
others, had lost 36 per cent, of their illuminating power at the end of 
1006 hours. 

The direct conversion of heat into electrical energy by Clamond’s 
stoves only produced 26 candle-power per pound of coke, as against 40 
candles per pound of anthracite in the usual way with incandescent j 
lamps. I 

It will he a surprise should not the direct conversion of heat into elec¬ 
trical power prove to have quite as many difficulties and as narrow 
limits as the conversion of heat into mechanical power by means of the 
steam-engine. 


Tlie Conversion of Electrical Power into Mechanical 

Power. 

The problem which just now is demanding of electricians their most 
earnest effort is the transmission of work by means of electricity. This 
effort will be repaid by the utilization of otherwise inaccessible water- 
powers, and the problems of locomotion will have their simplest and 
least objectionable solution when it is an accomplished fact. 

Marcel Deprez has recently transmitted 60 horse-power from Creil to 
Paris with a mechanical efficiency of 53 per cent. 

The expense attendant upon an experiment of this magnitude has 
been very great, but, the scientific possibility once proved, we can rely 
upon the progress of manufactures to reduce this expense and to define 
the limits within which power can be economically delivered. 







Electro-Dynamics. 


601 






Seventy-five per cent, of the indicated power of the engine is not an 
overestimate of the power required to move the cable alone for onr cable- 
cars on a road of two or three miles’ length, but it would at once condemn 
an electric railway, which should be made to yield a practical efficiency 
of over 50 per cent. 

Before discussing the details of the transmission of power, Ave Avill, 
with the aid of our previously-used hypothetical fluid and pipe, and Avith 
two pumps to represent the dynamo and motor, endeavor to make clear 
the laws controlling the transmission of poAver by electricity. 

Assume two pumps, Fig. 5, A and B, connected by a closed line of pipe, 
so that the fluid must be pumped round a closed circuit. Let the pump 
A be driven by means of any external poAver. Let the pump B be 
reversed and acting as a motor. Let each of these pumps haA r e a vertical 
standpipe projecting from its top, which will sIioav the head E or e 



resulting from its action. The pump A acts under the law that its head 
E is proportional to the speed at which it is driven. The motor B acts 
under a similar laAV that its counterhead e is proportional to the speed at 
Avhieh it is alloAved to run. The weight of fluid per second passing 
through the conduit is directly proportional to the difference of these 
heads, and inversely to the resistance. 

Let I equal the Aveight of fluid passing along the pipe each second. 

The fluid passing along the pipes between A and B, and through the 
pumps, Avill lose, each second, a certain amount of head per unit of 
Aveight because of the resistances. 

Let R equal this resistance. Then Ave can say 

The weight of fluid per second is then directly proportional to the effective 
head, and inversely proportional to the resistance. This is Ohm’s law, Avhieh 
for electricity is, The intensity of the current is directly proportional to the 
difference of potential, and inversely as the resistance. 

The loss of power per second in friction in the pipes is the loss of head 
multiplied by the Aveight per second. This is Joule’s law for loss in heat 
for electricity: 

Work per second of pump A — head X wt. per second. 


H U << H 


— EI -- 


E (E-e) 


R 


“ lost in pipe A B — I* R = 


R 


R — head lost per unit of Aveight per second. 


Ohm’s law: 

T E-e iWeight f_ diff. head ____ 

— R per second ~ (head lost per unit of Aveight per second 




















602 


Electko-Dynamics. 


Joule’s law: 


(E—e) 2 

Work lost per second = I ( E—e) — —^— = I 2 R. 


Work per second of motor B 

. “ (i << << « “ _g j_ 


R 

head X wt. per second. 
e(E—e) 

R * 


Theoretical efficiency = £ = — . 

Work of motor B per second a maximum for e ( E—e ) maximum; that 
E 

is, e= 0 . 


Greatest work of motor B = 
“ “ pump A = 


£2 
4 R * 
£f 
2£‘ 


The practical efficiency of this combination of pump and motor will 
be diminished because the head E will require a coefficient greater than 
unity, and the counterhead e a coefficient less than unity. 




€ (1—3") 


The value of 


1 — x 


E(l + X)' 
must be determined by experiment. 


1 + X 

It will at once be seen that when the motor is acting at its greatest 
horse-power the theoretical efficiency is 50 per cent, and its practical 

(1 — x) m € 

efficiency still less, for we must introduce as a factor of 

(1 -r A ) h 

On the other hand, if we increase the counterhead e, the efficiency of 
the motor B increases proportionally, but the weight of fluid per second 

E—e, 

becomes less and less, and the work of the motor B per second 


R 


decreases as — E ^ decreases. 
xi 

But the work of the pump A per second also decreases as 

decreases, and the lost work due to resistance to flow through 
(E— e)- 

pumps decreases as -—— decreases. 

si 

With a theoretical efficiency of 50 per cent, an electrical motor is doing 
the largest amount of work in horse-power of which it is capable, but it 
demands of the generating dynamo twice as much power as it gives out. 
With a greater theoretical efficiency, it does not turn out so much work 
per second, but it makes a demand of less than twice its work upon the 
generating dynamo. 

This statement requires to be modified somewhat because of the imper¬ 
fections of machines, and becomes more accurate in proportion to the 
perfection of the machines used. 

This work lost in resistance of the pipe is plainly a minimum for E=e 
and a maximum for e = o if R remains constant. Let us separate it into 
its component parts. 

We see from the above equation that - E - ~ ^ = constant, when R 

Sv 

varies as the square of the difference of the heads, or when the square 
root of R varies as the difference of heads. 


E(E-e) 

R 

pipe and 























Electro-Dynamics. 


603 


If now we assume the resistance of the pumps as trifling in comparison 
with that of a long pipe or pipes connecting them, we see that we must 
double the difference of heads (E — e) in order to have the same loss of 
work per second with a pipe four times as long. 

Increasing the resistance four times gives us only one-quarter the 
weight of fluid per second assumed to be passing through the pipe, hut 
doubling the difference of heads also doubles the weight of fluid per 
second, so that, under the altered conditions, we obtain one-half the 
weight of fluid per second, and twice the effective head. Therefore, the 
work per second lost in the pipe is 

(E-e) (E-e)* 

R R ’ 


2 (E-e) XI 


as before. 

The work done by the pump and motor, each working with twice its 
former head, remains the same as before, and their relative efficiency 
is the same. 

This is what Marcel Deprez meant when he said, 

“ The useful mechanical work and the efficiency remain the same whatever "be 
the distance of transmission, provided the electro-motive forces, positive and 
negative, vary proportionately to the square root of the circuit's resistance.” 

Your attention should be called to the deadly nature of the very high 
electro-motive forces demanded by this law in the case of great dis¬ 
tances. 

The experiments of Marcel Deprez on the transmission of power by 
electricity have been conducted on a larger scale than others have yet 
attempted. 

In 1881, at the Paris Electrical Exhibition, he exhibited in the Palace 
of Industry one dynamo furnishing power to 27 different pieces of appa¬ 
ratus. No measurements of efficiency were made, as the question of dis¬ 
tribution was the only one then to be solved. He. however, then stated 
that it was possible to transport a useful work of.10 horse-power 31 miles 
by means of an ordinary telegraph wire, with the expenditure of only 
16 horse-power on the generating dynamo, realizing 6234 per cent, 
mechanical efficiency. 

At the Munich Electrical Exhibition of 1S82, over a line of telegraph- 
wire 36 miles, he obtained an electrical efficiency of 39 per cent, and 
an actual mechanical efficiency of 30 per cent. In his experiments on 
the lines of the Chemin de fer du Nord, March 4,1883, he transported 5 - 6 
horse-power 834 miles over ordinary telegraph-wires with 9 7 horse¬ 
power at the generating dynamo, realizing an electrical efficiency of 
69 1 4 per cent, and a mechanical efficiency of 58'3 per cent. 

In his experiments announced October 16,1885, he obtained from his 
first 77 per cent, electrical and 47-7 mechanical efficiency. In the second 
experiment he obtained 78 per cent, electrical and 53‘4 mechanical 
efficiency by means of dynamometric measurements. The distance 
between these two points is 56 kilometres—about 35 miles. The speed 
of the generator varied from 170 to 190 times a minute, and there was no 
appreciable heating. 

Tabulated results of experiments of Marcel Deprez—Convection of 
work between Creil and Paris: 



First Experiment. 

Second Experiment. 


Generator. 

Motor. 

Generator. 

Motor 

Turns per minute. 

190 

248 

170 

277 

Difference of potential. . . . 

5469 volts. 

4242 

5717 volts. 

4441 

Current. 

7'21 amperes. 

7-21 

7-20 ampOres. 

7-20 

Work in field. 

9 20 II.-P. 

3-75 

10-30 H.-P. 

3-80 

Work in armature. 

53-59 “ 

41-44 

55-90 “ 

43-40 

Measured mechanical work . 

62-10 “ 

35-80 

61* 

40- 

Electrical efficiency. 

Mechanical efficiency .... 

----------—-- 

77 per cent. 

47 - 7 per cent. 

78 per cent. 

53’4 per cent. 

































604 


Electro-Dynamics. 


Resistance of line, 100 ohms. 

“ “ generator, 33 ohms. 

“ “ motor, 36 ohms. 

Diameter of copper wire, 5 millimetres. 


35-80 

62*10 + 9-20 + 3-75 


Ofy CA 

= 0 - 477, for first experiment. 


40 

61 + 10-30 + 3-80 


40 

—= 0 534, for second experiment. 
75+ 


The labors of Marcel Deprez have both theoretically and practically 
opened the way and proved the entire feasibility of transporting great 
amounts of power for long distances. Much remains and will yet be 
accomplished in the way of cheapening the first cost of apparatus 
required, and also of rendering it automatic. 

Perhaps the first condition to be placed upon a motor used in manu¬ 
factures is that its speed shall be regular under all variations of load. 
Now, we know that with a constant field intensity, II , and length, L, of 
armature wire, the speed, V, and the counter electro-motive force, e, vary 
together. 


e = IIL V ; 


e 

IIL 


— V = constant. 


We see. then, that if we demand a constant speed and cannot vary the 
length of the armature wire, the intensity of the field must vary with the 
counter electro-motive force. 

This can be accomplished by means of double enrollment, commonly 
called “compound winding,” patented by Marcel Deprez in 1881. 

We know how the lines of force of the field are led by iron cores sur¬ 
rounded by coils of wire to the spot where the armature in revolving 
can cut them. If the whole current generated in the armature is led 
through the coil around the magnet and then through the external cir¬ 
cuit, the winding is technically called series winding. If only a part of 
the current is taken oil' at the binding-posts of the machine, and led 
through the coils around the magnet and back to the armature, the wind¬ 
ing is technically called shunt winding. 

The resistance of the shunt-wound magnet-coils is usually much greater 
than the external circuit, but the number of turns also is greater, and so 
we attain a field of equal intensity. 

Compound winding consists of the joint use of these two methods. 

Mr. F. J. Sprague has recently (April 7, 1885) patented a very clever 
combination of shunt and series winding for the purpose of obtaining a 
constant speed of motion for a constant potential circuit, such as is ordi¬ 
narily used for incandescent lighting. 



constant = 



Let R s = resistance of shunt field coils. 

“ iV g = number of turns of shunt field coils. 

“ i? d = resistance of series field coils. 

“ iV d = number of turns of series field coils. 

“ E — potential at terminals of motor. 

“ I = intensity of current through series coils. 
“ R — resistance of armature. 

E — R&I = potential at shunt terminals = E a . 


E-R t I 
R s 

E — R,\I— e 
R 


= amperes in shunt coils. 


r 


= amperes in armature 


' J 


= amperes in series coils. 






















Electro-Dynamics. 


605 


From the first equation we have 





N s f - JV d 

Ag 




Eliminating, 


iVs 

JVd 


J2 + 
R 


The magnetizing currents in shunt and series windings are sent in 
opposite directions, and the number of shunt windings is to the number 
of series windings as the sum of the resistances of the series Windings 
and the armature is to the resistance of the armature. 

This condition produces a magnetic field whose intensity is directly 
proportional to the counter electro-motive force, provided the magnets 
have not reached saturation. 

Mr. Sprague, by ingenious devices, causes the currents to act together 
to start the motor with a very strong effort, and, once started, reverses 
one current and sets the contrary currents in the field coils to balancing 
eacli other, so as to produce a constant speed. 

For constant potential circuits this motor will not govern if its theoret¬ 
ical efficiency is less than 50 per cent. On the other hand, for constant 
current circuits such as are used for arc lighting, this motor will not gov¬ 
ern if the theoretical efficiency is greater than 50 per cent. We need not 
discuss it. 

To avoid sparking at the brushes, Mr. Sprague has added a third series 
coil, which causes, in the case of dynamos having consequent poles, a 
counter-distortion of the poles of the field magnet proportional to the 
increase of strength of the armature magnet. 

For economical reasons, motors running on arc circuits with a constant 
current should have other methods of governing than the use of com¬ 
pound reversed coils. 

Mr. Weston uses two methods for obtaining a constant speed. The first 
is by using belts upon reversed cone-pulleys, which, with the aid of a 
centrifugal governor, shift so as to retain a constant speed for the driven 
machine, whatever be the variations of speed in the motor. The second 
is to vary the intensity of the field by means of resistance controlled by 
a governor or other automatic device. In our equation of condition for 
a constant speed we observed two suggested methods of procuring this 
constant speed. The first was to vary the intensity of the field with the 
counter electro-motive force. The second was to vary the length of the 
wire in the armature coils. 

Th is latter is manifestly impossible with the ordinary forms of machines, 
although it is not impossible that part of the field might be cut oft', or the 
armature itself partially removed from a constant field. 

Another way is to vary the counter electro-motive force of the motor 
by shifting the brushes around the commutator, but this is usually pro¬ 
ductive of sparking, and results in injury to both brushes and commu¬ 
tator. 

The number of variations of this method is legion. 

For the purpose of locomotion special arrangements to produce a uni¬ 
form speed are not required. From all parts of the civilized world we 
learn the steady progress of the successful application of dynamic elec¬ 
tricity to problems of locomotion. 

In the transmission of power by electricity the epds to be reached can 
well be stated under these heads: 

(A) Each receiving apparatus should receive its part of the generated 
power, and, whatever be its action, should not influence other apparatus 
on the same circuit. 

(li) The efficiency must be independent of the number of apparatuses 
in actiop. 

( C) When a regular speed is desired, the regulation should be auto¬ 
matic and instantaneous, and should not require the intervention of an 
attendant. 















606 


The Construction of Ships. 


THE CONSTRUCTION OF SHIPS. 

From Mr. Fronde’s experiments, it would appear that the midship section 
has very little effect, if any, upon the resistance of ships to propulsion. 

Tliis resistance arises from two causes: 

(1) The skin resistance, due to the area and nature of the surface exposed 
to water. 

(2) The wave resistance, due to the dimensions of the waves formed by the 
vessel. 

At a speed in knots equal to the square root of the length of the vessel in 
feet, which should not be exceeded, its resistance in pounds is one-200tk of 
the whole weight of the vessel. 

The entire resistances of similar ships at corresponding speeds are as the 
cubes of their corresponding lengths or dimensions. 

The resistance increases enormously when the speed given above is ex¬ 
ceeded. 

Fronde’s estimate of the distribution of the indicated horse-power of 
marine-engines is as follows: 


Resistance of naked hull 

40 per 

cent. 

«i 

due to suction of screw 

16 

tt 

it 

ii 

“ friction “ 

04 

ii 

U 

it 

“ “ of machinery 

27 

u 

ti 

u 

“ air-pumps 

01 

it 

ii 

Loss by slip of screw 

12 

it 

it 



100 




Tt is impossible in the limits of a pocket-book to give more than useful 
memoranda for those fully qualified by education and experience to under¬ 
take the designs of vessels. The elaborate works of fc'cott-Russel and 
Rankine, as also the smaller works of Thearle and many others, should be 
carefully read. An engineer who undertakes the design of a ship without 
exhausting all the attainable sources of knowledge of naval architecture is 
not honest. It would be well if all engineers professing to undertake designs 
for structures upon which human life depends should be held liable to trial 
for their lives in case of failure through their lack of knowledge or vigilance. 
—W. D. M. 

FAST OCEAX PASSAGES. 

The Cunard steamer Etruria holds the championship of the Atlantic, 
having made the fastest ocean passage each way, in August, 1885. On the 
7th she arrived off the Fastnet Light, from Sandy Hook, in 6 d. 5 h. 35 in. 
Returning, she left Queenstown August 16th, and made the run to Sandy 
Hook in 6 d. 5 h. 46 m. From Fastnet to Queenstown the run is 2 h. 4? m. 
On several days the steamer ran 465 miles in the twenty-four hours, showing 
about the same speed as an ordinary railroad train, say 20 miles an hour. 
The following table shows some of the fast passages made before the Etruria 
changed the record: 

To New York. To Queenstown. 





D. 

ir. 

M. 

l). 

H. 

M. 

Oregon. 




10 

10 

6 

15 

57 

America. 




15 

41 

6 

14 

18 

Alaska. 



. 6 

21 

40 

6 

18 

37 

Servia. 



. 7 

0 

55 

7 

1 

25 

Urania. 



. 7 

7 

0 

7 

10 

54 

Britannic. 



. 7 

7 

11 

7 

12 

41 

Arizona. 



. 7 

8 

32 

7 

7 

48 

Austral. 



. 7 

16 

0 

7 

9 

0 

Gallia. 




16 

32 

7 

18 

38 


2 $" = area of greatest immersed section. 

The following tables will prove useful as giving approximate dimensions. 

















4 v^tront'sParabolic Construction ot\s'hif 







































































































































































To Construct a Displacement Scale. 


607 


TO CONSTRUCT A DISPLACEMENT SCALE. 

D = displacement of the vessel in cubic feet. 

& = displacement in cubic feet per inch of difference of draft, 
a = area of load water line in square feet, d = draft of water in feet. 
a — area of any water line at draft y and displacement x. 
n = exponent of the displacement scale. 


a d 



D 



£d. 

d" 

y = 

VxT 

b 

1 

s 

l 

• 

0 = 

D ?/ n_ \ 

d n 


12 d n 



Example. The area of the load water line of a vessel is a = 6400 square feet; 
draft ot water d = 17 leet, and the load draft displacement D = 80,500 cubic feet. 
Required the draft expouent n — ? and at what draft y the displacement is x — 
45,000 cubic feet ? 


n = 6400X 17 =1 
80500 


y = 17 


1 45000 

80500 


= 11.05 feet, the draft required 


Construct a scale as shown by the accompanying figure, and draw the ordi¬ 
nates x; the draft d being divided into eight equal parts. 

Assuming the displacement as unit, the ordinates x are found in the following 
table, opposite the given exponent n. 

After the exponent is known, the displacement can be expressed in tons, and 
the load draft displacement multiplied by the tabular number gives the displace¬ 
ment x at the corresponding draft y. 

Rule. Multiply the load draft displacement, expressed either in tons or cubic 
feet, by the tabular number for the given exponent and water line, and the pro¬ 
duct is the corresponding displacement. 


Displacement Scale. 


a d 



Ordinate Waterlines. 



Dead rise. 

n =- 


3 



5 



D 

1 

3 

4 

6 

7 

1.00 

.1250 

.2500 

.3750 

.5000 

.6250 

.7500 

.8750 

Flat bottom. 

1.05 

.1127 

.2333 

.3571 

.4830 

.6105 

.7393 

.8692 


1.10 

.1015 

.2176 

.3300 

.4665 

.5963 

.7287 

.8634 


1.15 

.0915 

.2031 

.3237 

.4506 

.5824 

.7183 

.8577 


1.20 

.0825 

.1895 

.3082 

.4353 

.5689 

.7080 

.8512 

•52 © 

t/i 

1.25 

.0743 

.1768 

.2935 

.4205 

.5557 

.6980 

.8463 

c £ 

1.30 

.0669 

.1649 

.2794 

.4061 

.5428 

.6880 

.8407 

cl 

1.35 

.0604 

.1539 

.2660 

.3923 

.5302 

.67S2 

.8351 


1.40 

.0544 

.1436 

.2533 

.3789 

.5179 

.6685 

.8295 

* 

1.45 

.0490 

.1340 

.2303 

.3660 

.5047 

.6589 

.8240 

& 

1.50 

.0447 

.1250 

.2297 

.3535 

.4941 

.6495 

.8185 


1.55 

.0398 

.1166 

.2187 

.3415 

.4826 

.6402 

.8130 


1.60 

.0359 

.1088 

.2082 

.3299 

.4714 

.6311 

.8076 


1.65 

.0323 

.1015 

.1982 

.3186 

.4605 

.6221 

.8023 

tl. iQ_ 

1.70 

.0291 

.0947 

.1887 

.3078 

.4498 

.6132 

.7969 

r—i 

1.75 

.0257 

.0884 

.1797 

.2985 

.4393 

.6044 

.7916 


1.80 

.0233 

.0824 

.1711 

.2872 

.4291 

.5958 

.7864 


1.85 

.0213 

.0769 

.1629 

.2774 

.4129 

.5873 

.7811 


1.90 

.0192 

.0718 

.1551 

.2680 

.4094 

.5789 

.7769 


1.95 

.0173 

.0670 

.1477 

.2588 

.4008 

.5706 

.7708 

Highest 

2.00 

.0156 

.0625 

.1406 

.2500 

.3906 

.5625 

.7656 

dead rise. 



























































60S 


Approximate Lengths of Vessels. 



Sharp Vessels. = 3. 

Length L, Beam B, aiul draft d==iB, all lit feet. 



PU.S 

m 

5 

I, = 

5B 

L = 
«R 

L = 
7B 

L 8B 

L = 
9B 

L = 
I0B 

Displace¬ 

ment. 

II 03 

L = 
6B 

I. = 
7B 

L = 
8B 

L = 
9B 

L = 
10B 

T 

L, 

L. 

1a 

L, 

L 

L 

T 

I, 

1a 

L. 

I. 

1a 

L, 

1 

16-6 

183 

203 

22-2 

210 

25-8 

1000 

166 

183 

203 

222 

240 

258 


210 

22'9 

25-6 

28-0 

30-3 

32 5 

IlOO 

171 

189 

210 

230 

248 

267 

3 

240 

26 "4 

29-3 

320 

34 6 

37 2 

liOO 

177 

194 

216 

236 

255 

274 

1 

26-1 

29-0 

32 3 

35-3 

38-2 

410 

1300 

181 

199 

222 

242 

262 

281 

5 

28-4 

314 

34-8 

3S-0 

42'0 

44-2 

1400 

186 

204 

226 

249 

269 

288 | 

6 

30-3 

33-3 

370 

404 

43-8 

47-0 

1500 

190 

210 

233 

255 

275 

295 

'7 

31 7 

34 9 

38-8 

42-4 

4G’0 

49-3 

1000 

195 

214 

238 

260 

281 

302 

8 

33-2 

30'6 

406 

444 

480 

51-4 

1700 

199 

218 

243 

26-5 

286 

308 

9 

310 

38 0 

42-2 

46-2 

50-0 

53-7 

1800 

202 

223 

248 

270 

292 

314 

10 

35-7 

39-3 

48-7 

47-8 

517 

55 - 5 

1900 

206 

227 

252 

275 

298 

320 

11 

36-9 

40-6 

451 

49-3 

53-4 

57-3 

2000 

210 

230 

256 

280 

302 

325 

12 

38 0 

419 

46-6 

509 

55'0 

591 

2100 

212 

234 

260 

285 

307 

330 

13 

39 0 

43-0 

47-8 

52-2 

56 5 

GO-6 

2200 

216 

238 

265 

290 

312 

335 

14 

400 

440 

490 

53-6 

58-0 

62-2 

2300 

220 

242 

269 

295 

317 

341 

15 

40 "9 

45 0 

50-0 

547 

590 

63-5 

2400 

223 

245 

272 

300 

322 

346 

10 

41-8 

4G 0 

51-2 

5G'0 

60-6 

65 0 

2500 

226 

248 

276 

304 

326 

351 

17 

42-7 

46-9 

52-2 

57 0 

61 8 

663 

2GOO 

229 

252 

280 

308 

330 

355 

18 

430 

47 8 

53-2 

58-0 

63 0 

67-7 

2700 

232 

255 

283 

311 

334 

360 

19 

44 4 

48 6 

54-2 

59 0 

642 

690 

2800 

235 

259 

287 

315 

338 

365 

30 

45-0 

49-5 

55-1 

60 0 

65-2 

70-0 

2900 

238 

261 

290 

318 

342 

369 

25 

48-5 

534 

59-2 

64-8 

70-2 

75-4 

3000 

240 

264 

294 

321 

347 

373 

30 

51 G 

57-0 

63-3 

69 0 

747 

80-2 

3100 

243 

267 

297 

324 

350 

377 

35 

543 

596 

66-4 

727 

78 6J 

84-3 

3200 

246 

270 

300 

327 

354 

381 

40 

568 

626 

69 5 

76-0 

82-2 

88-3 

3300 

24S 

272 

303 

330 

368 

385 

45 

59-2 

65 0 

72-3 

79-0 

85 - 5 

92-0 

3400 

250 

275 

306 

333 

362 

3S9 

50 

61-2 

67 2 

74 8 

81-8 

88 4 

95-0 

3500 

253 

278 

309 

337 

365 

393 

55 

63-1 

69-4 

77-3 

84-4 

91 4 

98-0 

3600 

255 

281 

312 

340 

369 

396 

60 

04-9 

71-4 

79-4 

868 

939 

101 

3700 

257 

283 

314 

344 

372 

399 

65 

GG 8 

73-5 

81-7 

89-3 

96-6 

104 

3800 

260 

285 

317 

347 

375 

403 

70 

68‘4 

75-3 

83-7 

91-6 

98-0 

107 

3900 

262 

288 

320 

350 

378 

406 

75 

70-1 

771 

858 

93-8 

101 

109 

4000 

265 

291 

3-'3 

353 

380 

409 

80 

71-6 

78-9 

87-6 

95-7 

103 

111 

4100 

268 

293 

325 

356 

383 

413 

85 

730 

80-5 

895 

97-7 

106 

114 

4200 

270 

296 

328 

359 

386 

417 

90 

745 

819 

911 

99 6 

108 

116 

4300 

272 

298 

331 

362 

389 

420 

95 

75-8 

83-4 

92 7 

111 

100 

118 

4400 

273 

300 

334 

365 

392 

423 

100 

77-0 

84-7 

94 4 

103 

112 

120 

4500 

275 

302 

337 

368 

395 

426 

110 

79-7 

87-6 

97-4 

lo7 

115 

124 

4600 

277 

304 

339 

370 

398 

429 

125 

83 0 

915 

100 

111 

120 

129 

4700 

279 

306 

331 

372 

401 

432 

150 

88-3 

97-0 

108 

113 

128 

137 

4800 

281 

308 

343 

374 

404 

4.35 

175 

930 

102 

114 

124 

134 

144 

4900 

283 

310 

345 

376 

407 

438 

200 

973 

107 

119 

130 

140 

151 

5000 

286 

312 

348 

378 

411 

441 

225 

101 

111 

124 

135 

146 

157 

5250 

2S9 

318 

354 

386 

418 

450 

250 

105 

115 

128 

140 

152 

163 

5500 

294 

323 

359 

392 

424 

457 

275 

108 

119 

132 

145 

156 

168 

5750 

298 

331 

364 

398 

430 

463 

300 

111 

122 

136 

149 

161 

173 

6000 

303 

336 

370 

404 

437 

469 

335 

114 

126 

140 

153 

165 

178 

6250 

307 

340 

375 

409 

443 

476 | 

350 

117 

129 

143 

157 

170 

1S2 

G500 

310 

345 

380 

414 

448 

482 

375 

120 

130 

147 

161 

173 

186 

6750 

315 

349 

384 

420 

454 

48S 

400 

122 

135 

150 

164 

177 

191 

7000 

319 

354 

389 

425 

460 

494 

450 

128 

140 

156 

170 

184 

198 

7250 

322 

358 

394 

430 

465 

500 

500 

132 

145 

161 

176 

191 

205 

7500 

326 

361 

398 

435 

470 

506 

550 

13G 

150 

166 

182 

197 

211 

7750 

330 

365 

402 

440 

475 

512 

600 

140 

154 

171 

188 

202 

217 

8000 

333 

370 

407 

445 

480 

517 

650 

144 

158 

176 

193 

208 

223 

8250 

337 

374 

410 

449 

485 

522 

700 

148 

162 

180 

197 

213 

229 

8500 

340 

377 

415 

453 

490 

527 

750 

151 

106 

185 

202 

218 

235 

8750 

343 

380 

419 

457 

495 

532 

800 

154 

170 

189 

206 

223 

210 

9000 

347 

384 

423 

461 

500 

537 

850 

158 

173 

193 

210 

227 

245 

925 0 

350 

388 

427 

466 

504 

542 

900 

160 

176 

196 

215 

232 

250 

9500 

353 

392 

430 

470 

509 

547 

950 

1G3 

1.^0 

200 

219 

236 

254 

10000 

359 

399 

438 

478 

517 

556 


u 










































Approximate Lengths of Vessels. 


G09 


Full Vessels. 


Length. L, Beam B, and draft d = lB, all in feet. 


c. d 

Ol 

5 

L = 
5B 

L = 
6B 

II ££ 

1 M 

L = 
8B 

L = 
9B 

L = 
10B 

Displace¬ 

ment. 

L = 
5B 

L = 
6B 

L = 
7B 

L = 
8B 

L = 
9B 

L = 
10B 

T 

L. 

L, 

L 

L. 

L, 

L, 

T 

L. 

L. 

L. 

L, 

L, 

Li 

1 

10-9 

14-G 

1G-2 

17'8 

19-2 

20-6 

lOOO 

109 

146 

162 

178 

192 

206 

3 

13-7 

18-3 

20-4 

22-4 

24-2 

26-0 

1100 

112 

151 

166 

184 

199 

213 

3 

15-7 

21-0 

23-4 

25-7 

27-7 

29-7 

1400 

116 

155 

173 

190 

205 

219 

4 

17-3 

23-2 

25-8 

28-3 

30-5 

32-8 

1300 

119 

160 

177 

195 

210 

225 

5 

18-7 

25-0 

27-7 

30-4 

32-8 

35-2 

1LOO 

122 

164 

180 

199 

216 

230 

6 

19-8 

26-6 

29 5 

324 

34-9 

376 

1500 

125 

167 

184 

204 

221 

235 

7 

20-8 

28-0 

31-0 

340 

3G-7 

39-4 

1600 

127 

171 

188 

208 

226 

240 

8 

21-8 

29-2 

324 

35 6 

38-4 

41-2 

1700 

130 

174 

193 

212 

230 

245 

9 

22-7 

30-4 

33-7 

37-0 

40-0 

42-8 

1SOO 

132 

177 

198 

217 

235 

250 

10 

23-5 

31-4 

34-9 

383 

41-3 

44-3 

1000 

135 

181 

201 

220 

239 

255 

11 

24-2 

32-4 

3G-0 

39-5 

42-6 

45-7 

3000 

137 

184 

205 

224 

243 

260 

33 

25-0 

33-3 

371 

40-7 

440 

47-2 

3100 

139 

188 

208 

228 

247 

264 

13 

25-6 

34-3 

381 

41-8 

45-1 

48-4 

3300 

141 

190 

211 

231 

251 

268 

14 

20-3 

352 

39-0 

42-9 

46-2 

49-6 

3.300. 

144 

193 

214 

234 

254 

272 

15 

268 

360 

39 9 

43-8 

47-2 

50 7 

3V00 

146 

195 

217 

238 

258 

276 

1G 

27-5 

368 

40-9 

448 

48-4 

51-8 

3500 

148 

198 

220 

242 

261 

280 

17 

28-0 

37 5 

41-8 

45'7 

49-4 

529 

3600 

150 

200 

223 

245 

264 

284 

18 

28-5 

383 

42-G 

46-6 

50-3 

540 

3700 

152 

203 

226 

248 

267 

287 

19 

29-0 

390 

43-4 

47-4 

51-3 

56-0 

3800 

154 

205 

228 

261 

270 

290 

30 

29 5 

39-G 

44-2 

48-2 

52-0 

58-0 

3000 

155 

208 

231 

254 

274 

294 

35 

31-8 

42-6 

45-4 

52 0 

56*1 

60-0 

3000 

157 

211 

234 

257 

277 

297 

30 

33 9 

45 5 

505 

55-3 

59-9 

64-0 

3100 

159 

213 

237 

260 

280 

300 

35 

35-6 

47-S 

531 

58-1 

62-9 

67-2 

3300 

160 

215 

240 

262 

284 

304 

40 

37 3 

500 

55-6 

608 

65-7 

70-4 

3300 

162 

218 

242 

264 

287 

307 

45 

38-8 

520 

57-8 

633 

68 4 

73-3 

3400 

164 

220 

245 

267 

290 

310 

50 

40-1 

53-8 

598 

65-5 

70-7 

75-7 

3500 

166 

222 

248 

270 

292 

313 

55 

414 

55-5 

Gl-7 

67-7 

73-1 

78-2 

3600 

167 

225 

250 

272 

295 

316 

60 

42-6 

57-1 

63-5 

G9-5 

75-2 

80-4 

3700 

169 

2-6 

252 

275 

297 

319 

65 

43-8 

58-7 

65-2 

71-5 

77-4 

82-7 

3800 

171 

228 

254 

278 

299 

322 

70 

45 0 

60-2 

G69 

73-4 

79-2 

850 

3000 

172 

230 

255 

280 

302 

325 

75 

46-0 

61-6 

68-6 

75-2 

81-2 

87-0 

4000 

173 

232 

258 

283 

305 

327 

80 

47-0 

63-0 

700 

770 

82-9 

88-8 

4100 

174 

234 

261 

285 

308 

330 

85 

47-9 

G4-2 

71-5 

78 2 

84-6 

905 

4300 

175 

236 

263 

287 

310 

333 

90 

48-8 

655 

72-8 

79-7 

863 

92-3 

4300 

177 

238 

265 

289 

312 

336 

95 

497 

GG-G 

74-2 

81-3 

87-8 

94-0 

4400 

178 

240 

267 

291 

314 

338 

100 

50-5 

67 S 

754 

82-7 

89-2 

95-6 

4500 

179 

242 

269 

294 

317 

340 

310 

52-2 

700 

78-0 

85-3 

92-2 

98-5 

4600 

181 

214 

271 

296 

320 

343 

135 

55-0 

73-0 

810 

89-0 

96-0 

103 

4700 

182 

246 

273 

29 S 

322 

345 

150 

57-8 

77-5 

86-4 

94-5 

102 

110 

4800 

184 

248 

275 

300 

324 

347 

175 

GO-8 

Sl-G 

90-8 

99-5 

108 

115 

4000 

185 

249 

276 

302 

326 

350 

300 

63 6 

85-4 

95-2 

104 

113 

120 

5000 

187 

250 

277 

304 

329 

352 

335 

66-2 

88-9 

99-0 

108 

117 

125 

5350 

190 

254 

278 

309 

334 

354 

350 

G8-7 

92 0 

102 

112 

121 

130 

5500 

193 

258 

287 

315 

339 

356 

375 

70-8 

95-0 

10G 

116 

125 

134 

5750 

196 

262 

291 

319 

345 

369 

300 

72-8 

97-8 

109 

119 

129 

138 

6000 

198 

266 

295 

324 

350 

374 

335 

74-0 

100 

112 

123 

132 

142 

6350 

200 

269 

300 

328 

354 

383 

350 

76-8 

103 

115 

126 

136 

146 

6500 

203 

273 

303 

332 

359 

384 

375 

78-7 

10G 

118 

129 

139 

149 

6750 

206 

276 

307 

336 

364 

390 

400 

80-3 

108 

120 

132 

142 

152 

7000 

209 

280 

311 

340 

368 

394 

450 

85'5 

112 

125 

137 

148 

158 

7350 

211 

283 

314 

344 

372 

399 

500 

86'4 

116 

130 

142 

153 

164 

7500 

213 

286 

318 

34S 

376 

404 

550 

892 

120 

133 

14G 

158 

169 

7750 

216 

289 

322 

352 

380 

407 

GOO 

91 9 

123 

136 

150 

162 

174 

8000 

218 

292 

325 

356 

3S4 

412 

650 

94-4 

127 

140 

154 

167 

178 

8350 

220 

296 

328 

360 

3S8 

417 

700 

96-6 

130 

144 

158 

170 

183 

8500 

222 

299 

331 

364 

392 

421 

750 

98-9 

133 

148 

1G2 

173 

187 

8750 

224 

301 

334 

367 

396 

425 

800 

10 L 

136 

151 

165 

177 

191 

0000 

226 

304 

337 

370 

400 

429 

850 

103 

139 

154 

169 

182 

195 

9350 

229 

307 

310 

374 

404 

432 

900 

105 

142 

157 

172 

186 

199 

9500 

231 

310 

314 

377 

408 

437 

950 

107 

144 

159 

175 

189 

203 

10000 

235 

315 

350 

384 

414 

446 


39 













































STEAMSHIP rERFORMAA'CE. 


Horsepower in Steamship Performance. 


Displace¬ 
ment in 
tons. 

1 

2 I 

Nautical in 

3 | 4 

ilea or Is 

5 

nots pe 
6 

r hour. 

7 

8 

9 

10 

T 

11 

H 1 

ii 

11 

ii 

ii 

ii 

11 

ii 

II 

1 

0-004 

0-035 

0-118 

0-280 

0'550 

0-949 

1-50 

2-24 

3-20 

4-38 

3 

0-007 

0-055 

0-190 

0-444 

0-870 

1-51 

2-40 

355 

5-08 

6 96 

3 

0-009 

0-075 

0248 

0-598 

114 

1-98 

3-12 

479 

6-91 

912 

4 

0010 

0-084 

0 300 

0-673 

1-40 

2-40 

3-80 

5-39 

8 06 

ll-lo 

5 

0-012 

0-102 

0348 

0-818 

1-52 

2-78 

440 

6"55 

936 

12 2 

6 

0-014 

0115 

0-390 

0-924 

1-8 L 

3-12 

4-96 

7-39 

10 6 

14-5 

7 

0-016 

0-128 

0435 

1-025 

2-01 

3 48 

5-50 

8-20 

11-7 

161 

8 

0-017 

0138 

0-479 

1-125 

2 20 

3-80 

601 

8-96 

12-8 

17 5 

9 

0-019 

0 151 

0-501 

1-211 

2-38 

412 

651 

969 

13-8 

190 

10 

0-020 

0-161 

0-552 

130 

2-54 

4-42 

6 98 

10-4 

14-9 

203 

11 

0022 

0175 

0590 

1-40 

2-72 

4-70 

746 

111 

15-9 

21-8 

13 

0-023 

0185 

0 624 

1-48 

2-88 

4-99 

790 

11 8 

16-8 

23-0 

13 

0-024 

0195 

0-654 

1-56 

3-04 

5-25 

833 

12-5 

17-7 

243 

11 

0024 

0198 

0-690 

1-62 

3-18 

5-52 

8"75 

130 

18 6 

25-4 

15 

0026 

0-213 

0-725 

1-70 

3-32 

5-80 

9-20 

13-6 

19-5 

26 6 

16 

0-028 

0223 

0-780 

1-78 

3-49 

604 

9-55 

14-2 

20-4 

27-9 

17 

0-029 

0*236 

0785 

1-89 

3-64 

6-28 

9-95 

150 

21-2 

29-1 

18 

0-030 

0-242 

0-815 

1-94 

3-78 

6'52 

10-3 

15 5 

22-0 

30-2 

19 

0 031 

0-250 

0-850 

2-GO 

390 

6-80 

10-7 

16-0 

22 8 

31-2 

30 

0-032 

0-258 

0-875 

2'06 

402 

7-00 

11*1 

165 

23-0 

32-2 

35 

0 038 

0-300 

1015 

2-40 

4 14 

812 

12-9 

19-2 

24-2 

33-1 

30 

0042 

0-33S 

1 14 

270 

530 

9-18 

14 6 

21-6 

31 0 

42-4 

35 

0-017 

0-375 

1-26 

300 

fa-89 

10-1 

162 

24-0 

31-2 

471 

*0 

0-050 

0-409 

1-39 

3-27 

6 41 

11-1 

17 6 

20-2 

375 

51-3 

45 

0-056 

0-445 

150 

356 

6-95 

12 0 

19 0 

28-5 

40-5 

55 G 

50 

0056 

0474 

1-61 

3-79 

744 

12 9 

20-5 

30-3 

43 2 

595 

55 

0-062 

0-501 

172 

4-06 

7 95 

138 

21-8 

32-5 

46-2 

63-6 

60 

0-067 

0-538 

1-80 

4-30 

841 

14-4 

231 

34-4 

491 

67 3 

65 

0-071 

0-570 

1-90 

456 

888 

151 

24-4 

36-5 

51-8 

71-0 

70 

0-074 

0597 

2-02 

477 

9-36 

16-2 

25-5 

38-2 

544 

74-9 

75 

0-078 

0-625 

2 12 

5-00 

9 77 

16-9 

26-8 

40-0 

56-8 

78-0 

80 

0081 

0*650 

2-20 

5-20 

10-2 

17 6 

28 0 

41-6 

5*1 

81-6 

85 

0-085 

0 680 

2-30 

5-44 

10-6 

18 4 

29-2 

43-5 

62 0 

850 

90 

0-0S8 

0705 

238 

5 64 

11 0 

19 1 

30 5 

45-2 

64-5 

88-4 

95 

0-088 

0710 

2-49 

5 6S 

11 4 

19-9 

31-3 

470 

66'6 

91-5 

10O 

0094 

0755 

2-56 

604 

11-8 

20-5 

32-4 

48-4 

68 5 

94-5 

110 

0-101 

0 810 

2-73 

6-48 

12 6 

219 

34 6 

51-8 

732 

101 

135 

0-109 

0877 

2-98 

702 

13-7 

23-8 

*j i *5 

56-2 

80-0 

110 

150 

0124 

0 990 

3’o8 

7-72 

15 5 

27 0 

428 

6L-7 

90-5 

124 

175 

0138 

110 

3-72 

8-81 

17 2 

29-8 

47 2 

70-5 

100 

138 

300 

0150 

1-20 

4 06 

9-60 

18-8 

32-5 

51 5 

76-9 

110 

150 

335 

0-1G2 

1 30 

4-39 

10-4 

20-2 

351 

66-0 

83-3 

118 

162 

350 

0175 

1-40 

4-70 

11-2 

21-9 

37 6 

69-8 

89 2 

127 

175 

375 

0188 

1-50 

5 04 

11-9 

23-2 

40-3 

63-8 

95-2 

136 

186 

300 

0196 

1-57 

6-31 

12 6 

245 

425 

67-5 

100 

142 

196 

335 

0 201 

1-66 

5-63 

13-3 

26 0 

45 0 

712 

106 

152 

208 

350 

0 220 

1-75 

5 91 

14 0 

27 4 

47-3 

75-0 

112 

159 

219 

375 

0-228 

1-82 

6 12 

14-6 

28-6 

49-0 

78-4 

117 

166 

229 

400 

0-240 

1-91 

6 42 

15-3 

29 8 

51-4 

81-7 

122 

172 

238 

450 

0250 

2-06 

6-98 

16-5 

32-2 

55-8 

88 5 

132 

188 

258 

500 

0-276 

2-21 

7-45 

17-7 

34 6 

59-6 

94-3 

141 

200 

276 

550 

0295 

2-36 

7 98 

18-9 

369 

63-8 

101 

151 

215 

295 

600 

0-312 

2-50 

840 

200 

390 

67-2 

107 

160 

220 

313 

650 

0330 

2-64 

8 90 

21-1 

41-2 

712 

113 

169 

240 

329 

700 

0-348 

2-78 

9 32 

22-2 

43-3 

74 6 

119 

177 

250 

337 

750 

0-362 

2-90 

9 80 

23-2 

45-2 

78-4 

124 

186 

264 

352 

800 

0-380 

3-03 

10-2 

24-2 

47-3 

81 5 

130 

194 

274 

378 

850 

0-394 

315 

10 6 

25-2 

49-2 

85 0 

135 

202 

288 

394 

900 

0410 

328 

11-0 

26 2 

fall 

88-1 

140 

210 

296 

409 

950 

1 0-422 

3 41 

11 4 

27-3 

631 

91-8 

146 

218 

310 

<445 
























































Steamship Performance. 


611 


Horsepower in Steamship Performance. 


Displace¬ 
ment in 
tons. 

11 

12 

13 

Nautical miles or knot 

14 | 15 | 16 

s per ho 

17 

ur. 

18 

19 

20 

T 

11 

11 

ii 

H 

11 

11 

H 

ii 

n 

H 

1 

585 

7'59 

9-63 

12-0 

14-8 

179 

216 

25-6 

30-1 

351 

3 

9-28 

12-0 

15 3 

19 1 

23-5 

284 

34 2 

40 6 

47-8 

54-7 

3 

122 

15-8 

200 

250 

30-8 

38-3 

44-8 

53-3 

62-6 

730 

4 

148 

19 2 

24 4 

3( -3 

37 4 

431 

543 

64-5 

759 

88-4 

5 

17-2 

22-2 

28 3 

35-2 

43-4 

52-4 

63 0 

74-9 

880 

97 8 

6 

194 

25-1 

319 

39 7 

49-0 

59-1 

71-1 

84-5 

992 

116 

7 

21-4 

27-8 

35-3 

44-0 

54-0 

655 

790 

937 

no 

128 

8 

234 

30-4 

38 6 

48-1 

59-3 

68-7 

86-2 

102 

121 

140 

9 

253 

32 9 

41 8 

52-1 

640 

77-5 

93-2 

110 

130 

152 

io 

27 2 

35 3 

44 8 

55-8 

688 

83 2 

100 

119 

140 

163 

11 

29-0 

37-6 

47-8 

59-7 

735 

89-0 

107 

lz7 

150 

174 

12 

307 

399 

60 6 

632 

77-7 

944 

113 

134 

158 

184 

13 

324 

42-0 

53 3 

66-6 

820 

996 

120 

142 

167 

194 

14 

34 0 

44-2 

56 0 

700 

860 

105 

126 

149 

176 

203 

15 

35G 

46-3 

58-7 

735 

90-0 

109 

131 

156 

183 

213 

16 

37-2 

48 3 

613 

76-5 

94-0 

114 

137 

163 

192 

223 

17 

387 

60-2 

638 

796 

98-0 

120 

143 

170 

200 

233 

18 

40-2 

52-2 

662 

82-7 

102 

124 

148 

176 

207 

242 

19 

41-7 

54-0 

687 

85*8 

106 

128 

154 

182 

215 

250 

20 

432 

560 

71 0 

88-9 

111 

132 

159 

189 

222 

258 

25 

500 

65-0 

82-5 

103 

127 

154 

184 

194 

258 

265 

30 

565 

734 

93-2 

117 

143 

173 

208 

248 

291 

339 

35 

626 

81-3 

103 

130 

159 

192 

230 

274 

322 

377 

40 

684 

88-8 

113 

141 

173 

209 

252 

300 

350 

410 

45 

74-0 

96 2 

122 

152 

188 

228 

273 

3.4 

382 

445 

50 

794 

103 

131 

164 

201 

242 

293 

346 

410 

476 

55 

84-6 

110 

140 

174 

215 

260 

312 

370 

437 

509 

60 

900 

117 

149 

185 

226 

285 

330 

393 

464 

538 

65 

947 

123 

156 

195 

240 

292 

849 

414 

488 

568 

70 

99 6 

130 

1C4 

206 

252 

306 

367 

437 

512 

599 

75 

’104 

135 

171 

214 

264 

320 

383 

455 

536 

624 

80 

109 

141 

180 

224 

276 

333 

400 

467 

561 

653 

85 

113 

147 

187 

234 

287 

348 

417 

496 

584 

680 

90 

118 

153 

194 

243 

298 

362 

433 

516 

607 

707 

95 

122 

158 

201' 

251 

309 

376 

448 

533 

629 

732 

100 

126 

164 

207 

259 

318 

387 

464 

551 

648 

756 

110 

135 

175 

222 

277 

340 

414 

495 

588 

693 

807 

125 

146 

190 

241 

300 

370 

450 

539 

640 

753 

878 

150 

165 

215 

273 

342 

420 

494 

609 

724 

852 

992 

175 

183 

238 

302 

378 

464 

564 

675 

802 

946 

1100 

200 

200 

260 

330 

412 

506 

615 

737 

875 

1027 

1201 

225 

217 

281 

368 

447 

548 

666 

800 

947 

1118 

1300 

250 

232 

301 

384 

478 

688 

714 

855 

1016 

1200 

1400 

275 

248 

322 

409 

510 

627 

762 

912 

1087 

1286 

1490 

300 

262 

340 

432 

540 

662 

806 

966 

1146 

1347 

1573 

325 

277 

360 

457 

1 570 

700 

852 

1010 

1213 

1428 

1665 

350 

290 

378 

480 

600 

737 

896 

1073 

1276 

1500 

1750 

375 

305 

395 

502 

627 

770 

936 

1122 

1332 

1570 

1830 

400 

317 

412 

522 

654 

803 

976 

1170 

1402 

1632 

1907 

450 

343 

446 

567 

708 

870 

1060 

1265 

1500 

1770 

2065 

500 

368 

478 

607 

759 

932 

1131 

1358 

1611 

1896 

2213 

550 

393 

510 

648 

810 

995 

1210 

1450 

1720 

2025 

2362 

600 

415 

540 

684 

856 

1036 

1280 

1532 

1820 

2140 

2500 

650 

440 

570 

724 

905 

1111 

1350 

1618 

1923 

2265 

2636 

700 

460 

599 

759 

938 

1166 

1417 

1700 

2016 

2373 

2770 

750 

483 

627 

797 

995 

1220 

1485 

1780 

2113 

2490 

2900 

800 

503 

654 

830 

1038 

1274 

1548 

1857 

2206 

2593 

3026 

850 

525 

680 

866 

1080 

1330 

1620 

1935 

2300 

2710 

3152 

900 

545 

708 

898 

1123 

1380 

1675 

2009 

2385 

2803 

3274 

950 

565 

734 

933 

1170 

1430 

1740 

2080 

2478 

2920 

3400 




































Steamship Performance, 


612 



Horsepower in Steamship Performance. 


Diaplace- 


Nautical miles or knots per hour. 


tons. 

1 

2 1 

3 

4 

5 

6 

7 

8 

9 

10 

T 

11 

II 

ii 

ii 

ii 

ii 

ii 

11 

ii 

II 

1000 

0438 

3-50 

11-8 

28-0 

54-9 

94-6 

150 

225 

318 

439 

1100 

0'456 

3-75 

12-5 

30-0 

58-4 

100 

160 

239 

338 

467 

1200 

0-500 

4-00 

13-4 

320 

62 0 

107 

170. 

254 

359 

495 

1300 

0'515 

4-12 

140 

330 

65-3 

112 

179 

267 

378 

523 

1100 

0-548 

4-38 

14-9 

350 

68-7 

119 

189 • 

281 

398 

549 

1500 

0562 

4-50 

15-5 

360 

719 

124 

197 

295 

417 

575 

1000 

0-578 

4-62 

16-2 

370 

75-0 

130 

206 

307 

435 

600 

1700 

0-594 

4-75 

16-9 

38-0 

781 

135 

215 

320 

453 

625 

1800 

0625 

5-00 

17-5 

400 

81-2 

140 

224 

332 

470 

649 

1900 

0-634 

6 25 

181 

42-0 

84-2 

145 

231 

345 

488 

673 

2000 

0*700 

5-60 

1S-8 

44-0 

87-0 

150 

239 

356 

504 

696 

2100 

0719 

5-75 

19 4 

46 0 

90-0 

155 

247 

£69 

521 

720 

2200 

0-735 

5-88 

200 

47-0 

92-7 

160 

255 

380 

537 

741 

2300 

0765 

612 

20-6 

49 0 

956 

1G5 

262 

391 

654 

764 

2400 

0788 

6-28 

211 

50 2 

984 

170 

270 

4o2 

569 

786 

2500 

0-805 

614 

21 8 

51-5 

101 

174 

277 

414 

685 

808 

2600 

0-828 

6 62 

22-4 

53.0 

104 

179 

285 

424 

600 

826 

2700 

0-851 

6-81 

23-0 

545 

106 

184 

292 

436 

616 

850 

2800 

0-872 

6-98 

23-5 

55-8 

109 

188 

299 

446 

631 

871 

2900 

0-876 

712 

240 

571 

111 

192 

30G 

457 

646 

893 

3000 

0-909 

735 

24-6 

58-8 

114 

197 

313 

467 

660 

913 

3100 

0 931 

7-45 

251 

59-8 

117 

201 

320 

478 

676 

933 

3200 

0952 

7-62 

25-6 

61-0 

119 

205 

327 

488 

690 

952 

3300 

0 972 

7-78 

26 1 

62-2 

121 

209 

334 

498 

704 

972 

3400 

0 992 

7 94 

26-8 

63-5 

124 

214 

340 

6o8 

718 

992 

3500 

1*01 

8-10 

27-2 

64-8 

127 

218 

347 

618 

733 

1010 

3000 

1-03 

8-25 

27-8 

66-0 

129 

222 

354 

528 

746 

1025 

37 00 

1-05 

839 

28-2 

671 

131 

226 

360 

538 

759 

1049 

3800 

1-08 

8-60 

27-8 

68-5 

133 

230 

367 

648 

774 

1070 

3900 

109 

8"7() 

28-9 

69-6 

135 

234 

373 

658 

787 

1087 

4000 

111 

8-85 

29-9 

70 8 

138 

238 

380 

567 

801 

1105 

4100 

113 

9-01 

30-4 

711 

140 

242 

3!-G 

577 

814 

1122 

4200 

114 

9-14 

30-9 

731 

142 

24G 

392 

686 

827 

1141 

4300 

116 

9-30 

314 

744 

145 

250 

398 

595 

840 

1160 

4400 

1-18 

9-42 

31-9 

75-5 

147 

2£4 

404 

604 

863 

1179 

4500 

119 

9-50 

32-4 

76-5 

150 

258 

410 

613 

8*6 

1198 

4000 

1 22 

9-72 

32-8 

77-7 

152 

261 

416 

622 

879 

1216 

4700 

1-23 

9-86 

33-4 

78-9 

154 

26G 

422 

631 

881 

1232 

4800 

1-25 

100 

33-9 

80-0 

156 

270 

428 

640 

904 

1248 

4900 

1-28 

101 

34-4 

811 

158 

274 

434 

649 

916 

1265 

5000 

1-30 

103 

34 8 

82-7 

160 

277 

440 

658 

929 

1282 

5250 

1-32 

10-6 

35-6 

85-0 

165 

283 

455 

670 

959 

1324 

5500 

1-36 

109 

36 4 

87-5 

171 . 

290 

469 

700 

990 

1867 

5750 

1-40 

11-2 

37 - 5 

900 

17G 

298 

483 

721 

1024 

1408 

0000 

1-42 

114 

38 0 

92-8 

181 

303 

497 

742 

1050 

1448 

6250 

1-47 

119 

40-2 

95-2 

188 

322 

512 

762 

1065 

1488 

6500 

1-52 

12-2 

41 2 

97-8 

191 

330 

526 

782 

1**78 

1526 

0750 

1-56 

12-5 

42-4 

100 

19G 

339 

540 

802 

1123 

1567 

7000 

1-60 

12-9 

43-2 

103 

202 

34G 

554 

822 

1174 

1616 

7250 

1 64 

131 

44-4 

105 

205 

355 

506 

842 

1198 

1644 

7500 

1-68 

135 

45 5 • 

108 

210 

3 64 

679 

861 

1226 

1682 

7750 

1-72 

13 8 

46 5 

110 

215 

372 

599 

879 

1253 

1719 

8000 

1*75 

140 

47-4 

112 

220 

379 

603 

899 

1280 

1757 

8250 

1-78 

14-2 

48 4 

115 

224 

387 

615 

918 

1306 

1793 

8500 

1-81 

14-5 

49-4 

116 

229 

395 

628 

929 

1333 

1829 

8750 

1 84 

14 9 

50-0 

119 

233 

403 

640 

955 

1554 

1865 

9000 

1-88 

15-2 

51-1 

122 

238 

411 

653 

973 

1385 

1902 

9250 

1-92 

15 4 

52-2 

124 

242 

418 

668 

991 

1411 

1937 

9500 

1-95 

150 

63-2 

126 

24G 

426 

683 

1008 

1437 

1972 

10000 

205 

16 4 

651 

131 

255 

411 

714 

1044 

1488 

2012 






































Steamship Performance, 


613 


Displace- 


Horsepower in Steamship Performance. 

Nautical miles or knots per hour. 


tons. 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

T 

II 

II 

. H 

ii 

II 

ii 

II 

II 

H 

11 

1000 

585 

759 

963 

1206 

1480 

1798 

2157 

2560 

3008 

3514 

1100 

622 

806 

1024 

]2S4 

1574 

1913 

2295 

2723 

3203 

3736 

1400 

660 

858 

1090 

1360 

1670 

2030 

2435 

2890 

3400 

3967 

1300 

696 

903 

1147 

1432 

1758 

2136 

2564 

3043 

3576 

4178 

1400 

732 

950 

1204 

1508 

1850 

2248 

2697 

3200 

3762 

4394 

1500 

706 

995 

1264 

1580 

1938 

2355 

2825 

3252 

3943 

4605 

1000 

800 

1038 

1317 

1648 

2020 

2458 

2948 

3500 

4113 

4803 

1700 

833 

1083 

1374 

1718 

2107 

2561 

3072 

3646 

4286 

5006 

1800 

864 

1123 

1422 

1784 

2188 

2660 

3140 

3785 

4448 

5195 

1000 

897 

1166 

1479 

1850 

2270 

2760 

3310 

3928 

4615 

5390 

4000 

927 

1205 

1527 

1913 

2345 

2854 

3420 

4060 

4770 

5570 

4100 

958 

1247 

1582 

1979 

2382 

2948 

3535 

4195 

4935 

5762 

4400 

988 

1284 

1628 

2037 

2500 

3038 

3642 

4325 

5084 

5935 

4300 

1017 

1324 

1680 

2102 

2578 

3134 

3755 

4460 

5241 

6120 

4400 

1047 

1360 

1723 

2160 

2646 

3220 

3860 

4580 

5386 

6290 

4500 

1077 

1400 

1777 

2222 

2725 

3313 

3970 

4715 

5542 

6470 

4000 

1102 

1435 

1820 

2280 

2796 

3400 

4075 

4835 

5655 

6637 

4700 

1131 

1473 

1870 

2338 

2868 

3486 

4180 

4960 

5832 

6813 

4800 

1169 

1508 

1911 

2395 

2935 

3508 

4280 

5076 

5970 

6970 

4900 

1189 

1545 

1960 

2452 

3010 

3655 

4385 

5200 

6115 

7142 

3000 

1215 

1582 

20 JO 

2508 

3075 

3740 

4485 

5318 

6255 

7300 

3100 

1242 

1614 

2048 

2565 

3145 

3822 

45S5 

5440 

6394 

7470 

3400 

1268 

1648 

2092 

2616 

3210 

3905 

4680 

5550 

6525 

7622 

3300 

1296 

1683 

2134 

2671 

3280 

3985 

4775 

5670 

6666 

7781 

3400 

1320 

1717 

2178 

2725 

3343 

4063 

4870 

5784 

6784 

7936 

3500 

1347 

1750 

2220 

2779 

3408 

4143 

4965 

5893 

6936 

8090 

3000 

1373 

1783 

2264 

2830 

3175 

4222 

5060 

6010 

7061 

8250 

3700 

1398 

1815 

2303 

2881 

3534 

4300 

5155 

6115 

7184 

8400 

3800 

1422 

1848 

2348 

2941 

3606 

4385 

5250 

6238 

7333 

8563 

3900 

1446 

1880 

2385 

2986 

3660 

4453 

5340 

6336 

7444 

8696 

4000 

1473 

1912 

2427 

3038 

3725 

4530 

5430 

6444 

7580 

8847 

4100 

1497 

1944 

2468 

3086 

3785 

4610 

5520 

6550 

7700 

8988 

4400 

1520 

1975 

2507 

3137 

3850 

4680 

5610 

0655 

7830 

9141 

4300 

1545 

2008 

2546 

3186 

3910 

4750 

5700 

6761 

7950 

9285 

4400 

1568 

2037 

2585 

3238 

3970 

4825 

5790 

6865 

S072 

9432 

4500 

1593 

2070 

2624 

3286 

4025 

4900 

5875 

6970 

8195 

9572 

4000 

1614 

2100 

2664 

3333 

4087 

4935 

5960 

7070 

8320 

9710 

4700 

1639 

2130 

2702 

3382 

4145 

4970 

6045 

7172 

8437 

9850 

4800 

1663 

2160 

2740 

3431 

4202 

5112 

6130 

7275 

8555 

9990 

4900 

1686 

2190 

2779 

3478 

4260 

5193 

6215 

7375 

8673 

10120 

5000 

170S 

2220 

2817 

3525 

4321 

5253 

6300 

7475 

8792 

10250 

5450 

1760 

2293 

2909 

3640 

4414 

5426 

6507 

7723 

9081 

10601 

5500 

1822 

2365 

3000 

3755 

4608 

5600 

6715 

7972 

9370 

10953 

5750 

1876 

2436 

3090 

3868 

4744 

5767 

6917 

8204 

9652 

11269 

0000 

1930 

2507 

3180 

3981 

4880 

5935 

7120 

8436 

9935 

11586 

0450 

1982 

2574 

3261 

4094 

5013 

6096 

7313 

8519 

10203 

11902 

0500 

2035 

2642 

3352 

4207 

5146 

6258 

7505 

8603 

10472 

12218 

0750 

2088 

2710 

3438 

4320 

5281 

6419 

709S 

8986 

10741 

12534 

7000 

2141 

2778 

3524 

4434 

5417 

6580 

7892 

9370 

11010 

12851 

7450 

2191 

2842 

3606 

4531 

5542 

6733 

8076 

9587 

11265 

13152 

7500 

2241 

2907 

3688 

4629 

566S 

6886 

8260 

9805 

11521 

13453 

7750 

2290 

2971 

3770 

4276 

5794 

7039 

8445 

10022 

11776 

13754 

8000 

2340 

3036 

3852 

4824 

5920 

7192 

8628 

10240 

12032 

14056 

8450 

2488 

3098 

3931 

4923 

6042 

7340 

8806 

10451 

12280 

14345 

8500 

2636 

316L 

4011 

5023 

6164 

7488 

8984 

10662 

12528 

14634 

875 0 

2784 

3223 

4095 

5123 

6286 

7637 

9162 

10823 

12776 

14922 

9000 

2933 

3286 

4170 

5222 

6408 

7785 

9340 

11084 

13024 

15211 

945 0 

2879 

3346 

4247 

5343 

6516 

7926 

9512 

11289 

13364 

15493 

9500 

2826 

3407 

4324 

5465 

6645 

8068 

9685 

11494 

13505 

15775 

10000 

2720 

3529 

4478 

5708 

6882 

8351 

10030 

11904 

13987 

16340 




























614 


Stability of Vessels. 


To find tbe Moment of Stability of a Vessel by Experiments. 


TF= weight in tons placed on deck at a distance 
r from the centre-line and h feet above the load- 
water line, when the vessel-is in equilibrium; v = 
careen angle, d = depth in feet of the centre of 
gravity of the vessel under meta-centre, mo¬ 
ment of stability in foot-tons. 



Q = W(r cos. v-\- h sin. v ), 
- Q Sin. v — - Ct 


2 7 • 

sin. v 


Td 


Example. The weight JF= 15 tons, the centre of gravity of which is placed at r = 
12 feet from centre on deck and h = 8 feet above the water, which careens the 
vessel to an angle v = 2°. The displacement T= 4288.8 tons. Required, the mo¬ 
ment of stability Q, and depth centre of gravit}' dl 

Q = 15(12 X cos. 2° -{- 8 X sin. 2°) = 184.05 foot-tons, 

184.05 

and d = 40 ^ 8 X 0 0 3 4 9 ~ feet, the depth of the centre 
of gravity of the vessel, under meta-centre. 

Moment of Wind oil Sails Careening a Vessel in Sailing. 


Let F denote the force of wind in tons, acting at 
right angle to the vessel on the centre of gravity of 
all the sails, = l feet above the centre of gravity of the 
displacement. Then the moment of the wind will 
be— Q = FI = Td 6in. v. 

Example. The centre of gravity of all the sails 
being! = 35 feet above the centre of gravity of the 
displacement of a vessel of T— 4288.8 tons. The 
force of wind on all the sails F= 7 tons. The depth 
of the centre of gravity of the vessel, under meta¬ 
centre d = 1.223 feet, as found by experiments. Re¬ 
quired, the moment Q of the wind, and to what angle 


-the vessel will be careened? 



<2 = 7X35 = 245 foot-tons, 




245 


and, sin. v =—— = — 

’ Td 4288.8X 1-223 

= sin. 2° 4CK 40 // , the careen. 


= 0.04071 


r • 





































































Tonnage Measurement. 


615 


Tonnage of Vessels.—Old U. S. Measurement. 

T= tonnage of vessel. L = length of the vessel in feet, from the fore part of the 
stem to the after part of the stern-post, measured on the upper deck. B = greatest 
beam in feet,Pleasured above the main-walls, d — depth of the vessel in feet. For 
double-decked vessels, halt the beam B is taken as the depth d. For single-decked 
vessels, the depth is taken from the underside of deck plank to the ceiling of the hold. 

Example. L = ISO feet, B — 30 and d = 15, for a double-decked vessel. Re¬ 
quired, the tonnage? 

T = (L — 0.6 B) = — * 15 (180 — 0.6 X 30) = 7C7.4 tons. 

Custom-House New Tonnage L<aw, May 6, 1864. 

An Act to regula te the admeasurement of tonnage of ships anil vessels of the U. 3. 

Beit enacted by the Senate and House of' Representatives of the United States of 
America in Congress assembled. That every ship or vessel built within the United 
States, or that may be owned by a citizen or citizens thereof, on or after the first 
day of January, eighteen hundred and sixty-five, shall be measured and registered 
in the manner hereinafter provided; also every ship or vessel that is now owned 
by a citizen or citizens of the United States, and shall be remeasured and reregis¬ 
tered upon her arrival after said day at a port of entry in the United States, and 
prior to her departure therefrom, in the same manner as hereinafter described: 
Provided, That any ship or vessel built within the United States after the passage 
of this Act may be measured and registered in the manner herein provided. 

Sec. 2. And be it farther enacted. That the register of every vessel shall express 
her length and breadth, together with her depth, and the height under third or 
spar deck, which shall be ascertained in the following manner: The tonnage deck, 
in vessels having three or more decks to 'the hull, shall be the second deck from 
below ; in all other cases the upper deck of the hull is to be the tonnage deck. The 
length from the forepart of the outer planking, on the side of the stem, to the 
af ter part of the main sternpost of screw steamers, and to the after part of the rud¬ 
der-post of all other vessels measured on the top of the tonnage deck, shall be ac¬ 
counted the vessel’s length. The breadth of the broadest part on the outside of the 
vessel shall be accounted the vessel’s breadth of beam. A measure from the under 
side of tonnage deck plank, amidships, to the ceiling of the hold (average thick¬ 
ness) shall be accounted the depth of hold. If the vessel has a third deck, then 
the height from the top of the tonnage-deck plank to the under side of the upper- 
deck plank shall be accounted as the height under the spar deck. All measure¬ 
ments to be taken in feet and fractions of feet; and all fractions of feet shall be 
expressed in decimals. 

Sec. 3. And be it further enacted. That the register tonnage of a vessel shall be 
her entire internal cubic capacity in tons of one hundred cubic feet each, to be 
ascertained as follows: Measure the length of the vessel in a straight line along 
the upper side of the tonnage deck, from the inside of the inner plank (average 
thickness) at the side of the stem to the inside of the plank on the sterntimbers 
(average thickness), deducting from this length what is due to the rake of the bow 
in the thickness of the deck, and what is due to the rake of the stern timber in the 
thickness of the deck, and also what is due to the rake of the stern timber in one- 
third of the round of the beam ; divide the length so taken into the number of 
equal parts required by the following table according to the class in such table to 
which the vessel belongs: 

Table of Classes. 

Class I.—Vessels of which the tonnage length according to the above measure¬ 
ment is fifty feet or under, into six equal parts. 

Class 2.—Vessels of which the tonnage length according to the above measurement 
is above fifty feet, and not exceeding one hundred feet long, into eigiit equal part';. 

Class 3.—Vessels of which the tonnage length according to the above measure¬ 
ment is above one hundred feet long, and not exceeding one hundred and fifty 
feet long, into ten equal parts. 

Class 4.—Vessels of which the tonnage length according to the above measure¬ 
ment is above one hundred and fifty feet, and not exceeding two hundred feet long, 
into twelve equal parts. 

Class 5.—Vessels of which the tonnage length according to the above measure¬ 
ment is above two hundred feet, and not exceeding two hundred and fifty feet 
long, into fourteen equal parts. 














616 


Tonnage Measurement. 


Class 6.—Vessels of which the tonnage length according to the above measure¬ 
ment is above two hundred and fifty feet long, into sixteen equal parts. 

Then, the hold being sufficiently cleared to admit of the required depths and 
breadths being properly taken, find the transverse area of such vessel^at eacn point 
of division of the length as follows: 

* Measure the depth at each point of division from a point at a distance of one- 
third of the round of the beam below such deck, or, in case of a break below a line 
stretched in continuation thereof, to the upper side of the floor timber, the inside 
of the limber strake, after deducting the average thickness of the ceiling, which 
is between the bilge planks and limber strake; then, if the depth at the midship 
division of the length do not exceed sixteen feet, divide each depth into four equal 
parts: then measure the inside horizontal breadth, at each of the three points of 
division, and also at the upper and lower points of the depth, extending each meas¬ 
urement to the average thickness of that part of the ceiling which is between the 
points of measurement; number these breadths from above (numbering the upper 
breadth one. and so on down to the lowest breadth); multiply the second and 
fourth by four, and the third by two; add these products together, and to the sum 
add the first breadth and the last, or fifth; multiply the quantity thus obtained by 
one-third of the common interval between the breadths, and the product shall be 
deemed the transverse area; but if the midship depth exceed sixteen feet, divide 
each depth into six equal parts, instead of four, and measure as before directed, the 
horizontal breadths at the five points of division, also at the upper and lower points 
of the depth ; number them from above as before; multiply the second, fourth and 
sixth by four, and the third and fifth by two; add these products together, and to 
the sum add the first breadth and thu last, or seventh; multiply the quantities 
thus obtained by one-third of the common interval between the breadths, and 
the product shall be deemed the transverse area. 

Having thus ascertained the transverse area at each point of division of the ves¬ 
sel, as required above, proceed to ascertain the register tonnage of the vessel in 
the following manner: 

Number the areas successively one, two, three, etc., number one being at the 
extreme limit of the length at tine bow, and the last number at the extreme limit 
of the length at the stern ; then whether the length be divided, according to table, 
into six or sixteen parts, as in classes one and six, or any intermediate number, as 
in classes two, three, four and five, multiply the second and every even-numbered 
area by four, and the third and every odd-numbered area (except the first and 
last) by two; add these products together, and to the sum add the first and last if 
they yield anything; multiply the quantities thus obtained by one-tliird of the 
common interval between the areas, and the product will be the cubical contents 
of the space under the tonnage deck; divide this product by one hundred, and the 
quotient, being the tonnage under the tonnage deck, shall be deemed to be the 
register tonnage of the vessel, subject to the additions hereinafter mentioned. 

If there lie a break, a poop, or any other permanent closed-in space on the upper 
decks, on the spar deck available for cargo or stores, or for the berthing or accom¬ 
modation of passengers or crew, the tonnage of such space shall be ascertained as 
follows: 

Measure the internal mean length of such space in feet, and divide it into an 
even number of equal parts, of which the distance asunder shall be most nearly 
equal to those into which the length of the tonnage deck has been divided; meas¬ 
ure at tlie middle of its height the inside breadths—namely, one at each end and 
at each of the points of division, numbering them successively, one, two, three, etc.; 
then to the sum of the end breadths, add four times the sum of the even-numbered 
breadths and twice the sum of the odd-numbered breadths, except the first and 
last, and multiply the whole sum by one-third of the common interval between 
the breadths; the product will give the mean horizontal area of such space; then 
measure the mean height between the plank of the decks, and multiply by it the 
mean horizontal area; divide the product by one hundred, and the quotient shall 
be deemed to be the tonnage of such space, and shall be added to the tonnage 
under the tonnage decks, ascertained as aforesaid. 

If a vessel has a third deck, or spar deck, the tonnage of the space between it 
and the tonnage deck shall be ascertained as follows: 

Measure in feet tlio inside length of the space, at the middle of its height, from 
the plank at the side of the stem to the plank on the timbers at the stern, and 
divide the length into the same number of equal parts into which the length 
of the tonnage deck is divided; measure (also at the middle of its height) the in- 

* Chapman’s rule, p. 114. 










Tonnage Measurement. 


617 


side breadth of the space at each of the points of division, also the breadth of the 
stem and the breadth at tlie stern ; number them successively one, two, three and 
so forth, commencing at the stem; multiply the second and all other even-num¬ 
bered breadths by four, and the third and all the other odd-numbered breadths 
(except the first and last) by two; to the sum of these products add the first and 
last breadths, multiply the whole sum by one-third of the common interval be¬ 
tween the breadths, and the result will give, in superficial feet, the mean horizon¬ 
tal area ot such space; measure the mean height between the plank of the two 
decks, and multiply by it the mean horizontal area; and the product will be the 
cubical contents of the space; divide this product by one hundred, and the quo¬ 
tient shall be deemed to be the tonnage of such space, and shall be added to the 
other tonnage of the vessel, ascertained as aforesaid. And if the vessel has more 
than three decks, the tonnage of each space between decks, above the tonnage 
deck, shall be severally ascertained in manner above described and shall be added 
to the tonnage of the vessel, ascertained as aforesaid. 

In ascertaining the tonnage of open vessels the upper edge of the upper strake 
is to form the boundary line of measurement, and the depth shall be taken from an 
athwartship line, extending from edge of said strake at each division of the length. 

The register of a vessel shall express the number of decks, the tonnage under 
the tonnage deck, that of the between decks, above the tonnage deck ; also that of 
the poop or other enclosed spaces above the deck, each separately. In every 
registered United States ship or vessel the number denoting the total registered 
tonnage shall be deeply carved or otherwise permanently marked on her main 
beam, and shall be so continued ; and if at any time it cease to be so continued, such 
vessel shall no longer be recognized as a registered United States vessel. 

Sec. 4. And be it further enacted. That the charge for the measurement of ton¬ 
nage and certifying the same shall not exceed the sum of one dollar and fifty cents 
for each transverse section under the tonnage deck; and the sum of three dollars 
for measuring each between decks above the tonnage deck; and the sum of one 
dollar and fifty cents for each poop, or closed-in space available for cargo or stores, 
or for the berthing or accommodation of passengers, or olficers and crew, above the 
upper or spar deck. 

Sec. 5. And be it further enacted, That the provisions of this act shall not be 
deemed to apply to any vessel not required by law to be registered, or enrolled, or 
licensed, and all acts and parts of acts inconsistent with the provisions of this act 
are hereby repealed. 

Englisli Tonnage Measurement. 

Divide the length of the upper deck between the after part of the stem and the 
fore part of the stern-post into 6 equal parts , and note the foremost, middle and 
aftermost points of division. Measure the depths at these three points in feet and 
tenths of a foot, also the depths from the under side of the upper deck to the ceil¬ 
ing at the limber strake; or in case of a break in the upper deck, from a line 
stretched in continuation of the deck. For the breadths, divide each depth into 5 
equal parts, and measure the inside breadths at the following points, viz.: at .2 
and .8 from the upper deck of the foremost and aftermost depths, and at .4 and .8 
from the upper deck of the amidsliip depth. Take the length, at half the amidship 
depth, from the after part of the stem to the fore part of the stern-post. 

Then, to twice the amiSship depth, add the foremost and aftermost depths for 
the sum of the depths; and add together the foremost upper and lower breadths, 3 
times the upper breadth with the lower breadth at the midship, and the upper and 
twice the lower breadth at the after division for the sum of the breadths. 

Multiply together the sum of the depths, the sum of the breadths, and the length, 
and divide the product by 3500, which will give the number of tons, or register. 

If the vessel has a poop or half deck, or a break in the upper deck, measure the 
inside mean length, breadth and height of such part thereof as may be included 
within the bulkhead; multiply these three measurements together, and divide the 
product by 92.4. The quotient will be the number of tons to be added to the re¬ 
sult as above ascertained. 

For Open Vessels. —The depths are to be taken from the upper edge of the upper 
strake. 

For Steam Vessels. —The t mnage due to the engine-room is deducted from the 
total tonnage computed by the above rule. 

To determine this, measure the inside length of the engine-room from the fore¬ 
mast to the aftermost bulkhead ; then multiply this length by the midship depth 
of the vessel, and the product by the inside amidship breadth at .4 of the depth 
from the deck, and divide the final product by 92.4. 




618 


Centripetal Propeller. 


CENTRIPETAL PROPELLER 


I 


Tiie Centripetal Propeller lias, since the year 1851, fought its way through the 
usual obstructions to success, and is now approved and adopted by the most ad¬ 
vanced engineers in Europe and America. 

Fronde’s experiments have revealed to us losses in the propeller of which we 
have no general mathematical theory. Pending such discovery Nystrom’s approxi¬ 
mate formula? may be used.—W. D. M. 

The propellers constructed by John Roach for the Pacific Mail Steamship Com¬ 
pany are upon the centripetal principle, a full description of which is given in a 
work entitled “Education and Shipbuilding,” published in the year 1866, by II. C. 
Rail'd, Philadelphia. 

The helicoidal or propelling surface in the common propeller is formed by a 
straight generatrix at right angle to the axis; whilst in the centripetal propeller 
that surface is formed by a spiral generatrix constructed in an angle to, Formula 7. 
In practice this angle can be assumed to be, 




v: = 30° for the fore-edge, and 

w'= 45° for the after-edge of the propeller. 


The difference between the angles xv and w' makes the pitch expanding from the 
centre to the periphery. 

Having given the spirals a and e, the spirals b, c and d are obtained by dividing 
the angles into four equal parts, as will be understood by the illustration. 

A straight generatrix inclined to the axis wall give the same helicoidal surface 
as that of the curved generatrix at right angles to the axis; but the inclination 
of the straight generatrix must be according to Formula 8. 

The dotted lines f g h i represent a centripetal propeller with straight inclined 
generatrix. Propellers constructed either as the dotted or drawn lines, or between 
the two cases, will produce the same propelling effect in the wafer. When the 
propeller is constructed between the two cases represented on the drawing, the 
blades will appear curved in both views. 

The length L of the propeller should he from 0 2D to 0 25X), and the pitch from 
15 D to 21). For very sharp vessels constructed lor speed, and when the draft of 
water is over one-half the beam, the pitch may be made 2'5D. 

One quarter of the pitch is set off on the centre line from 0 to 8, and the helix 
constructed in the ordinary way. The outer edge of the blades should not follow 
the true helix, hut he made slightly concave, as shown in the drawing, which 
makes the pitch expanding in the direction of the axis. 

The mean pitch of the propeller should he calculated by Formula 3, making 
r = 0-7 A’. 

Example 1. The diameter of a propeller is 10 feet 6 inches, and the angle 
W= 58° at the periphery. Required the pitch P— in feet? 

P = cot. 58° X 3'14 X 10-5 = 20'6 feet. 


Example 2. The propeller on riate XI. is of dimensions I) =15 feet, L — 5 
feet, W= 57° 30', the slip is 3S per cent, or S=0'38. W hat power is required to 
drive it 40 revolutions per minute, II—1 


153X403 / \ 

H= - - (5X0'38Xcos.57°30'-f011 J =514.8 IP, nearly. 

Example 3. A propeller of diameter D = 12 feet, angle TF=64°, and length 
L — 3 feet 6 inches, is to he driven by a steam engine of 450 IP, the slip S — 0 - 28. 
How many revolutions will it make per minute, n = ? 


78 450 

n = ---» /-- = 61 revolutions 

12 \ (3-5X0-28Xcos.64°+0-ll) 


per minute, 



















Plate XL 




















































I 








Formulas for Propellers 


619 










Pitch. 

P—ttDcoI.W 1 
360 L 




Areas. 


A = 


D 2 L m 

r ’ 

JD m, 


- - 9 

- - 10 


^( l +4 


o = 


2-5 D 2 


V w I) 2 -f P 2 


- - - 12 


Horsepower ami Revolutions. 


71= 


P 2 n 3 
480000 


(zScos. TF+0’11^, 13 n == 


H 


LScos. W-\- 0-11 


, 14 


Horsepower of Friction. 


JR, L km n 3 


59,400,000 P 


-(311*7 P 4 + 26-42 P 2 P 2 + - - -15 


D = diameter, R — radius, L — length, and P — pitch of the propeller in feet. 
W = angle of the blades to the centre line. 
v = projecting angle of each blade. 
w = centripetal angle for the curved generatrix. 

< f> — angle of inclination of the straight generatrix, 
a = projecting area of all the blades. 

A = helicoidal surface of the propelling side of all the blades. 
a — helicoidal surface of one whole convolution. 

O = acting area at right angles to the axis. All areas in square feet. 
x — length of any helix at radius r, and in — number of blades. 

X = length of external helix of the blade. 
n = number of revolutions per minute. 

II = horsepower required to drive the propeller. 
h — horsepower required for friction in the water. 
k — friction coefficient. See page 448. 


The pitch, of the propeller is equal to the tabular number opposite the 
given angle IF, multiplied by the diameter. 


w 

Pitch. 

W 

Pitch. 

W 

Pitch. 

W 

Pitch. 

W 

Pitch. 

W 

Pitch, 

so 

5-45 

40 

3-74 

50 

2-63 

60 

1-81 

70 

1*14 

80 

0-55 

31 

5*23 

41 

3-62 

51 

2-54 

61 

1-74 

71 

Ml 

81 

0-50 

32 

5*03 

42 

3-50 

52 

2-45 

62 

1-67 

72 

1-02 

82 

0-44 

33 

4-85 

43 

3*27 

53 

2-37 

63 

1-60 

73 

0-96 

83 

0*37 

34 

4-66 

44 

3-20 

54 

2-28 

64 

1-53 

74 

0-90 

84 

0-33 

35 

4-50 

45 

3-14 

55 

2-20 

65 

1-46 

75 

0-84 

85 

0-27 

36 

4*33 

46 

3*09 

56 

2-12 

66 

1-40 

76 

0-78 

86 

0-22 

37 

4-17 

47 

2-93 

57 

2-04 

67 

1-33 

77 

0-72 

87 

0-16 

38 

4-02 

48 

2-83 

58 

1-96 

68 

1-27 

78 

0-67 

88 

0-11 

39 

3-88 

49 

2-73 

59 

1-89 

69 

1-20 

79 

0-61 

89 

0-06 

















































620 


Counting Beats of Seconds. 


Approximate Comparative Values of Metals per Pound 

Avoirdupois. 

Prices of Metals. Prices of Mf.tals. 


Value per Pound 

Metal. Condition. Avoi V dupots . 

Vanadium....Crvst. fused.$4,792.40 

Rubidium.Wire. 3,261.60 

Calcium.Electrolytic. 2,446.20 

Tantalum.Pure.....*.. 2,446.20 

Cerium.Fused globule... 2,446.20 

Lithium.Globules. 2,228.76 

Lithium.Wire. 2,935.44 

Erbium.Fused. 1,671.57 

Strontium.Electrolytic. 1,576.44 

Indium.Pure. 1,522.08 

Ru thiu in. 1,304.64 

Columbiuin.. .Fused. 1,250.28 

Rhodium. 1,032.84 

Barium.Electrolytic. 924.12 

Thallium.*.. 738.39 

Osmium. 652.32 

Palladium. 498.30 

Iridium... 466.59 

Uranium. 434.88 

Gold. 299.72 

Titanium.Fused. 239.80 

Tellurium. “ . 196.20 


Metal. 


Condition. 


Value per Pound 
Avoirdupois. 

Chromium.Fused.$196.20 

Platinum. “ 122.31 

Manganese. “ 108.72 

Molybdenum. 54.34 

Magnesium.Wire and tape. 45.30 

Potassium.Globules. 22.65 

Silver. 18.60 

Aluminium.Bar. 16.30 

Cobalt.Cubes. 12358 

Nickel. “ 3.80 

Cadmium. 3.26 

Sodium. 3.26 


Bismuth.Crude. 

Mercury. 

Antimony. 

Tin..*.. 

Copper r . 

Arsenic. 

Zinc. 

Lead. 

Iron. 


1.95 

1.00 

36 

25 

22 

15 

10 

6 

H 


COUNTING SECONDS. 

When the occurrence of a distant sound is not anticipated, we are unpre¬ 
pared to record the exact moment, and before an appropriate time-keeper 
can be procured an uncertain time has elapsed. 

With some practice seconds can be counted in the mind with tolerable cor¬ 
rectness without the aid of a time-keeper; which practice has been of great 
service to the author in astronomical observations—practice counting seconds 
by the aid of an oscillating second pendulum or by the second-hand on a i 
watch until the counting agree with the time-keeper, without attention to 
the pendulum or second-hand. With good practice the counting should not 
differ more than one second per minute. 

When an unexpected distant sound is hoard and its cause observed, we can 
always he ready to count seconds and thus determine the distance. 

In "astronomical observations at sea it is customary to keep a watch in the 
hand or to station an assistant at the chronometer to note the time when the 
observer says “stop;” but there are known cases when the captain has taken 
his observations without the aid of a watch or assistant, and walked slowly 
and comfortably to his cabin and noted the time of his observations from the 
chronometer, with no little amusement to other observers, who naturally 
supposed that, the captain’s observations could not be very correct, but to 
their surprise were found to be as correct as their observations with ordi¬ 
nary precautions. The captain counted in his mind the seconds, and de¬ 
ducted the sum from the time observed on the chronometer. 

The practice of counting seconds correctly is of great utility and service 
for estimating various movements. When the action is of very short dura¬ 
tion, say less than 3 seconds, it is best to count half seconds or even four 
times per second, and a short time may be determined with a correctness 
within a quarter of a second. 

















































































Sound. 


621 


SOUND. . 

Velocity of Sound through. Air. 

v = velocity in feet, per second. 
t = temperature of the air, I'ahr. scale. 

D = distance in feet the souud travels in the tirao T. 


v= 1089.42/1 + 0.00208(t — 32). 

Velocity of sound in water is about 4 times that in air, and 8 times that through 
solids. 

Intensity of sound is inversely as the square of the distance. 


• D = 1089.422+1 + 0.00208(4 — 82), 

T=~. 

v 

Example. A ship at sea was seen to fire a cannon, and 6.5 seconds afterward the 
report was heard; the temperature in the air was 60°. Required, the distance to 
the ship. 


D = 1089.42 X 6.5/1+ .00208(60°—32) = 7284 feet, or 1.38 miles. 


Descriptions of Sound. 

A powerful human voice in the open air, no wind, 

Report of a musket,. 

Drum,.. 

Music, strong brass band,. 

Cannonading, very strong,. 

In a barely observable breeze a strong human voice 
with the wind can be heard. 


Audible at a distance of 


Feet. 

Miles. 

460 

0.087 

16,000 

8.02 

10,500 

2 

15,840 

3 

575,000 

90 

15,840 

3 


Distance in Feet which Sound Travels in Air at Different 

Temperatures. 


Time 


Temperature of the Air, Fahrenheit Scale. 


ritfC. 

0° 

10° 

20° 

32° 

0 

O 

50° 

60° 

70° 

o 

© 

<X) 

90° 

100° 

1 

1000 

1064.2 

1075.7 

1089.4 

1098.5 

1109 

1120 

1131 

1142 

1153 

1164 

2 

1985 

2128 

2151 

2179 

2197 

2219 

2241 

2262 

2285 

2306 

2328 

3 

2978 

3193 

3227 

3268 

3295 

3328 

3361 

3393 

3427 

3459 

3492 

4 

3971 

4257 

4303 

4358 

4394 

4438 

4482 

4524 

4570 

4613 

4656 

5 

4964 

5321 

5378 

5447 

5492 

5548 

5603 

5655 

5712 

5766 

5821 

6 

5956 

6385 

6454 

6536 

6591 

6657 

6723 

6786 

6855 

6919 

6984 

7 

6949 

7449 

7530 

7626 

7689 

7767 

7844 

7917 

7997 

8072 

8148 

8 

7962 

8514 

8606 

8715 

8788 

8876 

8964 

9049 

9140 

9225 

9312 

9 

8934 

9578 

9681 

9805 

9886 

9986 

10085 

10180 

10282 

10379 

10476 

10 

9927 

10642 

10757 

10894 

10985 

11096 

11306 

11311 

11425 

11532 

11640 

11 

10920 

11706 

11833 

11983 

12083 

12205 

12326 

12442 

12567 

12685 

12804 

12 

11912 

12770 

12908 

13073 

13182 

13315 

13447 

13573 

13710 

13838 

13968 

13 

12905 

13835 

13984 

14162 

14280 

14424 

14567 

14704 

14852 

14991 

15132 

14 

13898 

14899 

15060 

15252 

15379 

15534 

15688 

15835 

15995 

16145 

16296 

15 

14891 

15963 

16135 

16341 

16477 

16644 

16809 

16966 

17137 

17298 

17460 

10 

15883 

17027 

17211 

17430 

17576 

17753 

17929 

18097 

18280 

18451 

18624 

17 

16S76 

17091 

18287 

18520 

18674 

18863 

19050 

19228 

19422 

19604 

19788 

18 

17889 

19156 

19363 

19609 

19773 

19972 

20170 

20360 

20565 

20757 

20952 

19 

18861 

20220 

20438 

20699 

20871 

21082 

21291 

21491 

21707 

21911 

22116 

20 

19854 

21284 

21514 

21788 

21970 

22192 

22412 

22622 

22850 

23064 

23280 



















































C22 


Musical. 


MUSICAL VIBRATIONS. 


Musical vibration is the most accurate measurement of small intervals 
of time. 

C = first term, or vibrations of the fundamental note. 

c — the last term, or vibrations of the octave above the fundamental note. 
2 C = c. 

« = number of double vibrations of any note in the musical scale whose 
number of terms, from C inclusive, is a. 

a= number of terms between C and n inclusive. 

r = ratio of vibrations between each note or term. 

Each term multiplied by tlie ratio r gives the next following term, when 
the progression is increasing. 


a-l 


Ratio r 




a-l 

Vibrations n = Cr. 


In the application of these formulas to the division of the octave into the 
chromatic scale of thirteen notes, we can assume any arbitrary number of 
vibrations of the fundamental note, say (7 = 32, and the octave will then 
vibrate c = 64. Making a = 13, we find the 


a-l 


13-1 


12 


Log. r — 


Ratior =•>/-£—V-§- = V 2 

log. 2 0.30102999566 


12 


12 


0.02508583 = log. 1.059462 the ratio. 


The proportionate vibration of any note whose number, from C inclusive, 
is a, will be 

n = Cr = 32 X 1.059462. 

The number a includes also the half-notes or sharps. 

The harmonic intonation of the diatonic scale is established as follows: 

CDEFGABc 


'§ 


¥ 


2 

Harmonic, 32 36 ’ 40 42 f 48 53*’ 60~ 64 

Tempered, 32 35.91S8 40.3175 42.7149 47.9458 53.8174 60.408 64 

Difference, 0.000 —0.0812 -[-0.3175 +0.0483 —0.0542 +0.4841 +0.408 0.000 

The actual number of double vibrations, per second, of the standard con¬ 
cert pilch now generally used was established by a Congress of Philosophers, 
which met in Stuttgart in the year 1834; namely, G v = 264, +' = 440, and 
C' = 52S. With this data the following table is calculated. 

The last column, “ Prop, length of waves,” shows the proportionate 
diameters of bells for the corresponding note, when the souml-bow is of a 
certain proportion to the diameter in all the bells. This column also shows 
how to divide the bridge on a guitar , when the whole length of the string 
being 1 or the unit. 

Ringing Bells. 

/) = diameter of the bell in inches. 

US = thickness of the sound-bow in inches. 

n = double vibration per second, corresponding to the pitch of tone. 

W — weight of the bell in pounds avoirdupois. 

— , „ 240000 lc , S 

Diameter, D = 


n 


k = 


Weight, IP = 0.3 D*S to 0.35 IT-S. 


D 










Harmonic and Tempered Intonations. 


623 


Stuttgart Harmonic Scale. 


a 

Vi 

0; 

H 


Keynote. 


13 

12 

11 

10 T 
9 
8 


7 

6 

5 

4 

3 

2 

1 


* 




C 

B 

A 

G# 

G 

n 

r 

E 


Double 

Vibra¬ 

tions. 

n r 


528 


33. 


495 

< 

467.5 

i 

440 

396 

374 


22 

22 

22 




A D » 


352 

330 


22 

22 


16.5 


813.5 


16.5 


D i 297 


16.5 


^ C# 280J5 


* C 


264 


16.5 


16.5 


Diffe 

Ratio 
of Pitch. 
n 

n' 

rence. 

In 

Vibration. 
n — n' 

Tempered C 
Seal 

Double 

Vibrations. 

n 

Geometric 

e. 

Prop. 
Length 
of waves 
264 

n 

1.00000 

0.000 

528 

29.635 

0.50000 

1.00680 

+ 3.365 

498.365 

27.970 

0.52973 

1.00616 

,1 

+ 2.895 

470.395 

26.402 

0.56123 

1.00908 

+ 3.993 

443.993 

24.919 

0.59461 

1.00257 

+ 1.074 

419.074 

23.519 

0.62996 

0.99887 

-0.447 

395.553 

22.201 

0.66742 

0.99827 

-0.648 

373.352 

20.954 

0.70711 

1.00113 

+ 0.398 

352.398 

19.787 

0.74915 

1.00794 

+ 2.611 

332.611 

18.660 

0.79370 

1.00144 

+ 0.451 

313.951 

17.621 

0.84090 

0.99774 

-0.670 

296.330 

16.638 

0.89090 

0.99714 

■ 

-0.808 

279.692 

15.692 

0.94388 

1.00000 

0.000 

264 

14.817 

1.00000 

























































G24 


Tempered Intonation of Musical Vibrations. 








































































Tempered Intonation of Musical Vibrations. 625 



40 
































































































626 Simple Elements witli Old ami New Equivalents (Chemistry), 


Name oe Elements. ‘'ifjj 1 * 

Aluminium. Al. 

Antimony. Sb. 

Arsenicum. As. 

Barium. Ba. 

Bismuth. Bi. 

Boron. Bo. 

Bromine. Br. 

Cadmium. Cd. 

Caesium . Cs. 

Calcium... Ca. 

Carbon. C. 

Cerium . Ce. 

Chlorine. Cl. 

Chromium. Cr. 

Cobalt. Co. 

Copper (Cuprum). Cu. 

Didymium. D. 

Erbium. E. 

Fluorine. F. 

Gallium. Ga. 

Glucinum. Gl. 

Gold (Aurum). Au. 

Hydrogen. H. 

Indium. In. 

Iodine. I. 

Iridium. Ir. 

Iron (Ferrum). Fe. 

Lanthanum. La. 

Lead (Plumbum). Pb. 

Lithium. Li. 

Magnesium. Mg. 

Manganese. Mil. 

Mercury. Hg. 

Molybdenum. Mo. 

Nickel. Ni. 

Niobium . Nb. 

Nitrogen. N. 

Osmium. Os. 

Oxygen.. O. 

Palladium . Pd. 

Phosphorus. P. 

Platinum. Pt. 

Potassium (Kalium). K. 

Rhodium. Rh. 

Rubidium. Rb. 

Ruthenium. liu. 

Selenium. Se. 

Silicon. Si. 

Silver (Argentum).... Ag. 
Sodium (Natrium)... Na. 

Strontium . Sr. 

Sulphur... S. 

Tantalum. Ta. 

Tellurium ... Te. 

Thallium. Tl. 

Thorium. Th. 

Tin (Stannum). Sn. 

Titanium. Ti. 

Tungsten (Wolfram). W. 

Uranium. U. 

Vanadium. . V. 

Yttrium. Y. 

Zinc. Zn. 

Zirconium. Zr. 


Old 

eqvlt. 

New 

eqvlt. 

8p. gr. 

13.7 

27.5 

2.50 

129.0 

122. 

6.70 

75. 

75. 

5.80 

68.5 

137.2 

4.70 

210.30 

210. 

9.80 

10.9 

11. 

2.00 

80. 

80. 

3.187 

56. 

111.6 

8.60 

132.4 

132.15 


20. 

39.9 

1.57 

6. 

12. 

3.52 

46. 

141.3 

5.5 

35.5 

35.5 

2.44 

26.3 

52.4 

6.8 

29.5 

58.6 

8.9 

31.7 

63.3 

8.9 

48. 

147? 


. 

170.6 


19. 

19.1 

1.31 


69.9 

5.956 

4.7 

9.25 

2.1 

196.44 

196.2 

19.34 

1 . 

1 . 

0.0692 

74. 

113.4 

7.2 

127. 

127. 

4.94 

98.6 

196.7 

22.40 

28. 

55.9 

7.8 

46. 

139. 


103.6 

206.4 

11.44 

7. 

7.022 

0.593 

12.16 

24. 

1.7 

27.40 

54.8 

8. 

100. 

200. 

13.59 

48. 

95.8 

8.6 

29.5 

58.6 

8.8 

48.8 

94. 


14. 

14.044 

0.971 

99.4 

198.6 

22.48 

8. 

16. 

1.1087 

53.2 

106.2 

11.5 

31. 

31. 

1.83 

98.6 

196.7 

21.5 

39. 

39.137 

0.855 

52.2 

104.2 

11. 

85.36 

85.2 

1.52 

52.11 

103.5 

8.6 

39.7 

78. 

4.8 

14. 

28. 

2.49 

108. 

108. 

10.5 

23. 

23.043 

0.972 

43.8 

87.2 

2.54 

16. 

32. 

2. 

68.8 

182. 

10.7 

64.5 

128. 

6.6 

204. 

203.6 

11.8 

59.5 

233.9 

7.7 

59. 

117.8 

7.3 

25. 

48. 

5.28 

92. 

184. 

19.1.3 

60. 

240. 

10.15 

68.5 

51.2 

5.5 

82.2 

89.6 


32.6 

64.9 

7. 

44.8 

1 90. 

4.15 


Remarks on the Elements. 

Light metal. Like zinc. 

White metal used in types. 
Metal, steel-gray lustre. 

White metal, fuses at red heat. 
Hard brittle reddish metal. 
Combination with potassium. 
Deep red volatile liquid. 

Very soft and ductile metal. 
Two strong blue lines in speetr. 
Light yellow malleable metal. 
Diamond. Graphite. Coal. 
Little known and less used. 

Gas, greenish-yellow color. 
Dark-gray metal, strong affinity 
Reddish-grav, magnetic metal. 
Yellowish-red ductile metal. 
Little known and less used. 
Classed as a metal. 

Found in fluorspar. 
Silver-white metal. 

Its salt has a sweet taste. 
Standard of value. 

Lightest of gases. 

Dark-blue lines in spectrum. 
Metallic bluish solid. 

Hard white metal. 

The most useful metal. 

Little known and less used. 

Soft and malleable metal. 

White metal, burns brilliantly. 
Burns brilliantly. 

Grayish-white metal. 

White liquid metal. 

White brittle metal. 

White, hard, ductile metal. 

Not generally known. 

Gas without color or taste. 
White and brittle metal. 

Gas, supports life and combus’n. 
Hard ductile white metal. 
Translucent solid easily ignited. 
Heaviest of all metals." 

Brittle metal, melts at 130°. 
White, hard metal. 

Metal little known. 

Most infusible of metals. 

A semi-metallic solid. 

Flint, quartz, glass, and clay. 
Metal of standard value. 
Bluish-white and soft metal. 
White metal like barium. 
Brimstone, widely used. 

Little used. 

Lustre of metal like sulphur. 
Green line in spectrum. 

Not used in the arts. 

White and malleable metal. 

Its oxide used for painting. 

An iron-gray metal. 

A steel-white metal. 

A metal little used. 

Found in Sweden in 1843. 

A bluish-white metal. 

In nature as silicate. 
























































































Chemical Compounds, with New Equivalents 


627 


Solids and Sails. 

Formulas. 

Commercial Names and Use. 

Aluminium sulphate. 

A1 2 (S0 4 ) 3 . 

Forpreparing salts of aluminium. 

Ammonium chloride. 

NHiCi. 

Sal ammoniac, for soldering. 

Arsenious acid. 

ASoOo. 

White arsenic, poisonous. 

Barium oxide.. 

BaO. 

Baryta, a gray powder. 

Barium sulphate... 

BaS0 4 . 

Heavy Spar. Fr. adult, wt. lead. 

Calcium oxide. 

CaO. 

Quick or caustic lime. 

Camphor. 


Used for making celluloid. 

Carbolic acid. 

G«H e a 

Used as a disinfectant. 

Carbonate of lime. 

CaO,C0 2 . 

Common limestone, marble. 

Chloride of lime. 

CaCl 2 0 2 . 

Bleaching powder. 

Chloride of sodium. 

CINa. 

Common salt. 

Copper sulphate. 

C 11 SO 4 . 

Blue stone or vitriol. 

Copper pvrites. 

Cu 9 S.Fe 2 S 2 . 

Pyramidal and tetrahedral crvs- 

Cuprous oxide.. 

Cu 2 0. 

Red oxide of copper. [tals. 

Gold chloride. 

AuC1 3 . 

Used in photography. 

Gold mercury. 

Au 2 Hg. 

Gold amalgam. 

Gun-cotton. 

C e H 7 (N0 2 ) 3 0 5 . 

Used as an explosive. 

Hydrogen sodium carb’te. 

HNaCO,. 

Baking powder, artificial yeast. 

Hydrogen potass, carb’te.. 

hkco 3 . 

Yeast for raising bread. 

Iron, ferric oxide. 

Fe.,0,. 

Red hematite, iron ore. 

Iron, ferric hydrate. 

Fe 2 H 6 0 6 . 

Yellow ochre, iron ore. 

Iron, magnetic oxide. 

Fe 3 0 4 . 

Loadstone, iron ore. 

Iron, bisulphide. 

FeS 2 . 

Pyrites, cube crystals. 

Iron, ferrous sulphate. 

FeS0 4 + 7H 2 0. 

Green vitriol, copperas. 

Indigo blue. 

c 8 h 5 no. 

Used in dyeing. 

Lead chromate. 

PbO,Cr0 3 . 

Chrome-yellow. 

Lead protoxide. 

PbO. 

Litharge, drver for oils. 

Lead chloride and oxide... 

(PbCl 2 .7PbO). 

Pigment, Turner’s yellow. 

Lead carbonate. 

PbO,C0 2 . 

White lead, paint. 

Lead sequi-oxide. 

Pb 3 0 4 . 

Minium, red lead. 

Lead sulphide. 

PbS. 

Galena, lead ore. 

Lapis lazuli. 

2AlP0 4 .MgII n 0 2 . 

Blue precious stone. 

Malachite. 

CuC0 v CuH 2 0 2 . 

Green precious stone. 

Manganese binoxide. 

Mu0 2 . 

For making chlorine and oxygen. 

Mercury chloride. 

HgCl 3 . 

Corrosive sublimate. 

Mercury sulphide. 

HgS. 

Cinnabar, ore of mercury. 

Oxalic acid. 

c 2 h 2 o 4 . 

A powerful poison. 

Paraffin. 

^27^54" 

For making candles. 

Potassium carbonate. 

Ko€0 3 . 

Used for making glass. 

Potassium chlorate. 

kcio 3 . 

For makingoxvgen in medicine. 

Potassium chromate. 

K 2 Cr0 4 . 

Used for bleaching. Calico print- 

Potassium cyanide. 

KCN. 

Used in photography. [ing. 

Potassium hyd. tartrate... 

HKC 4 II 4 0« 

Cream of tartar. 

Potassium nitrate. 

kno 3 . 

Saltpetre, prismatic crystals. 

Saccharose. 

I' 12 H 22 Gll* 

Cane-sugar, gum-arabic. 

Silver chloride. 

AgCl. 

Horn-silver, in photography. 

Silver nitrate . 

AgNOo. 

Lunar caustic. 

Silver cyanide. 

AgCN. 

Used in electro-plating. 

Sodium borate. 

Na 9 B 4 O 7 .10H 9 O. 

Borax, used as a flux. 

Sodium nitrate. 

NaN0 3 . 

Soda saltpetre, cubic crystals. 

Sodium carbonate. 

Na 2 C0 3 . 

Soda, used for making soap. 

Sodium oxide. 

NaO. 

Soda, oxide of natrium. 

Stannous chloride. 

S 11 CI 0 . 

Tin-salt, used in dyeing. 

Stannic oxide. 

SnOo. 

Tin-stone, cassiterite. 

Starch. 

CaHiot)** 

Used in washing. 

Stearic acid. 

C] H^36^2* 

Solid fat, candles. 

Strychnine. 

G 2 i h 22 n 2 o 2 . 

Strong poison. 

Sulphate of soda. 

NaO,SO, + 1UII 2 0. 

Glauber salt, colorless prisms. 

Sulphate of lime. 

Ca,S0 4 + 2H 2 0. 

Alabaster, gypsum, plaster Paris. 

Tannic acid. 


For tanning leather. 

Zinc chloride.. 

r /j I) 0 ) 9 • 

For preserving timber. 

Zinc sulphate. 

ZnS0 4 . 

White vitriol, used in medicine. 

Equal proportions of different atoms may be formed into different orders 

and make different substances, as cane-sugar and gum-arabic. 



































































628 


Chemical Compounds, with New Equivalents 




Liquids. 

Water. 

Alcohol. Ethyl. 

Methyl alcohol.. 

Ether. 

Chloroform. 

Glycerine. 

Nitro-glycerine. 

Oil of turpentine. 

Benzol. 

Nitro-benzol. 

Aniline. 

Carbon bisulphide. 

Nitric acid.. 

Sulphuric acid. 

Hydrochloric acid. 

Nitro-niuriatic acid. 

Citric acid. 

Oxalic acid. 

Quinic acid.. 

Quinine. 

Gases. 

Atmospheric air. 

Nitrous oxide. 

Nitric oxide. 

Carbonic acid. 

Carbonic oxide.. 

Carburetted hydrogen, 

Olefiant gas. 

Cyanogen. 

Ammonia. 

Cyanhvdric acid.. 

Hydrogen sulphide. 

Sulphurous anhydride 


Formulas. 

II oO. 
ColTgO.. 
CH 4 0. 
(C.,H 5 'oO. 
CHCl.v 
(C 3 H 5 )H 3 0 3 . 

c 3 h,n 3 o 9 . 

G10II 16 * 

<’ 0 h 9 . 

C 6 H 5 (N0 2 ). 

c«h 7 n. 

CSo. 

hno 3 . 

II.>S0 4 . 

HC1. 

HN0 3 + 2HC1. 
C 2 IIo0 4 2 ( H20. 

c 2 ok 2 iCo,. 


4N + 0. 


N 0 0. 

NO. 


C0 2 . 
CO 2 . 
C1I 4 . 
ColI 4 . 
NC. 
NH V ; 
HCN. 
1L>S. 

so 2 . 


Commercial Names or Use. 

The most abundant liquid. 

Spirit of wine, iutoxicatiug. 
Wood spirit. 

Used as a solvent, anaesthetic. 
Used as an anaesthetic. 

Much used in the arts. 

The most powerful explosive. 
Spirit of turpentine. 

Constituent of coal-tar. 

Forms the main portion of ani- 
For aniline colors. [line. 

A solvent, for India-ruhber 
Aqua-fortis, oxidizing agent. 

Oil of vitriol, much used. 
Muriatic acid. 

Aqua-regia, dissolves gold. 

Juice of lemons. 

A powerful poison. 

From Peruvian bark. 

For chills and fever. 




Not chemically combined. 
Laughing-gas. 

Extinguishes fire. 

Perfectly consumed coal. 
Suffocating, poisonous. 
Marsh-gas, fire-damp. 
Illuminating gas. 

Produces blue color. 

Hartshorn, volatile alkali. 
Prussic acid, poisonous. 

Used as a reagent in laboratory. 
Used for bleaching straw. 


Proportion of Compounds by Weight or Volume. 


Carbon. 

C. 

Hydrogen. 

11. 

Oxygen. 

6. 

Nitrogen. 

N. 

772 

133 

95 


780 

118 

102 


740 

103 

157 


760 

113 

127 


527 

129 

344 


432 

G8 

500 




280 

770 



210 

790 


1 

8 



2 

1 


853 

147 




Names. 


Olive oil, by weight. 

Spermaceti oil, “ . 

Castor oil, “ . 

Linseed oil, “ . 

Alcohol, “ . 

Sugar, “ . 

Atmosp. air. 

“ air hv volume ., 
Water, fresh, by weight.. 

“ “ “ volume. 

India-rubber by weight .. 


To Transform Atomic Formulas 


into Weights. 


Lu!e. Multiply together the equivalent (equiv.) and the exponent (exp.) 
of each substance, and the product is the proportion in the compound by 
weight. Divide each weight by its specific gravity, gives the proportions by 
bulk or volume. 

Example 1. The chemical formula for common alcohol is C 2 II 6 0. Required 
its proportioned parts by weight in 1000? 

Equiv. Kxp 


Carbon C 2 = 12 X 2 = 24 J 
Hydrogen II 8 * 1 X 6 = 6 > X 21*74- 
Oxygen O — 16 X 1 = 16 j 


1000 : 46 = 2P71 


52P76') 

180-44 >by weight. 
847*84 j 

1000 04 














































































Nitro-Glycerine. 


629 


NITROGLYCERINE, CJLNO, 

Nitro-glycerine is an oily liquid of the above composition, which is highly ex¬ 
plosive under peculiar circumstances, but can be set fire to and burned like alcohol 
without explosion. It explodes by concussion or pressure of about 2000 pounds to 
the square inch, or by the corresponding temperature of about 600° Fahr. suddenly 
applied. 

Small portions of nitro-glycerine, say half an ounce each, placed (any number) 
within a few feet of one another, if one of them is exploded, all the rest will explode 
instantaneously. Therefore, when a charge is to be exploded, care must bo 
taken that no more of it is in the neighborhood. 

It may appear strange that nitro-glyceriue can be so dangerous to handle, when 
it requires the enormous pressure of 2000 pounds to the square inch to explode it j 
but the fluid may be squeezed between surfaces of only one 10,000tli part of one 
square inch, when the pressure need be only 3 ounces to explode it. 

The charge of nitre-glycerine in blasting is exploded by a percussion cap placed 
on the end of a fuse and dipped into the liquid. The fuse explodes the fulminate 
in the cap, the concussion of which explodes the charge. On account of the action 
of uitro-glycerine being instantaneous, no tamping is required in the blast-hole, ex¬ 
cept water or loose sand, but even that is not necessary. This explosive is there¬ 
fore entirely unfit for use in firearms, which would be blown to pieces without dis¬ 
charge through the muzzle. 

The many and very serious accidents which have happened by unexpected ex¬ 
plosions of nitro-glycerine have caused it to be forbidden transportation on 
railroads and steamboats, for which a new form of the explosive has been invented, 
which consists in mixing sawdust and some other solid substances with nitro-gly¬ 
cerine, to the form of a moist brown powder, of nearly the same specific gravity 
as that of water. 

Dynamite. 

This powder is called dynamite, and is now manufactured by the nitro-glycerine 
inventor, Alfred Nobel, in Hamburg, and also by the Giant Powder Company, in 
San Francisco, California. 

The strength and instantaneous action of dynamite are precisely the same as 
those of nitro-glycerine, but it is much safer to handle, it is said—more so than 
common gunpowder. The dynamite powder is made up into cartridges of different 
sizes to suit the blast-hole, and is exploded by percussion caps like nitro-glycerine, 
and requires no tamping. It has been employed with great success in blasting im¬ 
mense masses of rock in the Andes, Peru. 

The price of dynamite is higher than that of gunpowder per weight, but its ex¬ 
ecution per price is much greater. The blast-holes for dynamite need be only one- 
half theeize of those for gunpowder, with equal execution. 

The instantaneous action of dynamite makes it far superior to gunpowder in 
blasting, but it is unfit for use in firearms. 

Any number of cartridges of dynamite placed in a deep blast-hole with tamp- 
ings of sand between, if one of them is exploded, all the rest will explode simulta¬ 
neously. Small cartridges are made for the percussion cap, and called primers, by 
which the principal charge is exploded. Should a charge fail to explode, put in 
a new fuse and primer. 

Blasting mnler Water. 

For this purpose the cartridges should be made of strong oiled paper and per¬ 
fectly water-tight, to save the dynamite from moisture. The cartridges should also 
be ballagted, so as to sink easy in water, which can be done by placing a lead ball 
in the bottom and pack the dynamite on the top, after which the cartridge is her¬ 
metically sealed with some varnish insoluble in water. The cartridges (any num¬ 
ber) are guided into the blast-hole through a tube, and finally the primer with the 
fuse, by which the whole charge is exploded. 

Dynamite is insoluble in water, but will not explode if moist with water. It 
freezes to a snowy mass at 40° Falir., but its explosive quality if not impaired 
thereby. At 212° the nitrogen evaporates and spoils the powder. 











630 


Cement, Concrete and Mortar. 


CEMENT, CONCRETE AND MORTAR. 

Roman Cement. Parker's analysis. 

One part of common clay to 2j parts of chalk, set very quick. 

Concrete. Eight parts of pebble or pieces of brick about the size of an egg, 
to 4 parts of scrap river-sand, and 1 part of good lime, mixed with water and 
grouted in, makes a good concrete. 

Lime Mortar. One part of river-sand to two parts of powdered lime, mixed 
with fresh water. 

Hydraulic Mortar. One part of pounded brick powder to two parts of pow¬ 
dered lime mixed with fresh water. This mortar must be laid very thick between 


the bricks, and the latter well soaked in water before laid. 

No. 1. Hydraulic Concrete, by Treussart. 

30 parts of hydraulic lime, measured in bulk before slacked. 

30 “ sand. 

20 “ gravel. 

40 “ broken stone, a hard limestone. 

This concrete diminishes one-fifth in volume after manipulation. 
The mortar is made first, and then mixed with gravel aud stone. 
No. 2. Another Concrete, by Treussart. 

33 volumes hydraulic lime unslacked. 

45 *• Puzzolauo (Pozzulano). 

22 “ sand. 

60 “ broken stone and gravel. 

Asphnlte Composition for street pavement, by Colonel Emy. 
2£ pints (wine measure) of pure mineral pitch. 

11 lbs. of Gaugeac bitumen. 

17 pints of powdered stone-dust, wood-ashes or minion. 


o 

Z, A 

. d o* 


n 

Zj 


> JS 
: 


Cements for Cast Iron. 


Two ounces sal-ammoniac, one ounce sulphur aud sixteen ounces of borings or 
filings of cast iron, to be mixed well in a mortar and kept dry. When required 
for use, take one part of this powder to twenty parts of clear iron borings or fil¬ 
ings, mixed thoroughly in a mortar', make the mixture into a stiff paste with a 
little water aud then it is ready for use. A little fine grindstone sand improves the 
cement. 

Or, one ounce of sal-ammoniac, to one hundred weight of iron borings. No heat 
allowed to it. 

The cubic contents of the joint in inches, divided by 5, is the weight of dry bor¬ 
ings in pounds Avoir, required to make cement to fill the joint nearly. 

Cement for Stone anil Brick Work. 

Two parts ashes, three of clay and one of sand, mixed with oil, will resist 
weather equal to marble. 

Brown Mortar. 

One part Thomaston lime, two of sand and a small quantity of hair. * 


Hydraulic Mortar. 

Three parts of lime, four Puzzolauo, one smithy ashes, two of sand and four 
parts of rolled stone or shingle. 


Crushing Weight in Pounds per Square Inch. 

on Portland cement , mixed with different proportions of sand , and at different age of 

the mixture in months. 


Age in 
months. 

0 

P 

1 

arts of sati 
3 

d to one of 
3 

cement. 

4 

5 

6 

3 

3809 

2490 

19)0 

1500 

1200 

950 

7^0 

6 

5280 

3550 

27 50 

2190 

1800 

1500 

1200 

9 

5980 

4150 

3350 

2700 

2280 

1800 

1440 

12 

61 Gp 

5150 

3850 

3010 

2450 

2050 

1300 


About °f this weight should be depended upon in practice. 

Some iron filings in a very weak solution of sal-ammoniac, mixed with Portland 
cement, increases its strength to double or more. 

































Bricks. 


631 


BRICKS. 


Dimensions. 

Common brick, 8 X 4£ X inches = 90 cubic inches. 

Front brick, -8* X 4£X2£ « =92.8“ “ 

When laid in a wall with cement , it occupies a space of — 

Common brick, 8£ X 4£ X 2f inches = 102 cubic inches. 
Front brick, SiX+iX “ = 111 “ “ 

"Weight and Bulk of Bricks. 


Tons. 

Pounds. 

Cub. ft. 

by it 
C. brick. 

Number 

self. 

F. brick. 

of bricks, 
in wall wit 
C. brick. 

i cement. 

F. brick. 

1 

2240 

22.4 

448 

416.6 

381 

347 

0.04464 

100 

1 

20 

18.6 

17 

15£ 

2.23 

6000 

50.00 

lOOO 

930 

850 

772 

2.4 

5376 

53.76 

1075 

lOOO 

914 

834 

2.62 

5872 

58.72 

1130 

1100 

1000 

913 

2.88 

6451 

64.51 

1240 

1200 

1100 

lOOO 


One perch of stone is 21.75 cubic feet. 

Acids for Soldering or Tinning. 

TIN. One part of muriatic acid, with as much zine as it will dissolve, then add 
two parts of water and some sal-ammoniac. 

BRASS and COPPER. One pound of muriatic acid, four ounces of zinc and 
five ounces of sal-ammoniac. 

ZINC. One pound of muriatic acid, two ounces of sal-ammoniac with all the 
zinc it will dissolve, then add three pints of water. 

IRON. One pound of muriatic acid, six ounces sperm tallow and four ounces of 
sal-ammoniac. 

GOLD and SILVER. One pound muriatic acid, eight ounces sperm tallow and 
eight ounces of sal-ammoniac. 

Silvering Metals. 

Ten parts of nitrate of silver, ten parts common salt, thirty parts cream of tar 
tar. Moisteu the powder with water when ready to apply. 

Glues. 

Jlice glue. Rice flour mixed in cold water and boiled in china or clay pot; stir it 
well during the boiling. This makes an excellent white glue. 

Houseblose glue. Dissolve the houseblose in strong alcohol, and apply it hot on 
the articles to be glued. This makes a very strong glue which is not soluble in 
water or moisture. 


Barrel Measure. 

A barrel of flour weighs 196 pounds. 
A barrel of pork. 200 pounds. 

A barrel of rice, 600 pounds. 

A barrel of powder, 25 pounds. 

A firkin of butter, 56 pounds. 

A tub of butter, 84 pouuds. 


14 pounds, 
28 pounds, 

4 quarters, 


1 stone. 

1 quarter. 
1 cwt. 


Busliel Measure. 

'The following are sold by weight per 
bushel: 

Wheat, beans and clover-seed, 60 
pounds to the bushel. 

Corn, rye and flax-seed, 56 pounds. 
Buckwheat, 52 pounds. 

Barley, 48 pounds. 

Oats, 35 pounds. 

Bran, 20 pounds. 

Timothy-seed, 45 pounds. 

Coarse salt, 85 pounds. 


Acre. 

A square of 208.75 feet each way is one acre. 
A circle of 235.5 feet in diameter is one acre. 






















632 


Values of Food. 


VALUES 

Comparative value of various foods 
as productive of dynamic work when 
digested iu the stomach: 

Cabbage.*. 1. 

Carrots. 1.2 

Egg, white of.:. 1.4 

Milk . 1.5 

OF FOOD. 

Value of stock food compared with 

10 pounds of good hay : 

Clover hay. 8 to 10 

Green clover.45 to 50 

Wheat straw. ..40 to 50 

Barley straw.20 to 40 

Oat straw.20 to 40 

Apples. 1.5 

Ale. 1.8 

Fish. 1.9 

Potatoes. 2.4 

Porter. 2.6 

Veal. 2.8 

Mackerel. 3.8 

Ham, lean. 4. 

Bread-crumbs. 5.1 

Pea straw. 10 to 15 

Potatoes.20 to 25 

Carrots (red).25 to 30 

“ (white).40 to 45 

Rye. 54 

Wheat. 46 

Oats. 59 

Peas and beans mixed. 45 

Buckwheat . 64 

Egg, hard-boiled. 5.4 

Egg, yolk. 7.9 

Sugar. 8. 

Isinglass. 8.4 

Bice. 8.9 

Pea meal. 9. 

Indian Corn. 57 

Acorns. 68 

Wheat bran .1.105 

Rye.109 

Wheat, pear and oat chaff..167 

Rye and barley mixed.179 

Arrowroot. 9.3 

Oatmeal . 9.3 

Cheese.10.4 

Cocoa.16.3 

Butter.17.3 

Fat of beef. 21.6 

Cod-liver oil. 21.7 

Time required for the full amount 
of cream to rise to the surface of new 
milk at different temperatures : 

Hours. Temperature, Fabr. 

10 to 12 77° 

18 to 20 68° 

24 55° 

36 50° 

Butter and cheese obtainable from 
100 pounds of milk : 

Pure butter. 3 lbs. 

Good cheese. 7.8 “ 

Common butter. 3.5 “ 

Common cheese.11.7 “ 

Skim-milk cheese.13.5 “ 

Good cream produces about ^ of its 
weight of butter. 

Cheese made from good milk con¬ 
tains 32 to 33 per cent, of water; that 
from skim-milk, about 60 per cent. 

An average good cow yields about 
one gallon of milk per day ; the very 
best yields two gallons, and the poor¬ 
est only half a gallon, per day. 

Percentageof alcohol in 100 parts of 
the following liquors (Prof. Braude): 

Scotch whiskey.51.53 

Irish “ *.53.9 

Rum.53.68 

Gin.51.6 

Unless food is thoroughly deprived 
of its living animalcules before it 
enters the stomach, its full nourish¬ 
ment will not be realized. The rudst 
effectual mode of destroying! he living 
principle is by application of heat by 
steaming, boiling, roasting, or smok¬ 
ing. 

An ox, to replace the daily loss of 
muscular fibre, requires from 20 to*24 
ounces of dry gluten or vegetable 
albumen daily. This would be sup¬ 
plied by 

120 lbs. turnips or 17 lbs. clover hay. 
115 “ wheat straw or 12 lbs. peas. 

75 “ carrots or 12 “ barley. 

67 “ potatoes or 10 “ oats. 

20 “ meadow hay or 5 “ beans. 

Burgundy.14.57 

Cape Mu seal.18.80 

Champagne (still).13.80 

“ (sparkling).12.61 

Cider.5.2 to 9.8 

Constant ia.19.75 

Gooseberrv wine.11.48 

Currant wine.20.50 

Port wine. 22.90 

Madeira wine.22.27 

Teneriffe wine.19.79 

Sherrv wine.19.17 

Claret wine.15.1 

Elder wine. 8.79 

Ale. 6.87 

Porter. 4.2 

Malaga wine.17.26 

Rhenish wine.12.8 

Small beer. 1 28 


































































































Yai.tjes of Food. 


633 


Proportion of Starcli in Vege¬ 
tables. 

Per cent. 


Arrowroot. 82.0 

Rice.79.1 

Rye meal.69.5 

Barley flour. 69.4 

Wheaten flour.66.3 

Indian corn meal. 64.7 

Oat meal. 58.4 

Peas.55.5 

Wheaten bread.47.4 

Potatoes.18.8 

Parsnips. 9.6 

Carrots.... 8.4 

Turnips. 5.1 

“Water in Various Foods. 

Beer and ale.91 

Buttermilk.88 

Skim milk.88 

New milk.86 

Skim cheese.44 

Cheese.36 

Cream.66 

White of egg.78 

Yolk of egg.78 

Fat beef.51 

Fat mutton..53 

Fat pork...;.39 

Indian meal.14 

Lean beef.72 

Lean mutton.72 

Oat meal....15 

Ox liver.74 

Parsnips.....82 

Pea meal.15 

Potatoes.75 

Poultry.74 

Pure butter and fats.15 

Rice.13 

Rye meal. 15 

Sugar. 5 

Veal;. 63 

White fish.78 

Sugar in Various Products. 

Raw sugar.95.0 

Treacle.77.0 

Buttermilk. 6.4 

Carrot. 6.1 

Parsnips. 5.8 

Oat. meal. 5.4 

Skim milk. 5.4 

New milk. 5.2 

Barley meal. 4.9 

Wheat flour. 4.2 

Rye meal. 3.7 

Wheaten bread. 3.6 

Potatoes. 3.2 

Turnips. 3.1 

Peas. 2.0 

Indian meal and rice. 0.4 


Percentage of Nutritive Ele¬ 
ments in Food. 

. Per cent. 


Raw cucumbers. 2 

“ melons. 3 

Boiled turnips. 4| 

Milk. 7 

Cabbage. 7£ 

Currants.10 

Whipped eggs.13 

Beets. 14 

Apples.15 

Peaches.20 

Boiled cod-fish.21 

Broiled venison.22 

Potatoes. Tl\ 

Fried veal.24 

Roast pork.24 

Roast poultry. 26 

Raw beef..26 

Raw grapes.27 

Raw plums...29 

Broiled mutton..30 

Oatmeal porridge..75 

Rve bread.79 

Boiled beans.87 

Boiled rice.88 

Barley bread.88 

Wheat bread...90 

Baked corn bread.91 

Boiled barley.92 

Butter. 92 

Boiled peas.93 

Raw oils.95 

Yield of Vegetables in Pounds 
per Acre. 

Hops. 442 

Wheat. 1260 

Barley. 1600 

Oats. 1840 

Peas. 1920 

Beans. . 2000 

Plums. 2000 

Cherries. 2000 

Onions. 2800 

Hay. 4000 

Pears. 5000 

Grass. 7000 

Carrot. 6800 

Potatoes. 7500 

Apples. 8000 

Turnips. 8400 

Cabbage.......10900 

Parsnips. 11200 

Mangel-wurzel.22000 

Fertilizing Properties of 
Manures. 

Peruvian guano.1000 

Human, mixed.-. 69 

Horse...,. 48 

Swine. 44 

Farm-yard. 30 

Cow. 26 






















































































































634 


Assaying. 


FIRE-ASSAY OF SILVER AND GOLD ORES. 

From actual practice by the author in California and South America. 

Assay Composition. 

Gold or silver ores, 400 grains. 

Litharge (oxide of lead), 500 “ 

Carbonate of soda, 240 “ 

Borax, 110 “ 

Charcoal, 20 “ 

Total. 1270 “ 

All the ingredients to be well powdered and mixed before placed in the cruci¬ 
ble. Should the ore contain much sulphur, stick a 3-inch nail in the assay. The 
more galena in the ore, the less litharge is required. Smelt the assay, cupel the 
lead and weigh the remaining button of precious metal. 

Should the button be pure silver, multiply the weight in grains by 100, and the 
product is the value of silver in dollars per ton of ore; if pure gold, multiply by 
1500, and the product is the value in dollars per ton of ore. 

When the button contains both gold and silver, the latter metal must be dis¬ 
solved in nitric acid, for which the alloy must contain at least 3 silver to 1 of gold, 
otherwise the acid will not dissolve it. In case the alloyed button does not con¬ 
tain sufficient silver, it is necessary to add what is required, and melt it into one 
button by blowpipe and charcoal. Hammer the button to a thin leaf and boil it 
in nitric acid ; when all the silver is dissolved, the pure gold remains solid. Wash 
the gold in clean water, dry and weigh it. 

Suppose the alloyed button to weigh 2.156 grains, and its color being between 
that of gold and silver, so as to suspect too little of the latter metal; then add, 
say, 2 grains of pure silver, and dissolve the button, weigh the remaining gold, 
which, for example, maybe 1.162 grains. Then 2.156 —1.162= 1.994 grains of 
silver in the assay. 

Silver, 1.994 + 100 = 199.40 dollars per ton. 

Gold, 1.162 -h 1500= 1743 “ 

Value of the ore. = 1942.40 “ “ 

About one per cent, of the precious metal is lost in the cupelling. 

This rule is sufficiently correct for practical purposes. 

North American Stamlard. 

p f Gold, 387 ounces, 8000 dollars, 
j Silver, 99 ounces, 128 dollars. 

Peruvian Standard. 

p f Gold, 1 ounce, *24.29 pesos = 19.43 soles. 

( Silver, 1 libra, 25.66 pesos = 20.53 soles. 

One peso = 4 francs; one sole = 5 francs. 


Assay Table I.—North and South American Measures. 

The table will answer for any system of assaying weights. 


Percentage 

Value of Metal per ton 

Value of Metal per quin- 

Silver per 
cajon. 

of metal in 

of Ore. 

tal of Ore. 

the ore. 

Gold. 

Silver. 

Gold. 

Silver. 

Per ct. 

Dollars. 

Dollars. 

Soles. 

Soles. 

Marcs. 

0.1 

602.9*24 

39.709 

31.690 

2.053 

12 

0.2 

1205.85 

79.418 

6*2.179 

4.106 

24 

0.3 

1808.77 

119.127 

93.269 

6.158 

36 

0.4 

2411.69 

158.836 

1*24.359 

8.211 

48 

0.5 

3014.62 

198.545 

155.448 

10/264 

60 

0.6 

3617.54 

238.254 

186.538 

12.317 

72 

0.7 

4*220.45 

277.963 

217.627 

14.370 

84 

0.8 

48*23.39 

317.672 

248.717 

16.4*22 

96 

0.9 

54*26.31 

357.381 

279 807 

18.473 

108 

i per cent. 

6029.24 

397.090 

310.896 

20.528 

126 

North American. 

South American. 



























Silver and Gold, 


635 


Suppose the assay to be 112 grammes, and the cupelled button weighs 0.657 of a 
gramme uf silver, then 0.657 X100:112 = 0.586 per cent. 


See Table 


0.5 =10.264 

0.08 = 1.642 
0.006= 0.012 


Soles per quintal. 


0.586 = 11.918 


See Table 


f 0.5 = 

1 0.08 = _ 

1 0.006 = 2.3S 

(. 0.586 = 232.61 


= 198.545 
31.767 

.382 


Dollars per 
of ore. 


ton 


Table II. For Gold and Silver. 



Wei 

Slits. 


Value 


Bulk. 


Avoirdupois. 

Trov. 

in dollars. 

Gold. 

Silver. 

Tons. 

Pounds. 

Ounces. 

Grains. 

Gold. 

Silver. 

Cub. ft. 

Cub. in. 

Cub. ft. 

Cub. in. 

I 

2000 

29166.6 

14 millions. 

602924 

39709 

1.0643 

2875.91 

3.060 

52S7.48 

0.0005 

1 

14.5833 

7000- 

301.46 

18.854 

- - 

1.43795 

_ - 

2.64284 

- - 

0.06857 

1 

480 

20.6718 

1.2929 

- - 

0.09859 

- - 

0.18129 

- - 

0.0002 

0.00283 

1 

0.04306 

0.00269 

_ - 

0.00020 

—- - 

0.00038 

- -* 

0.00332 

0.04837 

23.2202 

1 

1.000 

- - 

0.0524 

— - 

- - 

-- 

0.05304 

0.77316 

371.264 

l.oono 

1 

- - 

- —. 

- . - 

0.1401 

0.60085 

1201.7 

17524.8 

841190) 

362267 

- - 

1 

1728 

1.0000 

1728 

- - 

0.695 43 

10.1416 

4867 99 

209 645 

- - 

0.0U05S 

1 

0.00058 

1.00000 

0.32679 

653577 

9531.34 

4575043 

— 

12)76.4 

1.0000 

1728 

1 

1728 

- - 

0.378227 

5.51581 

2647.59 

— 

7.5095 

0.00058 

l.OoOO 

0.00058 

1 


Table III. Gold, Silver and Platinum. 

Weight in grains per square inch of sheet , thickness by Birmingham gauge for those 

metals, and in inches. 


Bir. G. 

Thick. 

Gold. 

Silver. 

Platin. 

Bir. G. 

Thick. 

Gold. 

Silver. 

Platin. 

No. 

inches. 

grains. 

grains. 

grains. 

No. 

inches. 

grains. 

grains. 

grains. 

1 

0.004 

20.68 

11.52 

25.50 

19 

0.063 

339.5 

184.7 

397.5 

2 

0.005 

26.93 

14.40 

31.26 . 

20 

0.069 

371.8 

201.9 

435.0 

3 

0.006 

82.19 

17.28 

38.0) 

21 

0.075 

404.0 

220.0 

471.8 

4 

0.008 

42.80 

23.52 

50.43 

22 

0.081 

436.1 

237.2 

509.2 

5 

0.010 

53.85 

29.28 

58.14 

23 

0.087 

468.8 

255.0 

548.6 

6 

0.012 

64.46 

35.04 

75.45 

24 

0.093 

500.0 

272.6 

586.0 

7 

0.014 

75.42 

40.80 

87.88 

25 

0.099 

533.3 

290.0 

625.0 

8 

0.016 

86.58 

46.56 

100.1 

26 

0.105 

506.6 

307.8 

663.0 

9 

0.018 

97.07 

52.00 

113.2 

27 

0.111 

596.1 

325.5 

697.3 

10 

0.022 

118.9 

64.32 

138.5 

28 

0.117 

630.0 

342 6 

735.0 

11 

0.025 

134.6 

72.96 

157.7 

29 

0.124 

673.3 

363 5 

783.1 

12 

0.029 

156 3 

84 96 

182.3 

30 

0.130 

701.5 

380.3 

817.0 

13 

0.033 

178.2 

96.43 

207.4 

31 

0.136 

730.0 

398.4 

855.0 

14 

0.038 

204.6 

111.3 

239.5 

32 

0.142 

769.5 

416.3 

892.0 

15 

0.043 

231.5 

125.8 

2705 

33 

0.148 

798.5 

433.3 

932.0 

16 

0.048 

258.8 

140.8 

3026 

34 

0.152 

837 6 

451.6 

970.0 

17 

0.053 

285.6 

155.3 

323.8 

35 

0.160 

865.7 

470.5 

1007 

18 

0.058 

312.6 

170.0 

365.3 

36 

0.166 

894.0 

4S6.0 

1047 


California Rule for Silver and Gold. 

It is an established custom in California to allow one per cent, for base metal in 
all gold and silver bars from the mines. The fineness is always stamped in parts of 
1000; that is, if a gold bar is stamped 900 fine, it is understood to contain— 

900 parts of pure gold, 

90 parts of pure silver, 

10 parts of base metal, 

in 1000 parts of the bar. 












































036 Gold and Silver. 


To Find tlie Value of Gold and Silver Bars. 

Example 1. Required, the value of the pure gold in a bar weighing 989 ounces 

and stamped 797 fine? 

y. ... (790 fine = 16.33.07) a 

From table | 7fine __ . 14.47 / dollars. 

Required value of the bar, 989 X 16.47.54 = 16294.17 dollars. 

Example 2. A gold bar weighing 366 ounces has been assayed and stamped to 
860 fine. Required, its total value? 

Metals. Bui. Fine. Ounces, per Ounce. Value. 

Gold, 366 X 860 = 314.76 X 20.67.18 = $6506.65.57. 

Silver, 366 X 130 = 47.58 X 1.27.29 = G0.57.00. 

Rase metal, 366 X 10 = 3.66 no value. _ 

Total amount 1000 = 366 Answer, $6567.22. 

The last two figures in the columns of Table IV. are decimals of a cent.. 

The fineness of gold is also expressed in carats, 24 for pure gold; that is, a piece 
of gold 18 carats fine is 18 X 1000 : 24 = 750 fine. 


Table IV.—Value of Gold and Silver, per ounce Troy, of 

Different Fineness. 


Finen. 
in 1000 . 

Gold. 

Silver. 

Finen. 
in 1000 . 

Gold. 

Silver. 

Fineness 
in 1000 . 

Gold. 

Silver. 


$ 

cts. 

s 

cts. 


$ 

cts. 

$ 

cts. 


8 

cts. 

$ 

cts. 

1 

0 

2.07 

0 

00.13 

290 

5 

99.48 

0 

37.49 

650 

13 

43.67 

0 

84.04 

■ 2 

0 

4 13 

0 

00.26 

SCO 

6 

20.16 

0 

38.79 

660 

13 

64.34 

0 

85 33 

3 

0 

6.20 

0 

00.39 

310 

6 

40.83 

0 

40.08 

670 

13 

85.01 

0 

86.63 

4 

0 

8.27 

0 

00.52 

320 

6 

61.50 

0 

41.37 

680 

14 

05.68 

0 

87.62 

5 

0 

10.33 

0 

00.05 

330 

6 

S2.17 

0 

42.67 

690 

14 

26.36 

0 

89.21 

6 

0 

12.40 

0 

00.77 

340 

r» 

i 

02.84 

0 

43.96 

700 

14 

47.03 

0 

90.51 

7 

0 

14.47 

0 

00.90 

350 

7 

23.51 

0 

45 25 

710 

14 

67.70 

0 

91.80 

8 

0 

16.54 

0 

01 03 

360 

7 

44.19 

0 

46.55 

720 

14 

88.37 

0 

93.09 

9 

0 

18.60 

0 

01.16 

370 

7 

04*86 

0 

47.84 

730 

15 

09.04 

0 

94.51 

10 

0 

20.67 

0 

01.29 

380 

7 

8553 

0 

49.13 

740 

15 

29.72 

0 

95.68 

20 

0 

41.34 

0 

02.59 

390 

8 

06.20 

0 

50.42 

750 

15 

50.39 

0 

96.97 

30 

0 

62.02 

0 

03.88 

400 

8 

26.87 

0 

51.72 

760 

15 

71.06 

0 

98.26 

40 

0 

82.69 

0 

05.17 

410 

8 

47.55 

0 

53.01 

770 

15 

91.73 

0 

99.56 

60 

1 

03.36 

0 

06.46 

420 

8 

68.22 

0 

54.30 

780 

16 

12.40 

1 

00.85 

60 

1 

24.03 

0 

07.76 

430 

8 

88.89 

0 

55.60 

790 

16 

33.07 

1 

02.11 

70 

1 

41.70 

0 

09.05 

440 

9 

09.56 

0 

56.89 

800 

16 

53.75 

1 

03 . 1.5 

80 

1 

65 37 

0 

10 34 

450 

9 

30.23 

0 

58.18 

810 

16 

74.42 

1 

04.73 

90 

1 

86.05 

0 

11.64 

460 

9 

£0.90 

0 

59.47 

820 

16 

95.09 

1 

06.02 

100 

2 

06 72 

0 

12.93 

470 

9 

71.58 

0 

60.77 

830 

17 

15.76 

1 

07.31 

110 

2 

27.39 

0 

14.22 

4S0 

9 

92.25 

0 

62.06 

840 

17 

36.43 

1 

08.61 

120 

2 

48.06 

0 

15.52 

490 

10 

12.92 

0 

63.35 

850 

17 

57.11 

1 

09.00 

130 

2 

68.73 

0 

16.81 

500 

10 

33.59 

0 

64.65 

860 

17 

77.78 

1 

11.19 

140 

2 

89.41 

0 

18.10 

610 

10 

54.26 

0 

65.94 

870 

17 

98.45 

l 

12.48 

150 

3 

10.08 

0 

19.39 

620 

10 

74 94 

0 

67-23 

880 

18 

19.12 

1 

13.78 

ICO 

3 

30.75 

0 

20.69 

630 

10 

95.61 

0 

68-53 

890 

18 

39.79 

1 

15.07 

170 

3 

52.42 

0 

21.98 

540 

11 

10.28 

0 

69-82 

900 

18 

60.46 

1 

16 36 

180 

3 

72 09 

0 

23.27 

550 

11 

36.95 

0 

71-11 

910 

18 

81.14 

1 

17.60 

190 

3 

92.76 

0 

24.57 

660 

11 

57.62 

0 

72-14 

920 

19 

01.81 

1 

18.95 

200 

4 

13.44 

0 

25.86 

570 

11 

78.29 

0 

73-69 

930 

19 

22.4S 

1 

2U.24 

210 

4 

34.11 

0 

27.15 

580 

11 

98.97 

0 

74,99 

940 

19 

43.15 

1 

21.51 

220 

4 

54.7S 

0 

28.44 

590 

12 

19.64 

0 

76-28 

950 

19 

63.82 

1 

22.83 

230 

4 

75.45 

0 

29 74 

600 

12 

40.31 

0 

77.58 

960 

19 

84.50 

1 

24.12 

240 

4 

96.12 

0 

31.03 

610 

12 

60.98 

0 

7S.87 

970 

20 

05.17 

1 

25 4 L 

260 

5 

16.80 

0 

32.32 

620 

12 

81.65 

0 

80.16 

980 

20 

25.84 

1 

26 71 

260 

5 

37.47 

0 

33.62 

630 

13 

02.33 

0 

81.46 

990 

20 

46.5.1 

1 

28 00 

270 

5 

5S.14 

0 

34.91 

640 

13 

23.00 

0 

82.75 

1000 

20 

67.18 

1 

29.29 

280 

■5 

78.81 

0 

36 20 








— 




































Chemistry. 637 



To Refine Silver. 


Dissolve the impure silver in nitric acid, add chloride of sodium (salt) sufficient 
to precipitate all the silver in form of chloride; then all the impurities will remain 
in solution. 

Filter, wash and dry the chloride of silver. Fuse in a crucible two weights of 
carbonate of potash, add gradually one weight of chloride of silver, raise the heat, 
aud the pure silver will melt and collect on the bottom. 

Tests for Metals in Solution with Acids. 

The reagents are placed in the liquid, which precipitates the metal in solution. 

REAGENTS. 

PRECIPITATES. 

SOLUTIONS. 

Sulphate of iron, 

Oxalic acid, 

Potash or soda, 

Gold, as brown powder, 
Gold in large flakes, 

Gold, yellow, 

Gold in 
aqua-regia. 

Potash or soda, 

Plate of copper, 

Muriatic acid, 

Common salt, 

Tincture of nutgall, 

Silver, dark olive, 
Metallic silver, 

White crude silver. 
White crude silver, 
Brown silver, 

Silver in 
nitric acid. 

Potash or soda, 
Ferro-prussiate of potash, 
Carbonate of potash, 

Blue cobalt, 

Green “ 

Red 

Cobalt in 
nitric acid. 

Pure water, 

Gallic acid, 

Potash or soda, 

White bismuth, 

Greenish yellow, 

White bismuth, 

Bismuth in 
nitric acid. 

Sulphate of soda. 
Sulphuric acid. 

Infusion of nutgall, 

| White lead, 

Lead in 
nitric acid. 

Plate of iron or zinc, 
Potash, 

Ammonia, 

Infusion of nutgall, 

Metallic copper, 

Green copper, 

Azure-blue copper, 
Brown copper, 

Copper in 
nitric acid. 

Pure water, 

Plate of iron. 

White antimony, 

Black antimony, j 

Antimony in 4 muriatic 
acid, 1 nitric acid. 

Plate of copper, 
iron, 

Gallic acid, 

Metallic mercury, ' 

Dark powder, 

Orange yellow, J 

Mercury in 

’ muriatic or nitric acid. 

Infusion of nutgall, 
Ferro-prussiate of potash, 
Ammonia, 

Black iron, "] 

Blue iron, 

Dark-red iron, J 

Iron in 
muriatic acid. 


Acltl Test for Strength ancl Quality of Iron and Steel. 

T1 1 is is a subject well worthy of attention by workers in iron and steel. The 
sample to be tested is filed smooth, or polished on all sides, and placed in di¬ 
lute nitric or sulphuric acid for 12 to 24 hours; then wash the sample and dry it. 
The action of the acid has revealed the structure of the sample, from which its 
quality can be decided with great precision. 

The best steel presents a frosty appearance; ordinary steel, honeycombed. Iron 
presents a fibrous structure in the direction in which it has been worked; the best 
iron shows the finest fibres. Should the iron be uneven, or made from a pile of dif- 
• ferent kinds of iron, all are exposed by the action of the acid. Hammered blooms 
show slag and iron ; gray cast iron shows crystals of graphic carbon ; other cast 
irons show different figures, all with marked characteristics. 







638 


Chemistry. 


Iron Pyrites, Snlphurets. 

There are two kinds of iron pyrites—namely, protosulphuret and bisulphuret, of 
which the latter is generally richest in gold. All iron pyrites are slightly mag¬ 
netic, but the gold seems to destroy the magnetism. The protosulphuret acts sen- 
biblv on the magnetic needle, whilst the bisulphuret does not, and may therefore 
be distinguished for gold. 

The presence of arseniuret of iron in sulphurets indicates richness in gold. 

Roasting of Sulphurets. 

When sulphurets contain magnesia, lime or arsenic, sufficient salt should ho 
added to chlorinate those substances, which then evaporate and go out through 
the chimney. The amount of those impurities should be ascertained beforehand. 
The salt should be well mixed with the ore before put into the furnace. Ten 
pounds of salt contain six pounds of chlorine and four pounds of sodium. 


Ten pounds of 

Those impurities are very t Magnesia, . 
injurious to chlorination of -< Calcium, 
the gold in the vat. Arsenic, . 


Pounds required. 
Chlorine. Salt. 


3.58 

5.78 

10.64 


6 

9.65 

17.6 


Chlorination of Gold in Roasted Sulphurets. 

Free gold is attacked and dissolved by chlorine gas, and forms two chlorides, 
namely, 


An. 844 parts of gold. 

Cl. 156 “ chlorine. 

An. Cl. 1000 protochloride of gold. 


An. 648.5 parts of gold. 

Clz. 351.5 “ chlorine. 

Au. Clz. 1000 tcrcliloride of gold. 


Gold-bearing sulphurets arc roasted for the purpose of obtaining the gold free 
for the action of chlorine gas. The combination is very slow, and requires the gold 
to be very tine for the prompt formation of chloride. In some ores, the gold is too 
coarse for chlorination, when it must be extracted by amalgamation. 


Composition for Making Chlorine Gas. 

For each ton of roasted ore in the vat are required 14 pounds of salt, 10 pounds 
of peroxide of manganese and 5 quarts of sulphuric acid. The composition should 
be constantly stirred in the gasometer, and kept to a uniform temperature of about 
180° Fa hr. The chlorine gas thus formed is led into the vat containing the ore. 

On account of chlorine gas being much heavier than air, the gasometer ought 
to be placed at a considerable height above the vat, to facilitate the chlorination 
of the gold. In California they place the gasometer below the vat, which is de¬ 
cidedly wrong. 

Chloride of gold is soluble in water, and can be washed out from the vat simply 
by pouring water on the top of the ore and running it into another vessel, where 
the gold is precipitated with sulphate of iron. 

Chloride of silver is not soluble in water, and remains in the ore in the vat. 
There is always some silver in gold sulphurets. 


Quartz Mills. 

Each stamp, weighing about 800 pounds, lilted one foot 60 times per minute, can 
crush one ton of quartz per 24 hours with a dynamic effect of two horse-power. 
This is the average performance. The custom-mill in Grass Valley, California, 
crushes quartz for about four dollars per ton. 

The stamps are generally divided into sets of four or five, working in one mor¬ 
tar, and called a battery. The shoes and dies in the battery are made of chilled 
cast iron. 

Most of the gold is collected by amalgamation in the battery. The pulp from 
the battery contains much gold, which is often allowed to run away, but generally 
the sulpliuret in the pulp is concentrated and roasted Cor chloriuation ; the rest of 
the pulp is ground in pans and the gold amalgamated. 

Amalgams. 

GOLD. One weight of mercury amalgamates with two weights of gold. 

SILVER. 10 silver to 19 mercury. 

7 “ “ 20 “ 

TIN. 1 tin to 3 mercury, for looking-glasses. 

1 tin, 1 lead, 2 bismuth, 10 mercury, for glass-globes. 

1 tin. 1 zinc, 3 mercury, for rubbers in electric machines. 













Optics. 


639 


OPTICS. 


Optics is that branch of philosophy which treats of the properties and motion 

of light. 


Mirrors. 

Example 1. Fig. 307. Before a concave mirror of r = 6 feet radios, is placed 
an object O = 1, at d — 1-75 feet from the vertex. Required the size of the 
image 1 = 1 


image I — 


0 r 
r —2 d 


1X6 

6—2X1-75 


- = 2*4 


Example 2. Fig. 308. Before a concave mirror of r — 5-25 feet radius, is placed 
an object 0 = 1, at D = 4'5 feet from the vertex. Required the size of the in¬ 
verted image 1 = ? 


image I = 


0 r 


1X5*25 

2X4-5—5-25 


Example 3. Fig. 309. Before a convex mirror of r —-1 '8 feet radius, is placed 
an object O = 1, at D = 3'15 feet from the vertex. Required the size of tlio 
image 1=1, and the distance in the mirror d — ? 

image 7= =0*222 distance d f ■ =-.0-699 ft. 

° 2X3-15+1-8 2X3-15 + 1-8 

Example 4. Fig. 310. A parabolic mirror is h = 1-31 feet high, and d = 2-16 
feet in diameter. Required the focal distance/= ? from the vertex. 

focal distance f= =• = 2*646 inches. 

J 16 h 16X1-31 


Optical Lenses. 


Example 5. Fig. 316. A double convex lens, of crown glass, having its radii 
E — r = 6 inches. Required its principal focal distance/ = ? 

For crown glass the index of refraction is m = 1-52. See table. 

f = — —®—. = 5’768 inchea 
J 2(1-52—1) 

Microscope 

Jjetiers uenotc. 

p =»= magnifying power of a lens. 

33 = limit of distinct vision. 

3 = limit of distinct sight, which for long-sighted eyes Is about 10 or 12 
inches, and near-sighted 6 to 8 inches. For common eyes take 
a= 10 inches. 

= limit distance of the object from the optical centre at distinct vision. 


Example 6 . Fig. 322. Required the magnifying power of a single microscope 
■sritb principal focal distance, / = 4-3 inches 1 


Mag. power p = 


a +/_ 10+4-3 
/ “ 4-3 


=3*325 times. 




























Optics. 


040 




Z> 






307 


Spherical Concave Mirror. 

and j 
r. 

Or 


r = radius, and f — \ r, focal distance of tho 
mirror. 

d r 


1 = 


. D=- 


r —2 d r —2 d 

The image disappears when d =f = 4 r. 



308 


I = 


Spherical Concave Mirror. 
0 r 

1 1J—r 


d = 


D r 


2 D—r 


When the object is beyond the focal 
distance the image will be inverted. 


309 


/ = 


Spherical Convex Mirror. 
Or _ t) r 


2 D+r 


d = 


2 I)+r 


310 


Parabolic Concave Mirror. 


f= 


d* 


1G h 


h — 


d ’ 


16/ 


311 


Hijpcrbolic Concave Mirror. 


Heat, Light, or Sound emanating from 
the foci of a hyperbola will be reflected 
divergently, from the concave surface. 


312 


Eliptic Concave Mirror. 


Emanating rays from either of the two 
foci in an elipse, will be reflected by the 
convex surface to the other foci. 










































OPTIC*. 


fill 


Astronomical Telescopes and Opera Glasses* 

Example 7. Fig. 325. The principal focal distance f = 0*65 inches of the 
ocular or eye-lens. F = 58 inches the principal focal distance of the objoctive- 

lens. 


Required the magnifying power of the telescope i = ? 

0 F 

f 0*6 5 


image I = 


= - - = 89*23 times the object. 


The telescope is to he used at the limited distance D *= 1380 feet and D — <». 
Required the proper lengths l = ? and micrometrical motion of the ocular or 
eve-leus? when the limit of distinct sight a — 10iu. F = 58 : 12 = 4'833 feet. 
f = 0*05 : 12 = 0 , 05416 feet. 


1 = 


1380X4*833 10X0*05416 


10+0*05416 


4*89035 

0*05386 


1380—4*833 

When D — 1380 feet, the length l =4*94421 feet. 


When D 


oo, 1 = 4*8333 + 0*05386 


4*88719 


0*05702 


v 

n 


Micrometrical motion of eye lens 0*68424 inches. 

_ii 

15 nearly. 


Table of Refractive Indices* 


Substances. 

Index* 

ni. 

Substances. 

Index. 

in. 

Cromato of Lead 

Realgar - 
Diamond - 
Glass, flint 

Glass, crown 

Oil of Cassia 

Oil of Olives 


2-97 

2-50 

2-55 

2-45 

1-57 

1-52 

1-63 

147 

Quartz- ... 

Muriatic Acid 

Water - 

Ice .... 

Hydrogen - 
Oxygen - _ - 

Atmospheric air - 

1.54 

1.40 

1.33 

1.30 

1.000138 

1.000272 

1.000294 



314 Prism. 

A ray of light a a' falling upon a transparent 
medium A (say a glass prism) will he transmitted 
through in the direction a 5, and delivered in the 
direction bb\ parallel a a'a". 

V — angle of incidence, v = angle of refraction. 

Index of refraction m = —. 

sin. v 

itift 

315 Given the direction of the incident ray a a' t 
angles e and r ,—to find the angles u and z,—or 
the direction of the rays b b'. 

cos. z = C0S ' e , cos. u-m cos. (180 — z—r). 
m 

x = 180 —(e+r+w). 

When e = u, the angle x is smallest. 

An eye in b' will see the candle in the direc¬ 
tion O' b b". 


41 










































Optic?. 


012 



316 
/- + 
/ = 


Double Convex Lens. 

ft r the principal 

(m— l)(B+r) focal distance. 

when E = r 


2(m — 1 ) * 
o ,= optical centre of the lens. 


Plano Convex Lem. 
r 


/" + 


m —1 


The optical centre is in the convex 
surface. 


318 


Convex-concave Lens [Meniscus.) 

/- + Br 


(m —l)(f?— r) 

Draw the radii IV and r* parallel to 
one another.—Draw n o 7 then o is th® 
optical centre. 


Double Concave Lens. 
Br 


/-- 


[m — 1 )(/?+/•)* 


320 


Plano Concave Lens, 
r 


/= - 


m —1 


The optical centre is in the concave 
surface. 


Concavo-convex Lens. 

f =z _ ft t 
J (m— 1) [E— r)‘ 

Draw' E and r' parallel to ono another. 
Draw n o , then o is the optical centre. 


1 





































































Optics. 


643 



Single Microscope. 

T: 0=*f: f — d, I = D 

J J » /_d , 

®=J^, *" 


= A£ 
/-^ 

a / 

*+/' 



323 

When the object 0 is beyond the focal 
distance the image / will be inverted. 


/: 0 —f: D—f I=—£, d = P-f. 
J Jr d—f } D-f 


j 



1 Diminishing Power of a Double 
Concave Lens. 

r- of 
f+D’ 


I: 0=f:f+D 

d- fi°=n. 


d - 

D+f 


325 


Astronomical Telescope. 




D 


> 


t 


/: O-Fif 
D F * nf 


/ = 


* D—F^ « + /’ 

* -f- for astronomical telescope, 


<* 


O F 

/ ’ 

if z> 


oo, 


d= D f 

a y F * 

i-F + ±f\ 

a+// 


— for opera-glasses. 


326 Opera Glass. 



Formulas are the same as for Astronomical Telescope. 















































644 


Electro-Dynamics. 


ELECTRO-DYNAMICS. 

TIio Conversion of Heat into Electrical Power. 


The direct conversion of heat into electrical energy has already had a 
partially successful, but not economical, solution in Clamond’s stoves. 

According to Cabanellas, a Clamond's stove consisting of 6000 elements, 
and burning 22 pounds of coke per hour, will give a current of 7 amperes, 
and 218 volts difference of potential. 

Cabanellas also states that the amount of light obtained was equal to 
about 560 standard English candles. 

This would give us nearly 26 candles per pound of coke. As we shall 
presently see, this is a result much less economical of fuel than can be 
obtained by the use of an engine and dynamo under very unfavorable 
circumstances. The liability to derangement and the first cost of Cla¬ 
mond’s Pile have prevented it from becoming commercially successful. 

The ohm, volt, and ampere are the practical British Association units 
used by electricians. 

The legal ohm is the resistance of a column of mercury 1 square milli¬ 
metre in cross section and 106 centimetres in length, at the temperature 
of melting ice. 

Ohm’s law is 


Intensity of current = 


Diff. of Potential 
Resistance ’ 


r E 
otI=r, 


from which we at once see that the resistance equals the ratio between 
the electro-motive force lost in the circuit and the intensity of the 
current. 

This is a constant for any solid so long as its form and temperature are 
not changed. 

The volt and ampere are more difficult to define, and perhaps can have 
their meaning made clear by making use of analogous hydraulic for¬ 
mulae. 

This is only a case of analogy. It cannot be said that electricity is a 
fluid, or that'there is a current of electricity, or that it flows one way or 
the other. 

The volt may be said to represent the pressure or head of the assumed 
current of electricity, and the ampere to represent the intensity or weight 
of the current passing in one second. Lord Rayleigh has carefully de¬ 
termined the weight of silver precipitated from a solution of nitrate of 
silver by one ampere. It is 0 06708 grammes per minute, or 4‘0248 grammes 
per hour. 

Referring to Fig. 1, if W equals the weight of water that passes the 



point A in a pipe in one second, and h the loss of head, we have, for the 
work done in one second, Wh foot-pounds. 

Again, if I represents the intensity of a current passing the point A in 
a second, and E the difference of potential in volts between B and A, we 
have for the work done in one second IE volt-amperes, or watts. If we 
divide W k by 550 foot-pounds, we obtain the horse-power. 






Electro-Dynamics. 


645 


In the case of the pipe, if it were level, the loss of work would be due 
to friction and transformed into heat. 

m 7 V ' 2 7 „r , IF v 2 M v- 

Thus h = — and 11 h —-= - —• 

2g 2 g 2 

Joule has shown us by experiment that the heating of a wire conduc¬ 
es 

tor is proportional to I 2 R = I E — , or, using the analogous hydraulic 


formula, the heating is, IF 2 X ■ 


h 

IF 


h 2 

Wh ~K- 


W 

IE 

Again, —g - = work per second in kilogrammetres, but an English 

horse-power equals 76 01 kilogrammetres per second, and, therefore, one 

horse-power = g ; . Returning again to the ohm, we have R = ^. 

That is, the resistance is the loss of electro-motive force per second and 
per unit of intensity which an electrical current experiences when pass¬ 
ing along a conductor. If this conductor is the standard quicksilver col¬ 
umn, R = 1 ohm. 

In an analogous manner we would say of a horizontal pipe conveying 
water that the resistance is the loss of head per pound and per second 
when passing through the pipe. 

The resistance of the various materials used as conductors for the elec- 
tricai current has been repeatedly and carefully determined. 

By the electro-motive force in volts is meant something similar to the 
head of water in feet or its pressure in pounds. 

By the intensity of a current in amperes is meant something similar to 
the weight of water passing in pounds per second. 

By the resistance in ohms is meant something similar to the loss of head 
of water per pound and per second. 

The dynamo electric machine is the newest and the most perfect of 
machines for the transformation of energy from one form to another. 
Like the turbine, its efficiency has been proved so great as to preclude 
all hope of further increase of practical value. Its cost may be reduced 
by improved processes of the machine-shop : we cannot do more. 

One reason for this rapid perfecting lies in the apparent obscurity of 
electrical phenomena, which has had the effect of repelling all but subtle 
and acute minds from their study. The right end of the thread once 
seized by such minds, they have followed the clue with such rapidity 
and thoroughness of apprehension as to leave nothing more for us to 
accomplish. 

The recent experiments of the Franklin Institute upon the dynamos 
of Weston and Edison have set the seal of absolute measurement, with 
as great exactitude as we can hope to reach, upon the ability of these 
machines to transform mechanical work into electrical work. 

Of the five dynamo electric machines which successfully withstood the 
severe conditions of the code, Weston’s mammoth incandescent lamp 
machine, of a rated capacity of 125 amperes and 160 volts, returned as an 
average of four tests, in the form of electrical energy, 06*56 per cent, of 
the mechanical power used to drive it; 89'37 per cent, of the mechanical 
power was available as electrical energy in the external circuit. 

Of the total mechanical power applied, about 1 per cent, was lost in 
friction of the armature shaft and resistance of the air to its rapid revo¬ 


lution. 

Two and one half per cent, only remains to be accounted for, and was 
presumably lost in the form of heat and eddy currents. 

Every precaution was taken to avoid results which would not appear 
in every-day use, and all of the machines were run under full load for 
ten hours before the measurements began, and so were at as high a tem¬ 
perature as would be reached in actual practice with the same atmo¬ 
spheric temperature. The performance of this particular machine only 
exceeded the least efficient of the machines tested by 2T per cent, total 
efficiency. 

(Continued on page 595.) 








646 


The New British Gauge. 


The New British Gauge. 

(Legal standard in England from Mar. 1,1884. Superseding all other Gauges.) 


Gauge 

Number 

Differ¬ 

ences. 

DIAM 

Inches. 

ETER. 

Centi¬ 

meters. 

A rea of 
Cross 
Section. 

Cm2. 

PURE CO 
(Soft 

Resistance. 

Ohms 
per meter. 

PPER WIRE 
Drawn). 

Conductivity. 
Meters 
per ohm. 

Weight of 
Wire. 

Density 8.90 
(Copper) 
Grammes 
per Meter. 

7/0 


.500 

1.270 

1.267 

.000135 

7402.1 

1127.4 

6/0 

36 

.464 

1.178 

1.090 

.000157 

6370. 

970.2 

5/0 • 

32 

.432 

1.097 

.945 

.000181 

5521. 

840.8 

4/0 

32 

.400 

1.016 

.811 

.000211 

4736. 

721.3 

3/0 

28 

.372 

.945 

.701 

.000244 

4098. 

624.2 

2/0 

24 

.348 

.884 

.613 

.000279 

3584. 

545.9 

0 

24 

.324 

.832 

.532 

.000322 

3107. 

473.2 

1 

24 

.300 

.762 

.546 

.000375 

2666. 

406.1 

2 

24 

.276 

.701 

.386 

.000444 

2253. 

343.2 

3 

24 

.252 

.640 

.322 

.000532 

1881. 

286.5 

4 

20 

.232 

.589 

.273 

.000628 

1592. 

242.5 

5 

20 

.212 

.538 

.228 

.000751 

1331. 

202.7 

6 

20 

.192 

.488 

.187 

.000916 

1092. 

166.3 

7 

16 

.176 

.447 

.157 

.00109 

917.8 

139.8 

8 

16 

.160 

.406 

.130 

.00132 

757.2 

115.3 

9 

16 

.144 

.366 

.105 

.00163 

614.9 

93.7 

10 

16 

.128 

.325 

.0829 

.00206 

484.6 

73.8 

11 

12 

.116 

.295 

.0682 

.00251 

398.3 

60.7 

12 

12 

.104 

.264 

.0548 

.00312 

320.3 

48.8 

13 

12 

.092 

.234 

.0429 

.00398 

250.6 

38.2 

14 

12 

.080 

.203 

.0324 

.00528 

189.5 

28.9 

15 

8 

.072 

.183 

.0263 

.00651 

153.5 

23.4 

16 

8 

.064 

.163 

.0208 

.00824 

121.3 

18.5 

17 

8 

.056 

.142 

.0159 

.0108 

92.7 

14.1 

18 

8 

.048 

.122 

.0117 

.0147 

68.2 

10.4 

19 

8 

.040 

.1016 

.00811 

.0211 

47.4 

7.19 

20 

4 

.036 

.0914 

.00659 

.0260 

38.4 

5.84 

21 

4 

.032 

.0813 

.00519 

.0330 

30.3 

4.62 

22 

4 

.028 

.0711 

.00397 

.0431 

23.2 

3.o4 

23 

4 

.024 

.0610 

.00292 

.0587 

17.05 

2.60 

24 

2 

.022 

.0559 

.00245 

.0698 

14.32 

2.18 

25 

2 

.020 

.0508 

.00203 

.0845 

11.84 

1.80 

26 

2 

.018 

.0457 

.00164 

.104 

9.59 

1.46 

27 

1.6 

.0164 

.0417 

.00136 

.125 

7.97 

1.21 

28 

1.6 

.0148 

.0376 

.00111 

.154 

6.48 

.988 

29 

1.2 

.0136 

.0345 

.000937 

.183 

6.46 

.834 

30 

1.2 

.0124 

.0315 

.000779 

.220 

4.55 

.693 

31 

.8 

.0116 

.0295 

.0006S2 

.251 

3.98 

.607 

32 

.8 

.0108 

.0274 

.000591 

.290 

3.45 

.526 

33 

.8 

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2.51 

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.8 

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2.09 

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36 

.8 

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1.71 

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37 

.8 

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1.37 

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38 

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.0152 

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1.06 

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39 

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1.248 

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40 

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1.466 

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41 

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1.742 

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42 

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2.109 

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2.611 

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44 

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3.300 

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45 

.4 

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4.310 

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46 

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5.818 

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47 

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8.475 

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48 

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13.23 

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49 

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23.42 

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50 

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33.78 

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E^ectricitt, 


G47 


ELECTRICITY. 


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Electro-Chemical Order of 

3impSe SuhstoMces. 
Electro-positive. 

Potassium. 

Sodium. 

Lithium. 

Barium. 

Strontium. 

Calcium. 

Magnesium. 

Aluminium. 

Uranium. 

Manganese. 

Zinc. 

Iron. 

Nickel. 

Cobalt. 

Cadmium. 

Lead. 

Tin. 

Bismuth. 

Copper. 

Silver. 

Mercury. 

Palladium. 

Platinum. 

Gold. 

Hydrogen. 

Silicon. 

Titanium. 

Tellurium. 

Antimony. 

Carbon. 

Boron. 

Tungsten. 

Molybdenum. 

Vanadium. 

Chromium. 

Arsenicum. 

Phosphorus. 

Iodine. 

Bromine. 

Chlorine. 

Fluorine. 

Nitrogen. 

Selenium. 

Sulphur. 

Oxygen. 

El ectro-neo ati v 


Order of Compounds. 

Electro-positive. 

Fur. 

Smooth glass. 
Woollen cloth. 
Feathers. 

Wood. 

Paper. 

Silk. 

Lac. 

Rough glass. 
Sulphur. 

Colton. 

Electro-negative. 


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Order of Conducting Power 

_/or Electricity, 

Metals, best conductors. 

Well-burnt charcoal. 

Plumbago. 

Concentrated acids. 

Powdered charcoal. 

Diluted acids. 

Saline solutions. 

Metallic ores. 

Animal fluids. 

Sea water. ' 

Spring water. 

Rain water. 

Ice above 13° Fahr. 

Snow. 

Living vegetables. 

Living animals. 

Steam. 

Saits soluble in water. 

Rarefied air. 

Vapor of alcohol. 

Moist earth and stones. 

Powdered glass. 

Flower of sulphur. 

Dry metallic oxides. 

Oils, the heaviest the 
best. 

Ashes. 

Transparent crystals. 

Ice below 13° Fahr. 

Phosphorus. 

Lime. 

Dry chalk. 

Caoutchouc. 

Camphor. 

Silicious stones. 

Dry marble. 

Porcelain. 

Baked wood. 

Dry gases and air. 

Leather. 

Parchment. 

Dry paper. 

Feathers. 

Hair. 

Wool. 

Dyed silk. 

Bleached silk. 

Raw silk. 

Diamond. 

Mica.. 

All vitrifications. 

Glass. 

Jet. 

Wax. 

Sulphur. 

Resins. 

Amber. 

Shellac. 

Gutta-percha, the worst 
conductor of all. 


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G48 


Geography. 


GEOGRAPHY. 



The Earth on which we live is a round hall or sphere, with a mean diameter of 
7914 statute miles. The whole surface of the earth is 196,800,000 square miles, of 
which only one-fourth or nearly 50,000,000 square miles is land, and about 
150,000,090 square miles water. 

Table of Area and Population of Ihe Wliole Eartli, 1883. 


Divisions of the Earth. 

Area in Square 
Miles. 

Population. 

Proportion to 
Square Mile. 

America,. 

14.491.000 

100,460,000 

7 

Europe,. 

3.700,000 

327,743,400 

87 

Asia, •••••• 

16,313.000 

795,591,000 

205,823,2G0 

49 

At i ica, • • • • • 

10,936.000 

20 

Ocean ica,. 

4.500.000 

31,619,000 

7 

Total, .... 

50.000,000 

1,461,242,660 

30 


About yyth of the whole population are bom ever-y year, and nearly an equal 
number die in the same time; making about one born and one dead per second. 

The annual increase of population per 1000 is about 6 in Europe and 19 in 
America. Europe loses and America gains by emigration. The annual increase 
of population in the whole world is about 6 per 1000. 

The Earth is not a perfect sphere, it is flatted at the Pole3. The following are 
her true dimensions in statute miles of 5280 feet. 


Dimensions of tlie Eartli. 


Diameter, 


Difference, . . 
Flatted, . . . 

Circumference, 


■■7898.8809 miles at the Poles. 

7911.92 miles mean, or in 45° lat. 
(7924.911 miles at the Equator. 

2G.0302 miles Poles and Equator. 

13.015 miles at each Pole. 

f 24802.480 miles round the Poles. 
24851.040 miles mean, or in 45° lat. 
(24884.22 miles round the Equator. 


To Find tlie Radius of tlie Eartli in Any Given Latitude. 

R = 3955.90(1 -f 0.001G4 cos.2£), statute miles. 






























Geography. 


649 


■" ■ ■■ —— --—.. - — -- - - • • - -- 

Population of Conn tries and Cities 1880. 

Hr. Amer., 

6 ,000,000 

Russia, . . . 

100,372,562 

Guanajuato, 

64,000 

Montreal, . . 

140,600 

St.Petorsb’rg, 

SO 1,900 

Cuba, .... 

1,409,800 

Quebec, . . . 

62,400 

Moscow, . . . 

001,969 

Havana, . . . 

220,000 

Toronto, . . . 

86,400 

England, 

24,608,391 

St. JagoCuba, 

105,000 

St.John, N.B. 

20,120 

London, . . . 

3,814,571 

Porto Rico, 

661,494 

Halifax, N. S. 

30,000 

Scotland, 

3,734,441 

C. America, 

2,750,000 

Ottawa, <4nt. 

27,400 

Edinburgh, . 

228,190 

Whites, . . 

150,000 

U. S. Amer., 

50,155,783 

Glasgow,. . . 

511,532 

Indians, . . 

1,500,000 

N. Y. & Brk., 

1,772,902 

Ireland, . . 

5,159,839 

Negroes,. . 

40,000 

Philadelphia, 

847,170 

Dublin, . . . 

249,486 

Mixed,. . . 

1 ,100,000 

Chicago, . . . 

503,185 

France, . . . 

37,672,048 

Guatemala, 

1,215,310 

St. Louis,. . . 

350,518 

Paris, .... 

2,225.900 

Guatemala,A. 

57,728 

Baltimore,. . 

332,313 

Germ.Emp. 

45,194,177 

St.Salvador 

600,000 

Boston,. . . . 

202,839 

Berlin, .... 

1,122,360 

St.Salvador,A. 

21,000 

Cincinnati, . 

255,139 

Austria,. . . 

26,096,860 

Nicaragua, 

400,000 

San Francisco 

233,959 

Vienna, . . . 

1,003,857 

Managua, . . 

11,000 

Washington, 

147,293 

Hungary, 

11,644,574 

Honduras, 

351,700 

Buffalo, . . . 

155,134 

Pest-Buda . . 

359,821 

Comavugua, 

9,000 

Newark, . . . 

136,508 

Holland, . . 

4,060,580 

Costa liica, 

185,000 

Louisville,. . 

123,758 

Amsterdam, 

326,196 

San Jose,. . . 

30,000 

Cleveland,. . 

160,146 

Bavaria, . . 

5,284,778 

S. America, 

32,000*000 

Pittsburg, . . 

156,389 

Munich, . . . 

230,023 

Wild Indians, 

3,600,000 

Jersey City,. 

120,722 

S witzerl’d, 

2,846,102 

Whites, . . 

10 ,000,000 

Detroit, . . . 

110,340 

Berne, .... 

30,000 

Negroes,. . 

650,000 

Milwaukee, . 

115,587 

Belgium,. . 

5,536,604 

Mixed,. . . 

15,000,000 

Providence, . 

104,857 

Brussels,. . . 

170,345 

U. S. Colom. 

2,950,017 

Albany, . . . 

90,758 

Spain, .... 

16,623,384 

Bogot a,.... 

60,000 

Rochester,. . 

89,306 

Madrid, . . . 

397,690 

Panama,. . . 

25.000 

Alleghany, . 

78,682 

Italy, .... 

28,459,628 

Venezuela, 

2,075,245 

Richmond, . 

63,600 

Rome, .... 

289,3^1 

Caraccas, . . 

50,000 

New Haven, 

62,882 

Greece, . . . 

1,679,775 

Equador, 

1,066,137 

Charleston, . 

49,984 

Athens, . . . 

63,374 

Quito, .... 

28,000 

Trov,. 

50,747 

Turkey, E,, 

21 ,000,000 

Guiana, . . . 

222,000 

Syracuse, . . 

51,792 

Const’tiuople, 

1,150,000 

Georgetown, 

28,000 

Indianapolis, 

75,056 

Turkey, A., 

16,463,000 

Brsizil, . . . 

9,930,478 

Worcester, . 

58,291 

Smyrna, . . . 

160,000 

Rio Janeiro, 

430,000 

Lowell,. . . . 

59,475 

Arabia, . . . 

9,000,000 

Bahia, .... 

155,000 

Memphis, . . 

33,592 

Mecca, .... 

62,000 

Slaves,. . . 

1,400,000 

Cambridge, . 

52,0(59 

Persia, . . . 

11,299,500 

Pern, . 

3,374,000 

Hartford, N . . 

42,015 

Teheran,. . . 

70,000 

Lima,. 

130,000 

Scranton, . . 

45,850 

A fgh’nist’n 

5,000,000 

Callao. 

40,000 

Reading,. . . 

43,278 

Cabul, .... 

65,000 

Bolivia, . . 

2,000,000 

Kansas Citv, 

55,785 

Beloochs’n, 

450,000 

Ltl PilZj • • • • 

23,000 

Mobile. 

29,132 

Ivelat, .... 

17,000 

Chili, .... 

2,136,720 

Portland , . . 

33,810 

Turkistun, 

7,000,000 

Santiago, . . 

129,807 

Wilmington, 

42,478 

Bokhara,. . . 

120,000 

Valparaiso, . 

97,737 

Toledo,.... 

50,137 

India, .... 

252,541,210 

Argentine, 

1,737,923 

Columbus,. . 

51,647 

Bombav, . . . 

753,000 

Buenos Ayres 

248,710 

Lawrence,. . 

39,151 

Calcut ta,. . . 

683,458 

Paraguay, 

293,844 

Utica,. 

33,914 

China, .... 

360,279,897 

Ascuncion, 

16,000 

Savannah,. . 

30,709 

Peking, . . . 

2 ,000,000 

Uruguay, 

438,245 

Nashville, . . 

43,350 

Canton, . . . 

1 ,200,000 

Montevideo, 

130,000 

Alaska, . . . 

32,500 

Hong-Kong, 

50,000 

Patagonia, 

1 ,200,000 

Sweden, . . 

4,567,300 

Japan, . . . 

36,357,368 

Antonio,. . . 

? 

Stockholm, . 

175,000 

Yeddo, .... 

2 ,000,000 

Austin 1 ia, 

2,271,245 

Gotheborg, . 

76,761 

Miaco, .... 

500,000 

Melbourne, 

65,860 

Norkdping, . 

28,000 

Barbary, . . 

2,890,000 

Wellington, 

5,000 

Malmo, . . . 

37,000 

Tunis, .... 

140,000 

Jamaica, . . 

580,804 

Gelle,. 

19,000 

Egypt, . . . 

5,252,000 

Kingston, . . 

40,000 

Norway, 

1,818,853 

Cairo, .... 

349,883 

Hayti, .... 

572,000 

Christiania, . 

119,407 

Jerusalem, . 

26,000 

Pt. au Prince, 

22,000 

Bergen, . . . 

39,271 

Mexico, . . . 

9,300,000 

Sa ml wit h I. 

75,000 

Denmark, 

1,969,039 

Mexico City, 

225,000 

Honolulu, . . 

14,000 

Copenhagen, 

234,850 

Puebla, . . . 

80,000 

\Y. Indies, 

4,000,000 




































650 


Latitude and Longitude. 


Latitude and Longitude of Places (from Greenwich.) 


America 

Latitude. 

Longitude. 


Latitude. 

Longitude. 

Atl. Coast. 

D. M. S. 


D M S. 


France. 

D. M. s. 


D. M. S. 


Quebec ... 

40.49. 

N 

71.16. 

W. 

Paris, Obs.. . 

48.50.13 

N. 

0.09.21 E. 

Halifax . . . 

44.158. 

(4 

63.35. 

(4 

Cherbourg . . 

49.38. 


1.37. 

it 

Chicago . . . 

42.00. 

ti 

87.35. 

44 

Marseilles. 

43.18. 


5 22. 

it 

Huston . . . 

42.21. 

it 

71.01. 

44 

Calais , . . 

50.58. 


1.51. 

it 

New York . . 

40.42. 

tl 

74.00.42 

4. 

Brussels. . . 

50 61. 


4.22. 

it 

Philadelphia . 

34.57. 

U 

75.10. 

44 

Antwerp . . 

51.13. 


4t24. 

it 

Cincinnati . . 

39.05. 

ii 

81.30. 

t4 

Italy. 





Bf. Louis . . 

38.15 3. 

« 

89.36. 

44 

Turin . . . 

45.04.06 


7.42. 

it 

Washington . 

38.515. 

a 

77.00.18 

44 

Florence . . 

43.46. 


11.16. 

ti 

Cli illusion. . 

32.42. 

t« 

79.54. 

44 

Leghorn . . 

43.32. 


10.18. 

it 

New Orleans . 

29.57.39 

tt 

90.00. 

44 

Rome . . . 

41.54. 


12.27. 

it 

Georgetown.Br. 

32.22.12 

tt 

46.37.06 

44 

Malta . . . 

35.54. 


14.30. 

it 

Nassau . . . 

25.05.12 

u 

77.21.12 

4* 

Naples . . . 

40.50. 


14.16. 

it 

Port-au-Prince 

19.4(5.24 

ti 

72.11.12 

44 

Palermo . . 

38.08. 


13.22. 

ii 

Porto Rico. 

18.29. 

it 

66.07.06 

44 

Venice . . . 

45.26. 


12.21. 

ii 

Kingston, Jain. 

17.58. 

u 

76.46. 

44 

Austria. 





Havana ■ . . 

23.09. 

a 

82.22. 

44 

Vienna . . . 

48.13. 


16.23. 

ii 

Vera Cruz . . 

19.12. 

a 

96.09. 

44 

Trieste . . . 

45.39. 

« 

13.46. 

ti 

Mexico, City . 

19.2(5. 

u 

99.05. 

44 

Pestli.... 

47.28. 

ii 

19.13. 

ii 

Colon, N O. . 

9.2 1. 

u 

79.55. 


Germany. 





Para ... 

1.28. 

s. 

48.29. 

44 

Berlin . . . 

52.31. 


13.24. 

if 

Rio Janeiro . 

22.5(5. 

a 

43.09. 

44 

Hamburg . . 

53.33. 

C< 

9.56. 

it 

Buenos Ayres 

34.150. 

tt 

58.22. 

44 

Cologne . . 

50,56. 

it 

6.58. 

it 

Cape Horn . . 

55.59. 

<4 

67.16. 

44 

Amsterdam . 

52.22. 

a 

4.51. 

ti 

Pac. Coast. 





Bremen . . . 

53.05. 


8.49. 

(» 

Valparaiso . . 

33.02. 

ft 

71.41. 

44 

Berne . . . 

40.57. 

a 

7.25. 

it 

Callao . . . 

12 0 4. 

4 ; 

79.13. 

4i 

Turkey. 





Lima* . . . 

12.02.34 

a 

79.06. 

44 

Constantinople 

41.01. 

u 

28.59. 

ii 

Cuzco* . . . 

13.31.43 

a 

74.15.50 


Kagusa . . . 

42.38. 

u 

18.07. 

it 

Payta . . . 

5.05. 

u 

81.10. 

44 

Salonica . . 

40.39. 

u 

22.57. 

ti 

Guayaquil . . 

2.13. 

a 

79.53. 

44 

Athens . . . 

37.58. 

a 

23.44. 

ii 

Panama. . . 

8.57. 

N. 

79.31. 


Smyrna . . . 

38.26. 

a 

27.07. 

it 

Acapulco . . 

16.55. 

4 i 

99.48. 

it 

Cairo . _ . . 

30.03. 

a 

31.18. 

it 

San Francisco 

37.47. 

Ci 

122.21. 

4^ 

Jerusalem. Pal. 

31.48. 

a 

37.20. 

it 

Alaska . . . 

58. 


158 


Russia. 





Behring’s Strait 

07°. 


170 


St. Petersburg 

,59.56. 

a 

30.19. 

ii 

China, lnd. 





Moscow. . . 

55.40. 

a 

35.33. 

it 

Peking . . . 

39.5 l. 

it 

116.28. 

E. 

Nish Novgorod 

56.20. 

a 

43 43. 

it 

Canton . . . 

23.07. 

it 

113.14. 

4t 

Cazan . . . 

55.48. 

u 

48.50. 

a 

Hongkong. . 

22.15. 

a 

114.12. 

44 

Archangel . . 

64.32. 

a 

40.14. 

a 

Honolulu . . 

21.19. 

a 

157.52. 

“ 

Jecatherinburg 

56.50. 

it 

60.21. 

it 

Jeddo . . . 

35.4) 

a 

139.43. 


Astracau . . 

4(3.21. 

a 

47.46. 

tt 

Owyhee, S. Isl. 

20.23. 

it 

155.54. 

W 

Odessa . . . 

40.27. 

it 

30.42. 

ii 

Calcutta . . 

22.34. 

it 

88 .20. 

E. 

Warsaw. . . 

52.13.05 

it 

21.02.9 

it 

Batavia . . . 

6.08. 

a 

106.50, 

4. 

Sweden. 





Sydney . . . 

154.00. 

s. 

151.23. 

44 

Stockholm. . 

69.21. 

a 

18.04. 

tt 

Melbourne . . 

37.4-i.36 

t. 

144 57.45 

it 

Gothenburg . 

57.42. 

a 

11.57. 

it 

Wellington . 

41.14 

it 

174.44. 

44 

Wisby, Gotland 

57.39. 

a 

18.17. 

it 

Africa. 





Christiania 

69.55. 

u 

10 52. 

it 

Cp. of G. Hope. 

34.22. 

a 

18.30. 

44 

Bergen . . . 

60.24. 

a 

5.20. 

it 

Morocco . . 

39.31. 

N. 

2.23. 

44 

Ystad . . . 

55.25. 

a 

13.50. 

tt 

Algiers . . . 

35.47. 

it 

3.04. 

44 

Haparanda 

65.49. 

a 

24.11. 

tt 

England. 





Copenhagen . 

55.41. 

u 

12.34. 

tt 

London, Tower 

51.31. 

it 

0.08. 

IV. 

Spain. 





Greenwich 

61.28.38 

.4 

0 0 0 


Madrid . . . 

40.25. 

a 

3.4*2. 

w. 

Liverpool . . 

53.22. 

44 

2.52. 

44 

Barcelona . . 

41.23. 

it 

2.11. 

E. 

Glasgow . . 

55.52. 

it 

4.16. 

44 

Gibraltar . . 

36.06. 

tt 

5.20. 

W 

Edinburgh . . 

55.57. 

il 

3.12. 


Carthagena . 

37.36. 

it 

1.01. 

it 

Bristol . . . 

51.27. 

it 

2.35. 

44 

Lisbon . . . 

38.42. 

it 

9.09. 

It 

Dover . . . 

51.08. 

44 

1.19. 

E. 

Oporto . . . 

41.11. 

a 

8.38. 

it 

Dubliu . . . 

53.23. 

44 

6.20 

W. 

Terra, Island 

27.47. 

it 

17.56. 

ii 


* Measured by the author. 






















651 


Difference of Longitude in Time. 


Difference of Longitude in Time Between Places. 



San 

Francisco. 

New York. 

London. 

Greenwich. 

St. Peters¬ 
burg. 

Canton, 

China. 


H. 

M. 

S. 

H. 

M. 

S. 

II. M 

S. 

H. 

M 

S. 

H. M. S. 

Amsterdam. 

8 

29 

19 

5 

15 

32 

0 19 

32 

1 

41 

44 

7 13 24 

Antwerp. 

8 

27 

17 

5 

13 

30 

0 17 

36 

1 

43 

40 

7 15 20 

Batavia. 

8 

42 

50 

11 

5G 

37 

7 07 

20 

5 

6 

4 

0 25 36 

Berlin. 

9 

3 

22 

5 

49 

35 

0 53 

35 

1 

7 

41 

6 39 21 

Boston. 

3 

25 

33 

0 

11 

46 

4 44 

14 

6 

45 

30 

11 42 50 

Buenos Ayres . . . 

4 

16 

19 

1 

2 

32 

3 53 

28 

5 

54 

44 

11 26 24 

Canton . 

8 

17 

17 

11 

31 

4 

7 32 

56 

5 

31 

40 

0 0 0 

Calcutta. 

9 

56 

53 

10 

49 

20 

5 53 

20 

3 

52 

4 

1 39 36 

Cairo. 

10 

l4 

59 

7 

1 

12 

2 5 

12 

0 

3 

56 

5 27 44 

Copenhagen .... 

9 

0 

3 

5 

46 

16 

0 50 

1 G 

1 

11 

0 

6 42 40 

Constantinople . . . 

10 

5 

43 

6 

51 

56 

0 55 

56 

0 

5 

20 

5 37 0 

Dublin. 

7 

41 

25 

4 

30 

38 

0 25 

22 

2 

26 

38 

8 18 18 

Florence . 

8 

54 

51 

5 

41 

4 

0 45 

4 

1 

16 

12 

6 47 52 

Gibraltar. 

7 

47 

5G 

4 

31 

32 

0 21 

28 

2 

22 

44 

7 54 24 

Gothenburg. 

8 

57 

38 

5 

43 

51 

0 47 

48 

i 

13 

28 

6 45 08 

Halifax. 

3 

54 

27 

0 

41 

40 

4 14 

20 

6 

15 

36 

11 47 16 

Hamburg. 

8 

49 

39 

5 

35 

52 

0 39 

62 

1 

21 

24 

6 53 04 

.Jecatherinburg . . . 

11 

48 

57 

7 

25 

20 

4 1 

16 

2 

0 

0 

3 31 40 

Jerusalem. 

10 

39 

7 

7 

25 

20 

2 23 

20 

0 

22 

4 

5 09 36 

Lima. 

1 

43 

2 

0 

12 

24 

5 08 

24 

7 

9 

40 

11 18 40 

London . 

8 

9 

50 

4 

56 

3 

0 0 

24 

2 

1 

40 

7 33 20 

Lisbon. 

7 

33 

11 

4 

19 

24 

0 36 

36 

2 

37 

52 

8 08 32 

Madrid. 

7 

55 

39 

4 

41 

52 

0 14 

8 

2 

15 

24 

7 47 04 

Melbourne. 

4 

2 

24 

7 

16 

11 

9 39 

51 

11 

41 

31 

2 7 0 

Naples. 

9 

G 

51 

5 

53 

4 

0 57 

4 

1 

04 

12 

6 35 52 

New Orleans .... 

2 

9 

37 

1 

04 

10 

6 0 

10 

8 

01 

26 

10 26 54 

New fork. 

3 

13 

47 

0 

0 

0 

4 56 

3 

6 

57 

19 

11 31 01 

Paris. 

8 

19 

7 

5 

5 

20 v 

0 9 

20 

1 

51 

50 

7 23 36 

Peking . 

8 

4 

21 

12 

41 

52 

7 45 

52 

5 

44 

36 

0 12 56 

Quebec. 

3 

2 

36 

0 

11 

11 

4 44 

49 

6 

46 

5 

11 19 53 

Rome. 

8 

59 

35 

5 

45 

48 

0 49 

48 

1 

11 

28 

6 43 08 

San Francisco .... 

0 

0 

0 

3 

13 

47 

8 9 

47 

10 

11 

3 

8 17 17 

St. Louis. 

2 

8 

46 

1 

5 

1 

6 1 

1 

8 

2 

17 

10 26 03 

St. Petersburg .... 

10 

11 

3 

G 

57 

16 

2 1 

16 

0 

0 

0 

5 31 40 

Stockholm. 

9 

22 

11 

G 

8 

24 

1 12 

24 

0 

58 

52 

6 20 32 

Turin. 

8 

40 

38 

5 

26 

51 

0 30 

48 

1 

30 

28 

7 02 08 

Washington. 

3 

1 

46 

0 

12 

1 

5 8 

1 

7 

9 

17 

11 19 53 

Wellington, N. Z. . . . 

5 

30 

22 

8 

44 

09 

11 38 

56 

10 

19 

48 

2 46 52 


To lietluce Longitude in Degrees into Time, and vice versa. 

Rule 1. Divide the number of degrees by 15, and the quotient is the corre¬ 
sponding hours. Should the degrees be less than 15, multiply them by 4, and the 
product will be minutes in time. The minutes of degrees multiplied by 4 will be 
seconds in time. The seconds of degrees divided by 15 will be seconds in time. 

Rule 2. The time in hours, minutes and seconds, multiplied by 15, will be the 
corresponding angle in degrees, minutes and seconds. The trigonometrical table 
contains the conversion of time and angle. 

Example 1. Required, the difference in time between Philadelphia and Paris? 

Longitude of Philadelphia, 75° 10' IP. 

“ “ Paris, . . 2 20 E. 

Difference in longitude 77° 30' divided by 15 will be 
5 h 10»t, the difference in time. When it is 12 o’clock in Philadelphia, it is 6/i 10 /h 
o’clock in Paris. 

Example 2. A vessel sails from New York to Liverpool; after she has been at 
sea about one week, her difference in time with New York is found to be 2 h 7m45s. 
Required, her longitude from New York ? 

15(2/i 7 45) = 31° 58' 15" from New York. 

The time is ahead in the east, from where the sun rises. The time is behind in 
the west, toward sunset._j 






















































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Distances in Europe. 


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Astronomy. 


G56 


A S T R 0 N 0 M Y. 

The matter constituting the heavenly bodies has probably been evenly distrib¬ 
uted in space from the beginning; the force of attraction gradually formed bodies, 
which accumulated into groups or nebulas, each of which finally became a planet¬ 
ary system with the largest body in the centre, now seen as stars. Ibis operation 
of forming or creating bodies and planetary systems is still and will for ever be 
continued in parts of the infinite space. Each star has a planetary system, and 
astronomers have even been able to observe some planets of the star Sirius. 

The magnitude of this operation, with the enormous dimensions, even within a 
single gronj^ can hardly be conceived by any human mind; for the long row of 
figures expressing a number of conceivable units of length or weight does not 
bring the real magnitude, within conception. 

The Sun. 

The sun is a dark body surrounded by a luminous substance in which spots are 
frequently seen. The spots are caused by meteors or other heavenly bodies falling 
into it. 

Moan distance from the earth to the sun, 95,000,000 miles, or 11,992 diameters of 
the earth. 

Inclination of the ecliptic to the equinoctial, 23° 28' 40". 

The sun subtends an angle from the earth of 32' 3". 

Horizontal parallax of the sun, 8.6". 

The Moon. 

Distance from five Earth to the Moon, 273,000 miles, = 30 diameters of Earth, 
or about 0.25 diameters of the Sun, or the diameter of the Sun is twice the diam¬ 
eter of the Moon’s orbit around the Earth. 

Diameter of the Moon, 2160 miles, or about 0.2729 of the diameter of the 
Earth. 

Volume of the Moon, 0.02024 of that of the Eartlu 

Density of the Moon, 0.5657 of that of the Earth. 

Mass of the Moon, 0.0114 of that of the Earth. 

Inclination of the Moon’s orbit to the ecliptic, 5° 8' 48". 

The Moon subtends an angle fc m the Earth of 31' 7". 

Mean sidereal revolution of the Moon, 27.32166 solar days. 

Mean synodical revolution of the Moon, 29.5305887 solar days. 

The Moon passes the meridian in periods of 24.814 hours, or 48m. 50s. later every 
day. 

Moon’s Ag;e is the number of days from the last new moon. 

Epaet. of the Year is the Moon’s age on the 1st of January of each year. 
See Almanac for tho 19th Century. 

The sum of the epact of the year and that of the month is tho moon’s age on 
the first of tho month. 


Epaet of the Month. 


Jan. 

Feb. 

March 

April. 

May. 

June. 

July. 

Aug. 

Sept. 

Oct. 

Nov. 

0 

2 

1 

2 

3 

4 

5 

6 

8 

8 

10 


To Eiml the Moon s Age on Any Given Day. 

Add together the epaets of the year and month and the date of the month ; the 
sum will be the moon's age. If it exceeds 30, reject that much, and the remainder 
is the moon’s age. 


















Astronomy. 


657 


Almanac for the 19tli Century. 




Dom. 

-♦-J 



Dom. 

4-» 



Dom. 


Yrs. 

Days. 

let¬ 

ter. 

W 

Yrs. 

Days. 

let¬ 

ter. 

ft 

w 

Yrs. 

Days, 

let¬ 

ter. 

ft 

W 

1800 

Saturd.* 

FE 

4 

1834 

Saturd. 

E 

20 

1868 

Sunday* 

ED 

6 

1801 

Sunday. 

D 

15 

1835 

Sunday. 

D 

1 

1869 

Monday. 

C 

17 

1802 

Monday. 

Tuesday. 

C 

26 

1836 

Tuesd.* 

CB 

12 

1870 

Tuesday. 

B 

2S 

1803 

B 

7 

1837 

Wednes. 

A 

23 

1871 

Wednes. 

A 

9 

1804 

Thursd.* 

AG 

18 

1838 

Thursd. 

G 

4 

1872 

Friday.* 

GF 

20 

1805 

Friday. 

F 

29 

1839 

Friday. 

F 

15 

1873 

Saturd. 

E 

1 

1806 

Saturd. 

13 

11 

1840 

Sunday* 

ED 

26 

1874 

Sunday. 

D 

12 

1807 

Sunday. 

I) 

22 

1841 

Monday. 

C 

7 

1875 

Monday. 

C 

23 

1808 

Tuesd.* 

CB 

3 

1«42 

Tuesday. 

P 

18 

1876 

Wedns.* 

BA 

4 

1809 

Wednes. 

A 

11 

1843 

Wednes. 

A 

29 

1877 

Thursd. 

G 

15 

1810 

Thursd. 

G 

25 

1844 

Friday.* 

GF 

11 

1878 

Friday. 

F 

26 

1811 

Friday. 

F 

6 

1845 

Saturd. 

E 

22 

1879 

Saturd. 

E 

7 

1812 

Sunday.* 

ED 

17 

1S46 

Sunday. 

I) 

3 

1880 

Monday* 

DC 

18 

1813 

Monday. 

C 

28 

1847 

Monday. 

C 

14 

1881 

Tuesday. 

B 

29 

1814 

Tuesday. 

B 

9 

1848 

Wedns.* 

BA 

25 

1882 

Wednes. 

A 

11 

1815 

Wednes. 

A 

20 

1849 

Thursd. 

G 

6 

1883 

Thursd. 

G 

22 

1816 

Friday.* 

GF 

1 

1850 

Friday. 

F 

17 

1884 

Saturd.* 

FE 

3 

1817 

Saturd. 

E 

12 

1851 

Saturd. 

E 

28 

1885 

Sunday. 

D 

14 

1818 

Sunday. 

D 

23 

1852 

Mond.* 

DC 

9 

1886 

Monday. 

C 

25 

1819 

Monday. 

C 

4 

1853 

Tuesday. 

B 

20 

18S7 

Tuesday. 

B 

6 

1820 

Wed ns.* 

BA 

15 

1854 

Wednes. 

A 

1 

1888 

Thursd * 

AG 

17 

1821 

Thursd. 

G 

26 

1855 

Thursd. 

G 

12 

1889 

Friday. 

F 

28 

1822 

Friday. 

F 

7 

1856 

Saturd.* 

FE 

23 

1890 

Saturd. 

E 

9 

1823 

Saturd. 

E 

18 

1857 

Sunday. 

D 

4 

1891 

Sunday. 

D 

20 

1824 

Monda.* 

DO 

29 

1858 

Monday. 

C 

15 

1892 

Tuesd * 

CB 

1 

1825 

Tuesday. 

B 

11 

1859 

Tuesday. 

B 

26 

1893 

Wednes. 

A 

12 

1826 

Wednes. 

A 

22 

I860 

Thurs.* 

AG 

7 

1894 

Thursd. 

G 

23 

1827 

Thursd. 

G 

3 

1861 

Friday. 

F 

18 

1895 

Friday. 

F 

4 

1828 

Saturd.* 

FE 

14 

1862 

Saturd. 

E 

29 

1896 

Sunday* 

ED 

15 

1829 

Sunday. 

D 

25 

1863 

Sunday. 

D 

11 

1897 

Monday. 

C 

26 

1830 

Monday. 

O 

6 

1S64 

Tuesd.* 

CB 

22 

1898 

Tuesday. 

B 

7 

1831 

Tuesday. 

B 

17 

1865 

Wednes. 

A 

3 

1899 

Wednes. 

A 

18 

1832 

Thursd.* 

AG 

28 

1866 

Thursd. 

G 

14 

1900 

Friday.* 

GF 

29 

1833 

Friday. 

F 

9 

11867 

Friday. 

F 

25 

1901 

Saturd. 

E 

11 


The day of the week opposite the year in the a'manac falls on the dates in this 
table. 


February, 

February,* 


January, 

January,* 

September, 


March, 


May. 


April, 


June. 

November. 

August. 

October. 

July. 

December. 


1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 






In leap years take January ,* February.* 

Example 1. On what day of the week will the 4th of July foil in the year 1880? 

See table 1880 = Monday, which answers to the 5th in the date table, conse¬ 
quently the 4th of July is on Sunday. 

Example 2. It is known to be Saturday in the middle of August, 1875. Required, 
the date of that day ? 

The year 1875 = Monday (see almanac), then August the 16th falls on Monday, 
and Saturday on the 14th. 

Example 3. A gentleman was born on the 8th of February, 1824. Required, 
the day of the week ? 

1824 = Monday, which fell on the 9th. The gentleman was consequently born 
on a Sunday. 


42 




























653 


Chronology. 


CHRONOLOGY. 

Our unit of time, year, is the period in which the earth makes one revolution 
around the sun, in reference to a fixed star. 

The unit day is the period in which the earth makes one revolution around its 
axis, in reference to the sun ; but as the earth moves in an ellipse in which the 
sun is in one focus, the apparent solar day is not a constant period—that one hun¬ 
dred solar days in the winter are about half an hour longer than one hundred 
Solar days in summer, fo'r which a mean day is assumed in reference to an imagi¬ 
nary sun which falls in with the real sun about the 15th of April and 24th of De¬ 
cember, when the mean time and apparent time are alike. The mean solar day is 
divided into twenty-four hours, of which the clock indicates twelve hours twice. 

Sidereal Time. 

The Sidereal Day is the interval of time in which any fixed star passes the me¬ 
ridian. The sidereal day is only 237). 56m. 4.09s. mean solar time, or the fixed stars 
pass the meridian, rise or set, 3m. 55.909s. earlier every day. 

A Sidereal Clock in an astronomical observatory marks twenty-four hours in 
the interval of time in which any fixed star passes the meridian. The Right As¬ 
cension of the heavenly bodies is tbe time when the body passes the meridian by 
the sidereal clock. The dial of a sidereal clock is divided into twenty-four hours, 
and the hours are numbered from one up to twenty-four. 

Years. 

The tropical year, or the periodical return of seasons, is 365.24224 days = 365c?. 
57). 48m. 49.536s. mean solar time. 

The civil year is 365 days, or nearly one-fonrth of a day too short, for which one 
day is added every four years, called leap year. But this addition makes one day 
too much in every 128.866 years, which error is corrected every fourth century 
which can be divided by 4 without a remainder. 

Leap years of the Christian era are those that can be divided by 4 without a 
remainder. 

In some countries these important corrections are not properly attended to, as 
in Russia and Greece, where the dates are now twelve days behind our Gregorian 
reckoning. 

The wild Indians of South America reckon their time by new moons, when all 
their festivals are celebrated. 

Dates. 

The civil date of the month commences at midnight. The astronomical date 
commences at noon. The mariner’s date (sea account) commences twelve hours 
before the civil date, and twenty-four hours before the astronomical date, or the 
mariner’s date is one day ahead of the astronomical date. 

Cycle of 1 He Sun is the period of twenty-eight years, at which the days of the 
week return to the same days of the month. 

Lunar Cycle or Golden Number is the period of nineteen years, at 
which the changes of the moon fall on the same days of the month. 

To Find tlie Golden Number. 

Add 1 to the given year, divide the sum by 19, and the remainder will be the 
golden number. If 0 remains, the golden number is 19. 

The age of the Julian period on the 1st of January, 1872, is 2,404,794 days. 

Creation of the World. 

Creation of the world, 4000 before Christ. Julius Africanus says 5508; Samaritan 
Pentateuch, 4700; Septuagint, 5872; Josephus, 4658; Talmudists, 51444 ; and others 
give different times; hut the Chinese tradition and history claim an antiquity of 
100,000 years before Christ. From geological formations, and from the working of 
rivers like that of Niagara, and the Dannbe cutting through the Alps at the Iron 
Gate, it can be estimated at millions of years. 




Almanac. 


G59 


Before Christ. 


YEA.RS. 

Deluge, 2348 (Hales), . . 3154 

Tower of Babel finished, . 2247 

Chinese Monarchy, . . 2203 

Egyptian Pyramids, . . . 2090 

Moses born, .... 1567 

Troy destroyed, .... 1180 

The compass discovered, . 1111 

Foundation of Rome, . . 753 

Maps and Geometry introduced, GU5 
Money coined at Rome, . . 576 

Hannibal crossed the Alps, . 219 

Time first measured by water, 155 

Carthage destroyed, . . 146 

Cresar invaded Britain, . . 51 

Caesarean era, ... 48 


Chronological Notes. 

R. C. YEA.RS. I After Christ. A. D. 


Beginning of Christian Era, 
Christ crucified, 

Destruction of Jerusalem,. 
Arabic numbers introduced, 
Mohammedan Era, 

New Style in England, . 
America discovered, . 

Pizarro conquered Inca, Peru, 
Lutheran religion, . 

New South Wales discovered, 
Australia discovered, . 
American great Republic, 
Slaves free in West Indies, 
Slaves free in Russia, . . 

Slaves free in America, . 


YEARS. 

0 

37 

69 

991 

622 

1752 

1192 

1530 

1527 

1606 

1622 

1775 

1834 

1861 

1866 


ASTRONOMICAL ALMANAC. 

From the Fvglish Nautical Almanac. 

The following tables show the sun’s right ascension and declination; also, the 
equation of time at Greenwich, apparent noon, for the year 1873: 

In leap years.at 6 h. A. M. 

First year after leap year.at noon. 

Second year after leap year.at 6 h. P. M. 

In the year before leap year, at midnight following the date. 

By the aid of the tables of correction the data can be found for any time and 
for any meridian. 

Example 1. Required, the sun’s R. A. on the 10th of April, 1874, at Greenwich, 
apparent noon? 

1874 is the second year after leap year, when the tabular data is for 6 o’clock in 
the evening. The daily variation of the sun’s R. A. is 3 m. 40 s. for the 10th of 
April, which for 6 hours will be 55 s. 

The sun’s R. A. on the 10th of April, . 1 h. 16 m. 24 s. 

Correction for 6 hours, subtract ... _ 55 s. 

The required R. A. will be . . . 1 h. 15 m. 29 s. 

Example, 2. Required, the sun’s declination on the 20th of September, in the 
leap year 1876, at 3 o’clock P. M., in longitude 75°, or 5 hours west of Greenwich ? 

The tabular data for leap years is at 6 o’clock A. M. 3 P. M. and 5 hours differ¬ 
ence in longitude make 14 hours of correction. 

The daily variation in the sun’s declination is 23'. 

20'= 11' 40") 

3 = 1 45 | 

Correction, 

. . 0° 58' 

. . . 13' 25" 


See table of correction, page 502. 


13' 25" for 14 hours. 

Sun’s declination 20th September, 
Subtract correction, . . , 

The required declination, . . 


. . . . 0° 44' 35" 

In leap years take the tabular data one day earlier in January and February. 









6G0 Astronomical Almanac. 



January. 

February. 

March 


|_ 

Date 

Rt. As.l Declin. 

Eq. tm. 

Rt. As. 

Declin. 

Eq. tm. 

Rt. As. 

Declin. 

Eq. tin. 

Date 


h. m. s. 

o ' 

ni. s. 

h. m. s. 

O » 

m. a. 

h. m. s. 

O ' 

111. 8. 


1 

18 48 49 

S 22 59 

+ 3 59 

21 0 59 

8 16 59 

+13 54 

22 49 58 

S 716 

+12 30 

1 

2 

18 5314 

S 22 54 

+ 4 27 

!2l 5 3 

S 16 42 

+14 2 

22 53 42 

8 7 3 

+ 12 18 

2 

3 

18 57 38 

S 22 48 

+ 4 55 

21 9 6 

S 16 24 

+14 8 

22 57 26 

8 6 40 

+12 6 

3 

4 

19 2 2 

S 22 41 

+ 5 23 

21 13 8 

8 16 6 

+14 14 

23 1 9 

8 617 

+11 52 

4 

5 

19 6 26 

S 22 35 

+ 5 49 

21 17 9 

8 15 48 

+ 1418 

23 4 62 

8 5 54 

+11 3S 

5 

6 

19 10 49 

8 22 27 

-f- 6 1<> 

21 21 10 

8 16 30 

+14 22 

23 8 35 

S 5 31 

+11 24 

6 

7 

1915 11 

S 22 20 

+ 6 42 

21 25 9 

8 15 11 

+14 25 

23 12 17 

S 5 8 

+ 1110 

7 

8 

19 19 33 

8 22 12 

+ 77 

21 29 8 

8 14 62 

+14 28 

23 15 58 

8 4 44 

+10 55 

8 

9 

19 23 55 

8 22 3 

+ 7 32 

2133 6 

S 14 33 

+ 14 29 

2319 39 

8 4 21 

+10 39 

9 

10 

19 28 10 

S 21 54 

+ 7 56 

2137 3 

S 1413 

+14 30 

23 23 20 

8 3 57 

+ 10 24 

10 

11 

19 32 36 

8 21 45 

+ 8 20 

2141 0 

S 13 53 

+14 30 

23 27 1 

8 3 34 

+ 10 7 

11 

12 

19 36 56 

S 21 35 

+ 8 43 

21 44 56 

S 13 33 

+ 14 29 

23 30 41 

S 310 

+ 9 51 

12 

13 

19 4115 

8 21 25 

+ 96 

2148 51 

8 13 13 

+ 14 27 

23 34 21 

8 2 47 

+ 9 34 

13 

14 

19 45 33 

S 21 14 

+ 9 27 

21 52 45 

S 12 53 

+14 25 

23 38 0 

S 2 23 

+ 917 

14 

15 

19 49 51 

S 21 3 

+ 9 48 

21 56 38 

S 12 32 

+14 22 

23 41 39 

8 159 

+ 90 

15 

16 

19 54 8 

8 20 52 

+10 9 

22 0 31 

S 1212 

+14 18 

23 45 19 

8 136 

+ 8 43 

16 

17 

19 58 24 

S 20 !0 

+10 29 

22 4 23 

S 11 51 

+ 1413 

23 48 57 

S 112 

-j- 8 25 

17 

18 

20 2 40 

S 20 28 

+10 4S 

122 8 14 

S 1130 

+ 14 8 

23 52 36 

8 0 48 

+ 88 

18 

19 

20 6 55 

8 2015 

+ 11 6 

|22 12 5 

8 11 8 

+ 14 2 

23 56 15 

S 0 24 

+ 7 50 

19 

20 

20 11 9 

8 20 2 

+ 11 24 

22 15 55 

S 10 47 

+13 56 

23 69 53 

SOI 

+ 7 32 

20 

21 

20 15 23 

S 19 49 

+1140 

22 19 44 

8 10 25 

+ 13 49 

0 3 32 

X 0 23 

+ 713 

21 

22 

20 19 35 

S 19 35 

+ 11 57 

22 23 33 8 10 3 

+ 13 41 

0 710 

N 047 

+ 6 55 

22 

23 

20 23 47 

S 19 21 

+ 12 12 

22 27 31 

8 9 41 

+13 32 

010 48 

X 1 10 

+ 6 37 

23 

24 

20 27 59 

S 19 7 

+12 27 

122 31 9 

S 919 

+13 23 

0 14 26 

N 134 

+ 618 

24 

25 

20 32 9 

S 18 52 

+12 40 

22 34 56 

8 8 57 

+ 13 14 

0 18 4 

X 157 

+ 60 

25 

26 

20 3619 

S 18 37 

+12 53 

22 38 42 

S 8 34 

+13 4 

0 21 42 

X 22L 

+ 5 42 

26 

27 

20 40 27 

S 18 21 

+ 13 6 

22 42 28 

8 812 

+ 12 53 

0 25 20 

N 2 44 

+ 5 23 

27 

28 

20 44 35 

8 18 6 

+ 13 17 

22 46 13 

S 7 49 

+12 42 

0 28 59 

N 3 8 

+ 55 

28 

29 

20 48 43 

8 17 49 

+13 28 




0 32 37 

N 3 31 

+ 4 47 

29 

30 

20 52 49 

S 17 33 

+13 37 




0 36 15 

X 3 55 

+ 4 28 

30 

31 

20 £6 54 

S 17 16 

+ 13 46 




0 39 53 J N 418 

+ 410 

31 

Date 

April. | 

May. 

June. 

Date 

l 

0 43 32 

N 4 41 

+ 3 52t 

2 84 51 

N 15 11 

— 3 4 

4 37 41 

N 22 6 

— 2 27] 

1 

2 

0 4710 

N 5 4 

+ 3 34 

2 38 41 

N 15 29 

— 3 11 

4 41 47 

X 22 14 

— 2 18 

2 

3 

0 50 49 

N 5 27 

+ 3 16: 

2 42 31 

N 15 57 

— 318 

4 45 53 X 2222 

— 28 

3 

4 

0 54 27 

N 5 50 

+ 2 58 

2 46 21 

N 16 4 

— 324 

4 50 0 

X 22 29 

— 158 

4 

5 

0 58 6 

N 6 13 

+ 2 41 

2 50 12 

N 16 21 

— 3 29 

4 54 7 

N 22 36 

— 148 

5 

6 

1 145 

N 6 35 

+ 2 23 

2 54 4 

N 16 38 

— 3 34 

4 5S14 

N 22 42 

— 137 

6 

7 

l 5 25 

X 6 58 

+ 2 6i 

2 57 56 N 16 55 

— 3 39 

6 2 22 

X 22 48 

— 1 27 

7 

8 

19 4 

N 7 20 

+ 1 401 

3 1 49 

N 17 11 

— 3 42 

5 6 29 

X 22 53 

— 115 

8 

9 

1 12 44 

X 7 43 

+ 132 

3 5 42 

N 17 27 

— 3 46 

5 10 37 

X 22 58 

— 14 

9 

10 

116 24 

X 8 5 

+ 1 15 

3 9 36 

N 17 43 

— 3 48 

514 46 

N 23 3 

— 0 52 

10 

11 

120 4 

X 8 27 

+ 0 59 

3 13 30 

N 17 58 

— 3 51 

5 18 54 

N 23 7 

— 0 40 

11 

12 

1 23 44 

N 8 49 

+ 0 43 

3 17 25 

N 18 13 

— 3 52 

5 23 3 

N 2311 

— 0 28 

12 

13 

1 27 25 

X 911 

+ 0 27 

3 21 21 

N 18 28 

— 3 53 

5 27 12 

N 2315 

— 016 

13 

14 

131 6 

X 9 32 

+ 012 

3 25 17 

N 18 43 

— 3 54 

5 31 21 

N 2318 

— 04 

14 

15 

134 48 

N 951 

— 0 3 

3 29 13 

N 18 57 

— 3 53 

5 35 30 

N 23 20 

+ 09 

15 

16 

138 30 

N 10 15 

— 0 18 

3 3311 

N 19 11 

— 3 53 

5 39 39 

N 23 22 

+ 0 22 

16 

17 

142 12 

N 1036 

— 0 32 

3 37 9 

N 19 24 

— 3 51 

5 43 49 

N 23 24 

+ 0 35 

17 

18 

145 55 

N 10 57 

— 0 46 

3 41 7 

N 19 38 

— 3 50 

5 47 58 

N 2326 

+ 0 48 

18 

19 

1 49 38 

X 11 18 

— 0 59 

3 45 6 

N 19 51 

— 3 47 

5 52 8 

N 23 27 

+ 1 1 

19 

20 

l 53 21 

N 11 38 

— 112 

3 49 6 

X 20 3 

— 344 

5 5617 

N 23 27 

+ 114 

20 

21 

157 5 

N 11 59 

— 1 25 

3 53 6 

X 20 15 

— 3 40 

6 0 27 

X 23 27 

+ 127 

21 

22 

2 0 49 

N 1219 

— 1 37 

3 57 6 

X 20 27 

— 3 36 

6 4 37 

N 23 27 

+ 140 

22 

23 

2 4 34 

N 12 39 

— 1 49 

4 1 8 

N 20 39 

— 3 32 

6 8 46 

N 23 26 

+ 153 

23 

24 

2 8 20 

N 12 59 

— 20 

4 510 

N 20 50 

— 3 26 

612 50 

N 23 25 

+ 26 

24 

25 

2 12 5 

N 13 18 

— 211 

4 9 12 

X 21 1 

— 3 21 

6 17 5 

N 23 24 

+ 219 

25 

26 

2 15 52 

X 13 33 

— 2 21 

4 13 15 

N 21 11 

— 3 15 

6 21 15 |N 23 22 

+ 231 

26 

27 

2 19 39 

N 13 57 

— 2 39 

4 17 18 

X 21 22 

— 38 

6 25 24 

N 23 20 

+ 2 44 

27 

28 

2 23 26 

N 1416 

— 2 40 

4 21 22 

N 21 31 

- O 0 

6 29 33 

X 2317 

+ 2 56 

28 

29 

2 27 14 

N 14 34 

— 2 48' 

4 25 26 

N 21 41 

— 2 53 

6 33 42 

N 2314 

+ 38 

29 

30 

2 31 2 

N 14 53 

— 2 56 

4 29 31 

| X 2150 

— 2 45 

6 37 50 

X 23 10 

+ 3 20 

30 

31 




1 4 33 36 

N 21 58 

— 2 36 




i 31 



















































































Astronomical Almanac. 


661 




July 

• 

August,. 

September. 

| 

Date 

Rt. As 

Declin. 

Eq. tin. 

Rt. As. 

Declin. 

Eq. till. 

Rt. As. 

Declin. 

Eq.tlU 

Date 


h. ra. s. 

O f 

11). s. 

h. m. s. 

O \ 

m. s. 

h. m. «. 

O • 

m. s. 


1 

C41 58 

N 23 6 

+ 3 32 

8 46 42 

N 1757 

+ 62 

10 42 4( 

N 810 

— 012 

1 

2 

6 46 6 

N 23 2 

+ 3 44 

8 50 35 

N 17 42 

+ 5 58 

10 46 18 

N 7 49 

— 0 31 

2 

3 

6 5014 

N 22 57 

+ 3 55 

8 51 27 

N 17 26 

+ 5 54 

10 49 55 

N 7 27 

— 0 50 

3 

4 

654 22 

N 22 52 

+ 45 

8 58 18 

N 17 11 

+ 5 49 

10 53 32 

N 7 4 

— 110 

4 

5 

6 58 29 

N 22 47 

+ 416 

9 2 9 

X 16 51 

+ 5 43 

10 57 8 

N 6 42 

— 130 

5 

6 

7 2 35 

N 22 41 

+ 4 26 

9 5 59 

N 16 38 

-f* 5 06 

11 0 45 

N G 20 

— 150 

6 

7 

7 6 42 

N 22 34 

+ 4 36 

9 9 49 

N 16 21 

+ 5 29 

11 421 

N 5 57 

- 210 

T 

« 

8 

710 48 

N 22 28 

+ 4 45 

913 38 

N 16 4 

+ 5 22 

11 7 57 

N 5 35 

— 2 30 

8 

9 

7 14 53 

N 22 20 

+ 4 54 

9 17 26 

N 15 47 

+ 514 

11 11 33 

N 512 

— 2 51 

9 

10 

7 18 58 

N 22 13 

+ 5 3 

9 21 14 

N 15 29 

+ 55 

11 15 9 

N 4 49 

— 3 12 

10 

11 

7 23 3 

N 22 5 

+ 511 

9 25 1 

N 1512 

+ 4 55 

11 18 45 

N 4 27 

— 3 33 

11 

12 

7 27 7 

N 21 57 

+ 518 

9 28 47 

N 1154 

+ 4 46 

11 22 20 

N 4 4 

— 3 54 

12 

13 

73111 

N 21 48 

+ 5 26, 

9 3231 

N 14 35 

+ 4 35 

11 25 56 

N 3 41 

— 415 

13 

14 

7 35 14 

N 21 39 

+ 5 32] 

9 3619 

N 1417 

+ 424 

11 29 31 

N 318 

— 4 36 

14 

15 

739 17 

N 21 30 

+ 5 391 

9 40 4 

N 13 58 

+ 413 

11 33 6 

N 2 54 

— 4 57 

15 

16 

7 43 19 

N 21 20 

+ 5 14' 

9 49. 49 

N 13 39 

+ 41 

11 36 42 

N 2 31 

— 5 18 

16 

17 

7 47 21 

N 21 10 

+ 5 50 

9 47 33 

N 13 20 

+ 348] 

1140 17 

N 2 8 

— 5 39 

17 

18 

7 5123 

N 20 59 

+ 5 54] 

9 51 16 

N 13 1 

3 35 i 

11 43 53 

N 145 

— 60 

18 

19 

7 5521 

N 20 49 

+ 5 59 

9 54 59 

N 12 41 

+ 3 22 | 

11 47 28 

N 122 

— 6 21 

10 

20 

7 69 24 

N 20 37 

+ 6 2' 

9 58 42 

N 12 21 

+ 38 

[1151 4 

N 0 58 

— 6 42 

20 

21 

8 3 24 

N 20 26 

+ 6 5] 

10 2 24 

N 12 1 

+ 2 54 

1154 39 

N 0 35 

— 7 3 

21 

22 

8 7 23 

N 2014 

+ 6 8] 

10 6 6 

N 11 41 

+ 2 39 

115815 

N Oil 

— 7 24! 

22 

23 

81122 

N 20 2 

+ 6 10] 

10 9 47 

N 11 21 

+ 2 24 

12 1 51 

S 012 

— 7 45 

23 

24 

S 15 20 

N 19 49 

+ 612! 

10 13 2s 

Nil 0 

+ 2 8 

12 5 26 

S 0 35 

— 85 

24 

25 

8 19 17 

N 19 36 

+ 613 

10 17 8 

N 10 40 

+ 1 52 

12 9 3 

S 0 59 

— 8 26 

25 

26 

8 23 14 

N 19 23 

+ 613 

10 20 48 

N 10 19 

+ 135 

1212 39 

8 1 22 

— 8 46 

26 

27 

8 27 10 

N 19 10 

+ 613 

10 24 28 

N 9 58 

+ 1 18 

121615 

8 1 46 

— 96 

27 

28 

831 6 

N 18 66 

+ 6 12 

10 28 7 

N 9 37 

+ 1 1 

1219 62 

S 2 9 

— 9 26 

28 

29 

8 35 1 

N 18 42 

+ 611 

10 3146 

N 915 

+ 0 43 

12 23 29 

8 2 32 

— 9 46 

29 

30 

8 38 55 

N 18 27 

+ 69 

10 35 24 

N S 54 

+ 0 25 

12 27 6 

8 2 56 

—10 5 

30 

31 

I 8 42 49 

N 18 13 

+ 6 6] 

10 39 2 

N 8 32 

+ 07 




31 

Date 

Octobei 


November. 

December. 

Date 

1 

12 30 43 

8 3 19 

—10 24 

14 27 2 

S 14 33 

—1617 | 

(16 3057 

8 21 53 

—10 40 

1 

2 

12 34 20 

S 3 42 

—1043 

14 30 58 

S 14 52 

—1618 

10 35 17 

S 22 2 

—10 17 

2 

3 

12 37 58 

S 4 fi 

—11 2 

143154 

S 1511 

—1618 

16 39 37 

S 2210 

— 9 53 

3 

4 

12 41 37 

S 4 29 

—11 20 

44 38 52 

S 15 30 

—1G 17 

16 43 58 

S 2218 

— 9 19 

4 

5 

12 45 15 

S 4 52 

—11 38 

14 42 50 

8 15 38 

—1615 

116 48 20 

S 22 26 

— 94 

5 

6 

12 48 54 

S 5 15 

—11 56 

14 46 49 

S 16 6 

—1613 

16 52 42 

8 22 33 

— 8 38 

6 

7 

12 52 33 

S 5 38 

—1213 

-14 50 49 

S 16 24 

—1610 

16 57 4 

S 22 40 

— 813 

7 

8 

12 5613 

S 6 1 

—12 29 

(14 54 50 

S 1641 

—16 6 

17 127 

S 22 46 

— 7 46 

8 

9 

12 59 53 

S 6 24 

—12 46 

14 58 51 

S 16 59 

—16 1 

17 5 51 

8 22 52 

— 7 19 

9 

10 

13 3 34 

S 6 47 

—13 2 

15 254 

S 17 16 

—15 55 

17 1015 

8 22 58 

— 6 52 

10 

11 

13 715 

S 7 9 

—13 17 

15 6 57 

S 17 32 

-15-18 

17 1439 

S 23 3 

— 6 24 

11 

12 

13 10 57 

S 7 32 

—13 32 

1511 1 

8 17 48 

—15 40 

1719 4 

8 23 7 

— 5 56 

12 

13 

13 14 39 

S 7 54 

—13 46 

1515 C 

S 18 4 

—15-32 

[17 23 29 

S 23 12 

— 5 28 

13 

14 

13 18 22 

S 817 

—14 0 

151912 

S 18 20 

—15 22 

[17 27 54 

S 23 15 

— 4 59 

1< 

15 

13 22 5 

S 8 39 

—1413 

15 23 19 

S 18 36 

—15 12 

17 32 20 

8 2318 

— 4 30 

15 

16 

13 25 49 

S 9 1 

—14 26 

15 27 27 

S 18 31 

—15 1 

17 36 46 

8 23 21 

— 41 

16 

17 

13 29 34 

S 9 23 

—14 38 

15 31 35 

S 19 5 

—14 49 

17 4112 

S 23 23 

— 3 31 

17 

18 

13 33 T9 

S 9 45 

—14 49 

15 35 45 

S 19 20 

—14 36 

17 45 38 

S 23 25 

— 31 

18 

19 

13 37 4 

S 10 7 

—15 0 

15 39 55 

S 19 34 

—14 23 

17 50 5 

S 23 26 

— 2 32 

19 

20 

13 40 51 

S 10 28 

—1510 

15 44 6 

S 19 47 

—14 8 

17 54 32 

S 23 27 

— 22 

20 

21 

13 44 3S 

S 10 50 

—1519 

151818 

S 20 1 

—13 53 

17 58 58 

S 23 27 

— 131 

21 

22 

13 48 26 

S 11 11 

—15 28 

15 52 30 

S 20 14 

—13 37 

18 3 25 

S 23 27 

— 1 1 

22 

23 

13 52 14! 

S 1132 

—15 36 

1556 44 

S 20 26 

—13 20 

18 7 52 

8 1-3 27 

— 0 31 

23 

24 

13 56 3 

S 11 53 

—15 44 

16 0 58 

S 20 38 

—13 3 

IS 1218 

S 23 26 

— 01 

24 

25 

13 59 53 

S 12 14 

—15 51 

16 513 

S 20 50 

— 12 44 

18 10 45 

8 23 24 

+ 0 20 

25 

26 

14 3 43 

S 12 34 

—15 57 

16 9 29 

S 21 2 

—12 25 

18 2111 

S 23 22 

+ 0 5S 

26 

27 

14 7 35 

S 12 55 

—16 2 

1G13 45 

S 2113 

—12 5 

18 25 38 

8 23 20 

+ 1 28 

27 

28 

14 11 27 

S 13 15 

—16 7 

1618 2 

S 21 23 

—11 45 

IS 30 4 

S 2317 

+ 1 58 

28 

29 

14 15 19 

S 13 35 

—1610 

16 22 20 

S 21 34 

—11 24 

18 34 30 

S 2313 

+ 2 27 

29 

30 

14 1913 

S 13 55 

—1614 : 

16 26 38 

S 21 43 

—11 2 

18 38 55 

S 23 9 

+ 2 56 

30 

31 

14 23 7 

S 1414 

—1616 1 

1 



18 43 21 

S *23 5 

+ 3 24 

31 


























































































662 


Astronomy 


Corrections in Minutes and Seconds of Right Ascension, 
Declination and Equation of Time for Hours up to 18. 


Variations in Seconds for 24 Hours. 



o 

5 

G 

7 

8 

9 

10 

15 

20 

25 

30 

35 

40 

45 

50 

55 

Cb | 

P 


ft 

tr 

// 

// 

tf 

// 

It 

1/ 

ft 

tt 

ft 

tt 

It 

1/ 

ft 

i 

1 

0 

0 

0 

0 

0 

0 

1 

1 

1 

i 

1 

2 

2 

2 

2 

15 

2 

0 

l 

1 

1 

1 

1 

1 

2 

2 

3 

3 

3 

4 

4 

5 

30 . 

3 

1 

l 

1 

1 

1 

1 

2 

3 

3 

4 

4 

5 

6 

6 

7 

45 

4 

1 

l 

1 

1 

2 

2 

3 

3 

4 

5 

G 

7 

7 

8 

9 

GO 

6 

1 

l 

1 

2 

2 

2 

3 

4 

5 

G 

7 

8 

9 

10 

11 

75 1 

6 

1 

2 

2 

2 

2 

3 

4 

5 

G 

8 

9 

10 

11 

12 

14 

90 

7 

1 

2 

2 

2 

3 

3 

4 

6 

7 

9 

10 

12 

13 

16 

16 

105 

8 

2 

2 

2 

3 

3 

3 

5 

7 

8 

10 

12 

13 

15 

17 

18 

120 

9 

2 

2 

3 

3 

3 

4 

6 

8 

9 

11 

13 

15 

17 

19 

21 

135 

10 

2 

3 

3 

3 

4 

4 

G 

8 

10 

13 

15 

17 

19 

21 

23 

150 

11 

2 

3 

3 

4 

4 

5 

7 

9 

11 

14 

16 

18 

21 

23 

25 

1G5 

12 

3 

3 

4 

4 

5 

5 

8 

10 

13 

15 

17 

20 

22 

25 

27 

180 

13 

3 

3 

4 

4 

5 

5 

8 

11 

14 

1C 

19 

23 

24 

27 

30 

195 

14 

3 

4 

4 

5 

6 

6 

9 

12 

15 

18 

20 

23 

26 

29 

32 

210 

15 

3 

4 

4 

5 

6 

6 

9 

13 

16 

19 

22 

26 

28 

31 

34 

225 

10 

3 

4 

5 

5 

6 

7 

10 

13 

17 

20 

23 

27 

30 

33 

37 

240 

17 

4 

4 

6 

6 

6 

7 

11 

14 

18 

21 

25 

28 

32 

35 

39 

255 

18 

4 

5 

5 

6 

7 

8 

11 

15 

19 

23 

26 

oO 

34 

37 

41 

270 


Variation in Minutes for 24 Hours. 



fcC 


c 

HH 

1 

2 

3 

4 

5 

G 

7 

8 

9 

10 

20 

<D 

P 


I 

I 

// 

I 

It 

/ 

91 

/ 

ft 

/ 

If 

i 

1/ 

/ 

// 

/ 

// 

/ 

It 

/ 

ft 


1 

2 


6 


7 


10 


12 


15 


17 


20 


22 


25 


50 

15 

2 

5 


10 


15 


20 


25 


so 


35 


40 


45 


50 

1 

40 

30 

3 

7 


15 


22 


30 


37 


45 


62 

1 


1 


1 

15 

2 

30 

45 

4 

10 


20 


30 


40 


50 

1 


1 

10 

1 

20 

1 

30 

1 

40 

3 

20 

60 

5 

12 


25 


37 


50 

1 

2 

1 

15 

1 

27 

1 

40 

1 

52 

2 

5 

4 

10 

75 

6 

15 


30 


45 

1 


1 

15 

1 

30 

1 

45 

2 


2 

15 

2 

30 

5 


90 

7 

17 


35 


62 

1 

10 

1 

27 

1 

45 

2 

2 

2 

20 

2 

37 

2 

55 

5 

50 

105 

8 

20 


40 

1 


1 

20 

1 

40 

2 


2 

20 

2 

40 

3 


3 

20 

G 

40 

120 

9 

22 


45 

1 

7 

1 

30 

1 

52 

2 

15 

2 

37 

3 


3 

22 

3 

45 

7 

30 

135 

10 

25 


50 

1 

15 

1 

40 

2 

5 

2 

30 

2 

55 

3 

20 

3 

45 

4 

10 

8 

20 

150 

11 

27 


55 

1 

22 

1 

50 

2 

17 

2 

45 

3 

12 

3 

40 

4 

7 

4 

35 

9 

10 

166 

12 

30 

1 


1 

30 

2 


2 

30 

3 


3 

30 

4 


4 

30 

5 


10 


180 

13 

32 

1 

5 

1 

37 

2 

10 

2 

42 

3 

15 

3 

47 

4 

20 

4 

52 

5 

25 

10 

50 

195 

14 

35 

1 

10 

1 

45 

2 

20 

2 

55 

3 

30 

4 

5 

4 

40 

5 

15 

5 

50 

11 

40 

210 

15 

37 

1 

15 

1 

52 

2 

30 

3 

7 

3 

45 

4 

22 

5 


5 

37 

G 

15 

12 

30 

225 

1C 

40 

1 

20 

2 


|2 

40 

3 

20 

4 


4 

40 

5 

20 

6 


6 

40 

13 

20 

240 

17 

42 

jl 

26 

2 

7 

.2 

50 

I 3 

32 

4 

15 

4 

57 

5 

40 

6 

22 

7 

6 

14 

10 

255 

18 

45 

11 

3C 

\2 

15 

13 


1 3 

45 

4 

SO 

6 

15 

6 


G 

45 

7 

30 

15 


270 


Explanation of the Sidereal and Solar Time Table. 

The Sidereal Time = Mean Solar -f Correction. 

Mean Solar Time = Sidereal — Correction. 

To Find the True Sidereal Time. 

The sun’s Right Ascension -(-or — the Equation of time is the Sidereal time a 
Greenwich, mean noon. 

The sign -f or — must be used as noted in the Astronomical or Nautical Almana* 
for the given day. For any other meridian or longitude from Greenwich, cor 
rect the sun’s R. A. and the equation of time, and perform the same operation. 











































































Refraction of the Heavenly Bodies in Altitude. 


663 


Alt. 

Refr. 

Alt. 

Refr. 

Alt. 

Refr. 

Alt. 

Refr. 

Alt. 

Refr. 

Alt. 

Refr. 

D.M. 

M. S. 

D.M. 

M. S. 

D. M. 

M. S. 

D. M. 

M. a. 

D. 

M. S. 

D. 

M. S. 

0. 0 

33. 0 

2.30 

16.23 

G.30 

7.52 

12.20 

4.16 

30 

1.33 

60 

0,33 

0. 5 

32.11 

2.35 

16. 4 

6.40 

7.41 

12.40 

4. 9 

.31 

1.35 

61 

0 32 

1 0.10 

31.22 

2.40 

15.45 

6.50 

7.31 

13. 0 

4. 3 

32 

1.31 

62 

0.30 

0.15 

30.36 

2.45 

15.27 

7. 0 

7.21 

13.20 

3.57 

33 

1.28 

63 

0.29 

0.20 

20.50 

2.50 

15. 9 

7.10 

7.12 

13.40 

3.51 

34 

1.24 

04 

0.28 

0.25 

29. 6 

2.55 

14.52 

7.20 

7. 3 

14. 0 

3.46 

35 

1.21 

6) 

0.27 

0.30 

28.23 

3. 0 

14.35 

7.30 

6.54 

14.20 

3.40 

36 

1.18 

66 

0.25 

0.35 

27.41 

3. 5 

14.19 

7.40 

6.46 

14.40 

3,35 

37 

1.16 

67 

0.24 

0.40 

0.45 

27. 0 

3.10 

14.05 

7.50 

6.38 

15. 0 

3.30 

38 

1.13 

68 

0.23 

26.20 

3.15 

13.48 

8. 0 

6.30 

15.3T 

3.23 

39 

1.10 

69 

0.22 

0.50 

25.42 

3.20 

13.33 

8.10 

6.22 

16./0 

3.17 

40 

1. 8 

70 

0.21 

0.55 

25. 5 

3.25 

13.19 

8.20 

6.15 

16.30 

3.11 

41 

1. 5 

71 

0.20 

1. 0 

24.29 

3.30 

13.05 

S3) 

6. S 

17. 0 

3. 5 

42 

1. 3 

72 

0.19 

1. 5 

23.54 

3.40 

1239 

8 40 

6. 1 

17.30 

2.59 

43 

1. 1 

73 

0.17 

1.10 

23.20 

3.50 

12.14 

8.50 

5.55 

18. 0 

2.54 

44 

0.59 

74 

0.16 

1.15 

22.47 

4. 0 

11.50 

9. 0 

5.49 

18.30 

2.49 

45 

0.57 

75 

0.15 

1.20 

22.15 

4.10 

11.23 

9.10 

5.43 

19. 0 

2.44 

46 

0.55 

76 

0.14- 

1.25 

21.44 

4.2) 

11.07 

9.20 

5.37 

19,30 

2.40 

47 

0.53 

77 

0.13 

1.30 

21.15 

4.30 

10.47 

9.30 

5.31 

20. 0 

2.36 

48 

0,51 

78 

0.12 

1.35 

20.46 

4.40 

10.28 

9.40 

5.26 

20,30 

2.32 

49 

0.50 

79 

0.11 

1.40 

20.18 

4-50 

10.10 

9.50 

5.20 

21. 0 

2.23 

50 

0.48 

80 

0.10 

1.45 

19.51 

5. 0 

9.58 

10. 0 

5.15 

21.30 

2.24 

51 

0.4G 

81 

0. 9 

1.50 

19.25 

5.10 

9.37 

10.15 

5. 8 

22. 0 

2.20 

52 

0.45 

82 

0. 8 

1.55 

18.59 

5.20 

9.21 

10.30 

5. 0 

23. 0 

2.14 

53 

0.43 

83 

0. 7 

2. 0 

18.35 

5.30 

9. 7 

10.45 

4.54 

24. 0 

2. 7 

54 

0.41 

84 

0. 6 

2. 5 

18.11 

5.40 

8.53 

11. 0 

4.47 

25. 0 

2. 2 

55 

0.40 

85 

0. 5 

2.10 

17.48 

5.50 

8 39 

11.15 

4 41 

26. 0 

1.56 

56 

0.38 

86 

0. 4 

2.15 

17.26 

6. 0 

8.27 

11.30 

4.35 

27. 0 

1.51 

57 

0.37 1 

87 

0. 3 

2.20 

17. 4 

6.10 

8.15 

11.45 

4.29 

28. 0 

1.47 

58 

0,36 

SS 

0. 2 

2.25 

16.44 

6.20 

8. 3 

12. 0 

4.23 

20. 0 

1.43 

59 

0.34 | 

S9 

0. 1 


Conversion of Sidereal & Mean Solar Times. 


Hour. 

Oorr. 

1 Mia 

Corr. 

Min. 

Corr. 

Sec. 

Corr 

See. 

j Corr. 

H. 

m. a. 

M. 

S. 

M. 

R. 

S. 

S. 

S. 

S. 

1 

0 9.8 

1 

0.2 

31 

5.1 

1 

0.0 

31 

0.1 

2 

0 19.7 

2 

0.3 

32 

5.2 

2 

0.0 

32 

0.1 

3 

0 29.5 

3 

0,5 

33 

5.4 

3 

0.0 

33 

0.1 

4 

0 39.3 

4 

0.7 

34 

5.6 

4 

0.0 

34 

0.1 

5 

0 49.1 

5 

0.8 

35 

5,7 

5 

0.0 

35 

0.1 

6 

0 59.0 

6 

1.0 

36 

5.9 

6 

0.0 

35 

0.1 

7 

1 8.9 

7 

1.1 

37 

G.l 

7 

0.0 

37 

0.1 

8 

1 18.7 

8 

1.3 

38 

6.2 

8 

0.0 

38 

0.1 

9 

1 28.6 

9 

1.5 

39 

6.4 

9 

0.0 

39 

0.1 

10 

1 38.4 

10 

1.6 

40 

6.6 

10 

0.0 

40 

0.1 

11 

1 48.2 

11 

1.8 

4L 

6.7 

11 

0.0 

4L 

0.1 

12 

1 58.1 

12 

2.0 

42 

6.9 

! 12 

00 

42 

0.1 

13 

2 8.0 

13 

2.1 

43 

7.0 

13 

0.0 

43 

0.1 

14 

2 17.8 

14 

2.3 

44 

7.2 

14 

0.0 

44 

0.1 

15 

2 27 6 

15 

2.5 

45 

7.4 

15 

0.0 

45 

0.1 

16 

2 37.5 

16 

2.6 

46 

7.5 

16 

0.0 

46 

0.1 

17 

2 47.3 

17 

2.8 

47 

7.7 

17 

0.0 

47 

0.1 

IS 

2 57.1 

18 

2.9 

48 

7.9 

18 

0.0 

48 

0.1 

19 

3 7.0 

19 

3.1 

49 

8.0 

19 

0.1 

49 

0.1 

20 

3 16.9 

20 

3.3 

50 

8.2 

20 

0.1 

50 

0.1 

21 

3 26.7 

21 

3.4 

51 

8.4 

21 

0.1 

51 

0.1 

22 

3 36.5 

22 

3.6 

52 

8.5 

22 

0-1 

52 

0.1 

23 

3 46.4 

23 

3,8 

53 

8.7 

23 

0.1 

53 

0.1 

24 

3 56.3 

24 

3.9 

54 

8.8 

24 

0.1 

51 

0.1 



25 

4.1 

55 

9.0 

25 

0.1 

55 

0.2 



26 

4.3 

56 

9.2 

26 

0.1 

56 

0.2 



27 

4.4 

57 

9.3 

27 

0.1 

57 

0.2 



28 

4.6 

58 

9.5 

28 

0.1 

53 

0.2 



29 

4.8 

59 

9.7 

29 

0.1 

59 

0.2 



30 1 

4.9 

GO 1 

9.8 

30 1 

01 

60 1 

0.2 


The Sun’s Parallax 
in Altitude. 


Altitude. 

D. 

0 

10 

20 

30 

40 

50 

&o 

60 

65 

70 

75 

80 

85 

00 


Parallax. 

5 

9 

9 

8 

8 

7 

6 
5 
4 
4 
3 
2 
2 
1 
0 


Explanation. 

The sun’s parallax 
must be added to the 
observed altitude. 





































































6G4 


Astronomy. 


LATITUDE AND APPARENT TIME 

By Altitude of the Heavenly Bodies . 

Notation of Letters. 


A = meridian altitude above horizon. 

D — declination, to be found in the Astronomical Almanac. 

I = latitude of the place of observation. 

L = angle of apparent time from noon. 

a = any altitude of the heavenly body, before or after noon. 


When the latitude and declination are of 

Same Name. 

Latitude, 1 = 90 + Z> — A. 
Altitude, A = 90 -f D — l. 
Declination, D — A + 1 — 90. 

Apparent Time, 

Cos.Z = sin.a sec.1 sec JO — tan J tan ,D. 


When the latitude and declination are of 

Different Names. 

Latitude, l — 90 — A — D. 
Altitude, A =90— l — D. 

Declination, D = 90 — A — l. 

Apparent Time, 

Cos.Z = sin.a seci sec.D + tan.Z tan.Z>. 


At sea the altitude is observe<l from the visible horizon of the ocean, from which 
must be subtracted the dip of horizon. (See table, page 131.) 


On land the horizon must be determined by a spirit-level, or more correctly by 
an artificial horizon of quicksilver, oil, syrup or some similar liquid. 

The refraction of light through the atmosphere (see table, page503) must also bo 
subtracted from the observed altitude. 

When the sun or moon is observed, the parallax must be added to the observed 
altitude. 

Latitude. 

Example 1. On the 7th of April, 1872, the sun’s lower limb was observed to be 
51° 42' 50" above the horizon at noon, in longitude about 45° west of Greenwich; 
the observation was made from the deck of a vessel 20 feet above the sea. Required, 
the sun's true altitude and latitudo of observation? 


The declination and latitude are both north or of same name. 


Dip of horizon for 20 feet, 4' 24" 

Refraction, 51°, .... 46 

0 ’s semi-diameter, . . . 16 00 

21 10 

Sun’s parallax, subtract . 6 

Correction to be subtracted, 21 4 

O’s observed altitude, . . 5 1 42 50 

G’s tree altitude, . . . 51° 21' 46" 


Declination Nant. Almanac, 7° 3'19" 
Correct. 45° W. long., add 2 48 

True declination, . . D= 7 6 7 

Add. 90 

97 6 7 

True altitude, subtract A = 51 21 46 
The required latitude, l — 45° 44' 21" 


Artificial Horizon. 

When the observation is made by a sextant through an artificial horizon, the 
observed angle must be divided by 2 for the altitude, and there will bo no correc¬ 
tion for dip of horizon, nor for semi-diameter, as the sun’s discs are brought to 
cover one another. When a regular quicksilver horizon is not at hand, some 
viscous liquid, like oil or molasses, in an open vessel, may be used in calm wjeather. 

In perfectly calm weather the altitude may be taken in a pool of water, which 
has been done by the author in South America. 














Astronomy. 


665 


Apparent Time. 

Example. On the 8th of February, 1872, the sun’s true altitude was found to be 
a = 30° 46' 29" in the afternoon, the latitude 2 = 38° 18' 38" N., and declination 
corrected D —15° 5' 10" S. liequired, the apparent and mean time of obser¬ 
vation ? 

The latitude and declinations are of different names. 

Cos L = sin.30° 46' 29" X sec. 38° 18' 38" X sec.l5° 5' 10" — 0.675349 

+ tan. 38° 18' 38" X tan.l5° 5' 10" = 0.21293 
Apparent time of obs., L — lh. 10m. 23s. = cos. 27° 20' 34" = 0.888279 
Equation of time, add 14 26 

Mean time, 2h. 3 m. 58s. 

The calculation with logarithms is set up as follows: 

Log. sin.30° 46' 29" = 9.70898 
“ sec.38 18 38 = 0.10531 Tan. = 9.89765 

“ “ 15 5 10 = 0.01523 Tan. = 9 .43067 

Logarithms, . . . 9.82952 9.32832 

Natural numbers, . 0.675349 0.21293 

Add for different names, 0,21293 

App. time, L — lh. 49 m. 22s. *= 27° 20' 34" = 0.888279 = cosine for hour angle. 

The hour angle in time can be read off directly for the cosine in the trigono¬ 
metrical tables. 

When the observation is made in the forenoon, subtract the cosine hour angle 
from 12 hours; or it can be read off directly from the tables by calling cosine for 
sine L. 

To Find tlie Longitude. 

Small differences in longitude can be obtained from actual measurement, as 
explained in Plane Sailing and Traverse Surveying. 

For great distances, it is necessary to know the simultaneous times at the two 
meridians between which the longitude is required. 

At sea, the time at a distant meridian is kept by a chronometer, generally regu 
lated for Greenwich mean time, and the difference in time between the two merid 
ians is the difference in longitude. 

Eclipses of Jupiter’s Satellites. 

The most simple astronomical observations for finding the Washington or Green¬ 
wich mean time are of the eclipses of Jupiter’s satellites, but, unfortunately, the 
tables for those eclipses, which are published in the English and American Nautical 
Almanacs, are not yet reliable, as has been found by the author in using these 
tables in the interior of South America. The observations of several eclipses in 
one locality dad not give the same difference in longitude. Pi-of. S. Newcomb says, 

“ The times of the eclipses of Jupiter’s satellites differ greatly in accuracy. Those 
of the first satellite are generally correct within the necessary errors of observa¬ 
tion ; the errors are larger with each succeeding one as you go out, and in the case 
of the outer one they are frequently several minutes in orror. The labor of con¬ 
structing tables is so great that no one person can undertake it. It would require 
many years’ labor.” 

This is a subject well worthy of attention at the National Observatory at Wash¬ 
ington. The eclipses of Jupitor’s satellites can be observed by an ordinary good 
spy-glass, even at sea, in calm weather, which would be of great importance over 
the whole world if accurate tables could be procured. Chronometers could then 
be corrected with great precision to Greenwich or Washington mean time in any 
part of the world, and difference of longitude could be determined without the aid 
of complicated, bulky, and expensive instruments, which few can afford to buy or 
know how to use. 

When the Pacific coast of South Amei-ica was first surveyed, the longitude was 
determined by the average time of a dozen chi-onometers, which gave very incor- 
i-ect results, as was afterward found, whilst the eclipses of jupitor’s satellites would 
have given tho longitude correct even if the tables erred, because the same eclipses 
could be observed in regular observatories at the same time, and proper corrections 
made from compared data. 

The Nautical Almanac is prepared some three years in advance, and it therefore 
requires very accurate data to locate the time of tho eclipses with pi'ecision. 








666 Astronomy. 


Elements of tile Planetary System. 


The 

eo 

Mean 

Sidereal 

Recent. 

Diara- 

Vel orbt 

Rota- 




principal 

C 

tv 

distance 

period, 

part, 

eter in 

Miles 

tion in 


Mass. 

Volume. 

Planets. 

5J 

fr. Sun. 

Daj's, 

sm. axis 

miles. 

per sec. 

hours. 




Sun.. . 

o 




882000 

• • 

607 48 

0.25 

355000 

1378000 

Mercury, 


0.3871 

87.969 

0.2055 

3140 

30.4 

24 05 

1.12 

0.00966 

0.062IS 

Venus, 


0.7233 

224.70 

0.006S 

7800 

22.3 

23 21 

0.92 

0.877 

0.9531 

Earth, . 

© 

1. 

365.25 

0.0168 

7912 

18.9 

24 0 

1. 

1. 

1. 

Mars, . 

cf 

1.5236 

686.08 

0.0933 

4100 

15.33 

24 37 

0.95 

0.1313 

0.13S4 

Jupiter, 

% 

5.2028 

4332.6 

0.0482 

87000 

8.31 

9 56 

0.24 

317.5 

1322.5 

Saturn, 

h 

9.5388 

10759 

0.0561 

79160 

614 

10 29 

0.14 

139.5 

996.2 

Uranus, 

V 

19.182 

30687 

0.0467 

34500 

4.33 

9 30 

0.24 

198. 

82.47 

Neptune, 


30.037 

60127 

0.0087 

41500 

3.45 

• * 

0.14 

20. 

143.5 


Position of some Stars of tlie 1st and 

Tide Table. 


Rise 

2d Ma 

grnitudes, dan. 1 

1885. 


h. m. 

feet. 

Name of Star. 

Mg. 

td. 

Right 

As’sion. 

Ann. 

Var. 

Decli¬ 

nation. 

Ann. 

Var. 

Albany, N. Y., 
Altona, Ger., 
Amboy, N. Y., 
Antwerp, Bel., 

— 0 40 
+ 3 12 
+ 5 27 
+ 2 18 









N. Hemifcpliere. 




North. 


Baltimore, 

— 1 52 

12 


h. m. s. 

s. 

O / " 

tt 

Belfast, Ire., 

+ 8 36 

a Andromedse, 

2 

0 2 27 

+ 3.09 

28 27 20 

+ 19.9 

Bergen, Nor., 

— 0 37 


a Polaris, 

2 

1 16 37 

+ 20.7 

88 41 44 

+ 19.0 

Bordeaux, Fr., 

+ 4 34 


a Ariel is, 

2 

2 0 41 

+ 3.36 

22 55 05 

+ 17.2 

Boston, Mass., 

+ 9 20 

12 

a Persi, 

2 

3 16 07 

+ 4.25 

49 27 03 

+13.1 

Boulogne, Fr., 
Buenos Ayres, 

+ 9 18 

19 

a Aldebaran, 

1 

4 29 19 

+ 3.43 

16 16 37 

+ 7.61 

+ 66 


a Capella, 

1 

5 8 12 

+ 4.42 

45 52 46 

+ 4.13 

Bremen, 

+ 8 43 


/3 Tauri, 
afi Castor, 

2 

5 19 01 

+ 3.79 

28 30 32 

+ 3.42 

Cadiz, Spain, 

— 0 22 


2.1 

7 27 16 

+ 3.84 

32 8 33 

— 7.48 

Calais, Fr., 

+ 9 42 

19 

a Proeyon, 

1 

7 33 17 

+ 3.14 

5 31 8 

— 8.97 

Calc'tta.Beng. 

+ 0 23 


)3 Pollux, 
a Regains, 

1.2 

7 38 17 

+ 3.68 

28 18 10 

— 8.35 

Charleston, 

+ 5 44 

5 

1.2 

10 2 15 

+ 3.20 

12 31 42 

— 17.4 

Cherbourg, Fr. 

-j- 8 23 

20 

y l Leonis, 

2 

10 13 38 

+ 3.31 

20 25 22 

— 18.0 

Capellenry,A. 

+ 73 

4 

a Great Rear, 

2 

10 56 37 

+ 3.76 

62 22 18 

— 19.4 

('ape G. Hope, 

+ 0 29 


y Great Bear, 


11 47 47 

+ 3.19 

54 20 02 

— 2O.0 

Cape Horn, 

+ 29 

9 

7) Great Rear, 

2 

13 43 05 

+ 2.37 

49 53 15 

— 18.1 

C. llenlopen, 

+ 6 41 

5 

a Arcturus, 

1 

14 10 25 

+ 2.73 

19 46 53 

— 18.8 

Dublin Bar. 

+ 9 5 

12 

a Coronae, 

2 

15 29 49 

+ 2.54 

27 6 08 

— 12.3 

Gibraltar, Sp., 

+ 0 13 


7] 11 erc.nl is, 

3.2 

16 38 57 

+ 2.26 

39 8 29 

— 6.69 

G lasgow, Scot. 

— 0 42 


a. 1 Ilerculis, 

Var. 

17 9 24 

+ 2.7(3 

14 31 20 

— 4.39 

i I amburg, 

+ 3 22 


a Vega, 

1 

18 33 03 

+ 2.03 

38 40 38 

+ 3.13 

Halifax, N. A., 

+ 5 42 

8 

a Altair, 

1.2 

19 45 10 

+ 2.93 

8 33 55 

+ 9.22 

Havana, Cuba, 

+ 7 41 

3 

S. Hemisphere. 



South. 


Havre, Fr., 
Hull, Eng., 

+ 7 44 
+ 4 22 

12 

18 

/3 Ceti, 

2 

0 37 49 

+ 3.01 

13 37 05 

+ 19.8 

Key West,U.S. 

+ 7 22 

2 

a Achernar, 

1 

1 33 25 

+ 2.23 

57 49 17 

+ 18.4 

Lisbon B. Port. 

+ 0 13 


/3 Higel, 

1 

5 9 06 

+ 2.88 

8 20 08 

+ 4.45 

| Liverpool, 

+ 9 16 

25 

5 Orionis, 

2 

5 26 08 

+ 3.06 

0 23 07 

+ 2.96 

New York, 

+ G 6 

5 

a Canopus, 

1 

6 21 24 4- 1.33 

52 37 59 

— 1.85 

i New Haven,A. 

+ 8 38 

17 

a Sirius, 
s Canis Major, 

1 

6 40 05 

+ 2.64 

16 33 33 

— 4.69 

1 Newcastle, E. 

+ 2 16 


2.1 

6 54 06 

+ 2.36 

28 48 59 

— 4.67 

l Norfolk, U. S., 

+ 5 14 

7 

i Argus, 

2 

9 14 05 

+ 1.60 

58 47 33 

— 14.9 

Ostend, Belg., 

+10 18 

16 

a Hydras, 

2 

9 21 56 

+ 2.94 

8 9 38 

— 1‘>.4 

Panama, 

+ 1 24 

24 

a 1 Crucis, 

1 

12 20 12 

+ 3.27 

62 27 42 

— 19.9 

Philadelphia, 

— 0 49 

C 

a Spica, 

1 

13 19 08 

+ 3.15 

10 33 39 

— 18.9 

Portsmouth, 



/3 Ceu tauri, 

1 

13 55 43 

+ 4.16 

59 49 03 

— 17.6 

Eng. & IJ. S.. 

+ 9 28 

10 

a- Centanri, 

1 

14 31 49 

+ 4.04 

60 21 46 

— 15.0 

Providence, 

+ 5 25 


/3 Librae, 

2 

15 10 49 

! + 3.22 

8 57 28 

— 13.5 

i Quebec, Can., 

+ 4 31 

17 

/3 1 Soorpii, 

2 

15 58 45 

1 + 3.48 

19 29 23 

— lu.2 

Queenstown. 

+ 2 54 


a An tares. 

1.2 

16 22 21!+ 3.67 

26 10 33 

— 8.37 

Rio Janeiro, 

+ 0 53 

6 

a Australis, 

2 

16 36 30 

+ 6.28 

68 48 52 

— 7.32 

Rotterdam, 

+ 1 38 


a Pavonis, 

2 

20 16 331 + 4,79 

57 6 08 

+ 11.1 

j Sandy Hook, 

+ 4 58 

6 

/3 Aquarii, 

3 

21 25 30 

1+3.16 

6 4 36 

+ 15.6 

| Valparaiso, 

+ 7 2 

5 

a Cm is. 

2 

22 0 59 

f 3.81 

47 31 02 

+ 17.2 

San Francisco, 

+ 8 27 

6 

a Fomalhaut, 

1.2 

22 51 181+.3.33 

30 33 53 

+ 19.0 

Washington, 

!+ 1 58 






























































Astronomy. 


667 


To Find the Meridian or True North by the North Star, 

JPolans. 

Polaris is not in the true north, but revolves in a circle of radius 1° 22' 24" 
co-declination on the 1st of January, 1872, which 
diminishes 19" every year; that on the 1st of Janu¬ 
ary, 1873, its co-declination will be 1° 22' 5". 

The position of Polaris is generally traced by the 
direction of the stars a and /3, in the Great Bear, 
which point nearly to the North Star. See figure. 

Polaris passes the meridian, or is true north when 
the star e in the Great Bear is perpendicular either 
over or under Polaris. In low latitudes the Polaris 
is near the horizon, and the star e cannot be seen 
when tinder, but must be observed at its upper 
transit. When the star e is horizontal with Polaris, 
substract the radius of the circle, and the remainder 
will be the true north, from which the variation of 
the compass is ascertained. There is no star near 
the South Pole from which a similar observation can 
be made. 


Table Showing how Much Earlier any Fixed Star Passes 

the Meridian, rises or sets, in number of days or nights up to 100. 


CD 


SO 



CD 


CD 


tn 







Xi 


S 




to 

£5 

H.M. S. 

. fc£) 

£ 

II. M. S. 


to 

2 

H.M. S. 

to 

ii 

H.M. S. 

to 

2 

H.M. S. 

l 

0 3 55.9 

11 

0 43 15.0 


21 

1 22 34.1 

31 

2 01 53.2 

45 

2 56 55.0 

2 

0 7 51.8 

12 

0 47 10.9 


22 

1 26 30.0 

32 

2 05 49.1 

50 

3 16 35.9 

3 

0 11 47.7 

13 

0 51 06.8 


23 

1 30 25.9 

33 

2 09 45.0 

55 

3 36 15.0 

4 

0 15 43.6 

14 

0 55 02.7 


24 

1 34 21.8 

34 

2 13 40.9 

60 

3 55 54.5 

5 

0 1!) 39.5 

15 

0 58 58.6 


25 

1 38 17.7 

35 

2 17 36.8 

65 

4 15 34.1 

6 

0 23 35.5 

16 

1 02 54.5 


26 

1 42 13.6 

36 

2 21 32.7 

70 

4 35 13.6 

7 

0 27 31.4 

17 

1 06 50.5 


27 

1 46 09.6 

37 

2 25 28.6 

75 

4 54 53.2 

8 

0 31 27.3 

18 

110 46.4 


28 

1 50 05.5 

38 

2 29 24.5 

80 

5 14 32.7 

9 

0 35 23.2 

19 

1 14 42.3 


29 

1 54 01.4 

39 

2 33 20.4 

90 

5 53 51.8 

10 

0 39 19.1 

20 

1 18 38.2 


30 

1 57 57.3 

40 

2 37 16.4 

100 

6 33 10.9 


The preceding table is for regulating a watch, clock or chronometer. The fixed 
stars set 3 m. 55.909 s. earlier every day, and by observing the time of setting 
over a sharp, distant object, as a hill, mountain or a house, the time-keeper can be 
regulated with great precision. 

Example. A fixed star is observed to set at 9 h. 35 in. 51 s. 

Twenty-five days after, the same star set at 7 h. 66 m. 49 s. 

Add the correction for 25 days, . . . 1 38 17.7 

Sum,. 9 35 06.7 

The time-keeper has lost 51 — 06.7 = 44.3 seconds in 25 days, or 1.772 seconds per 
day. 

To Find the Time When Any Stai* or Planet Passes the 

Meridian. 

Subtract the sun’s right ascension from that of the star, increased by 24 f neces¬ 
sary, and the remainder will be the apparent time when the star passes the 
meridian. The sun's K. A. must be corrected for longitude from Greenwich, and 
for time of observation. This is the best mode of finding the meridian and varia¬ 
tion of the compass, but the apparent time must be correctly known. 

To Find Which Star Passes the Meridian Near a Desired Time. 

Add the sun’s R. A. to the desired hour, and the sum will be the nearest R. A. 
of the star passing the meridian at that time. Reject 24 hours if necessary. Find 
in the table of stars the one which comes nearest to that R. A. 

























668 


High Water. 


TO APPROXIMATE THE TIME OF HIGH WATER. 

On account of the Moon’s orbit being an ellipse, in which the Earth is one of the 
foci, and that the major axis of that ellipse does not point to the Sun, Hut to a 
lixed constellation of stars, the actual time of high water and also that of passing 
the meridian are not equal for equal age l>f the Moon, but may differ as much as 15- 
minutes from the average in the accompanying table. Also, the force and direc-. 
tion of winds cause a still greater variation. 

Find first the Moon’s age for the given day, as described on page 496. Opposite 
the age in the table is the time of the day when the Moon is south, or passes the 
meridian, and in the following columns are the times of high water at London 
Bridge in the morning and afternoon. Add or subtract the time in the tide table, 
page 606, for any other location, and the sum or difference is the time of high water 
at that place. 


Moon. 


Quart’r 


New. 




•s 


Half. 

to 

e. 




CO 


Full. 

CO 

Q- 


co 


Half. 

& 

5 

►a 

g 

-I 

r+ 

<x> 

■-s 


Face 


High water at 
London Bridge. 


Age 

d. 

h. 

South. 

m. 

Age. 

Morning, 
h. m. 

Afternoon 
h. m. 

0 

12 

0p 

. m. 

0 

1 

59 

2 

7 

1 

12 

49 


1 

2 

21 

2 

36 

2 

1 

38 

it 

2 

2 

50 

3 

3 

3 

2 

26 

U 

3 

3 

18 

3 

33 

4 

3 

26 

it 

4 

3 

47 

4 

2 

5 

4 

4 

a 

5 

4 

16 

4 

31 

6 

4 

55 

a 

6 

4 

50 

5 

5 

7 

6 

42 

a 

7 

5 

24 

5 

45 

8 

6 

30 

u 

8 

6 

7 

6 

35 

9 

7 

19 

(( 

9 

7 

7 

7 

46 

10 

8 

8 

a 

10 

8 

33 

9 

25 

11 

8 

57 

a 

11 

10 

14 

10 

54 

12 

9 

46 

a 

12 

11 

28 

11 

54 

13 

10 

34 

a 

13 

Noon. 

0 

17 

14 

11 

23 

u 

14 

0 

40 

1 

0 

15 

12 

12 a 

in 

15 

1 

20 

1 

40 

16 

1 

1 

a 

16 

2 

1 

2 

22 

17 

1 

50 

a 

17 

2 

42 

3 

5 

18 

2 

38 

a 

18 

3 

26 

3 

48 

19 

3 

27 

a 

10 

4 

10 

4 

34 

20 

4 

16 

a 

20 

4 

55 

5 

19 

21 

5 

5 

a 

21 

5 

44 

6 

12 

22 

5 

54 

a 

22 

6 

44 

7 

21 

23 

6 

42 

a 

23 

7 

59 

8 

43 

24 

7 

31 

u 

24 

9 

31 

10 

15 

25 

8 

20 

a 

25 

10 

52 

11 

23 

26 

9 

9 

a 

26 

11 

49 

M’night. 

27 

9 

58 

a 

27 

0 

11 

0 

29 

28 

10 

46 

u 

28 

0 

4S 

1 

5 

29 

11 

35 

a 

29 

1 

23 

1 

36 

29i 

12 


;< 

29* 

1 

51 

2 

7 


High Water. 

Examples. 

Required, the time of 
II. W. in Philadelphia on 
the 8tli of Feb., 1874? 

Kpact, year, . 12 pg. 497. 
Epact, month, 2pg. 496. 
Date, February, 8_ 

Moon’s age, . 22 days. 

II.W.Lond., 6h.44m. 

Phi I a. subt., 0 49 pg, 506. 


H.W.Phila., 5ii.55m. 
in the morn., Feb. 8,1874. 

Required, the time of 
II. W. in Panama on the 
7th of October, 1873? 

Epact, year, 1 pg. 497, 
Epact, month, 8 pg. 496. 
Date in October/?_ 

Moon’s age, 16 days. 

II.W. Lohd., 21i. lm.a.m. 
Panama add 1_24 . pg. 50 6 

II.W. 


3h. 25m. a.m. 


in Panama. 


Elements of .Jupiter’s Satellites. 


Order 

Radius of 

Time in days 

Revolutions 

Mass, 

Diameter 

of 

orbit, that 

of one 

ar. Jupiter 

that of 

of satellite 

satellite. 

of Jupiter=1 

revolution. 

per vear. 

Jupiter=l 

miles. 

1st. 

60-185 

1.7691 

206.457 

0.000017 


2d. 

9.6235 

5.5512 

65.7952 

0.000028 


3.1. 

15.3502 

7.1546 

51.0499 

0.006088 


4th. 

26.9983 

16.68SS 

21.8865 

0.000043 



Number of Moons or Satellites to each Planet. 

Earth,]. Jupiter, 4. Saturn, with rings, 8. Uranus, 8 moons. 









































Soundings, 


669 


SOUNDINGS. 

To Reduce Soundings to Low Water. 

Letters denote — 

T— time in hours between high and low water. 

t — time in hoilrs from low water to the time when the soundings are taken. 
R r= vertical rise of tide in feet from high to low water, 
r = reduction of the sounding taken at the time, t , in feet. 

180 * 


v = 


T 


and r = \R (1 cos.v), 


— cos.v when v < 90 
-|- cos.v when v > 90 

Example. High water at 10 h. 15 m. p. 

Low water at 3 h. 45 m. ' 


m. 


Time T— 6 h. 30 m. “ 

The sounding taken at 5 h. 30 m. “ was 16 feet 6 inches. 

Timet = 1 h. 45 m. 

Vertical rise R = 9.75 feet. 

Required, the reduction r — ? and true sounding at low water? 

v = 180 * = 48° 27', cos.v = 0.0631. 

6.5 ’ 

Reduction r = JX 9.75 (1 — 0.6631) == 1.6419 feet. 

Sounding taken at 5 h. 30 m. was 16.5 feet. 

Reduction subtract r = 1.6419 

True sounding at low water, 14.8581 feet. 

Reduction for Soundings to Ijow Water. 

This table will answer for any unit of measure of rise. 


Rise 

Time of sounding in brs. and min. 

before or after that of high water. 

Rise 

R. 

0.30 

1 

1.30 

2 

2.30 

3 

3.30 

4 

4.30 

5 

1 5.30 

6 

R. 

1 

0.98 

0.94 

0.87 

0.78 

0.67 

0.55 

0.43 

0.31 

0.20 

0.12 

0.05 

0.01 

1 

2 

1.97 

1.88 

1.74 

1.56 

1.31 

1.10 

0.86 

0.62 

0.40 

0.24 

0.10 

0.02 

2 

3 

2.95 

2.82 

2.61 

2.34 

2.01 

165 

1.29 

0.93 

0,60 

0.36 

0.15 

0.03 

3 

4 

3.93 

3.76 

3.48 

3.12 

2.68 

2.20 

1.72 

1.25 

0.82 

0.46 

0.20 

0.04 

4 

5 

4.92 

4.70 

4.35 

3.90 

3.35 

2.71 

2.15 

1.56 

1.03 

0.5S 

0.25 

0-05 

5 

6 

5.91 

5.65 

5.22 

4.68 

4.03 

3.30 

2.58 

1.87 

1.23 

0.69 

0.30 

0.06 

6 

7 

6.90 

6.59 

6 10 

5.46 

4.70 

3.S4 

3.01 

2.18 

1.44 

0.81 

0.35 

0.07 

7 

8 

7.88 

7.52 

6.97 

6.24 

5.36 

4.40 

3.41 

2.50 

1.65 

0 93 

0.40 

0.08 

8 

9 

8.86 

8.47 

7.81 

7.02 

6.03 

4.94 

3.87 

2.80 

1.85 

1.04 

0.45 

0.09 

9 

10 

9.85 

9.41 

8.71 

7.79 

6.71 

5.52 

4.30 

3.12 

2.06 

1.16 

0.50 

0.10 

10 

11 

10.9 

10.3 

9.59 

8.59 

7.39 

6 05 

4.74 

3.43 

2.27 

1.28 

0.55 

0.11 

11 

12 

11.9 

11.3 

10.5 

9.37 

8.06 

6.60 

5.16 

3.74 

2.47 

1.40 

0.60 

0.12 

12 

13 

12.8 

12.2 

11.3 

10.1 

8.72 

7.14 

5.60 

4.05 

2.6S 

1.51 

0.65 

0.13 

13 

14 

13.8 

13.2 

12.2 

11.0 

9.40 

7.70 

6.02 

3.36 

2.89 

1.62 

0.70 

0.14 

14 

15 

14.8 

11.1 

13.0 

11 7 

10.0 

7.25 

6.15 

3.67 

3.09 

1.74 

0.75 

0.15 

15 

16 

15.8 

15.0 

14.0 

12.5 

10.7 

8.78 

6.88 

5.G0 

3.30 

1.S5 

0.80 

0.16 

16 

17 

16.8 

16.0 

14.8 

13.3 

11.4 

9.35 

7.31 

5.25 

3.50 

1.97 

0.85 

0.17 

17 

18 

17.8 

17.0 

15.7 

11.0 

12.1 

9.90 

7.75 

5.60 

3.70 

2 08 

0.90 

0.18 

18 

19 

18 7 

17.9 

16.6 

11.8 

12.8 

10.4 

8.17 

5.91 

3.81 

2.20 

0.95 

0.19 

19 

20 

19.7 

18.9 

17.5 

15.6 

13 4 

11.0 

8.60 

6.23 

4.11 

2.32 

1.00 

0.20 

20 

21 

20.7 

19.8 

18.3 

16.4 

14.1 

11.5 

9.04 

6.54 

4.32 

2.43 

1.05 

0.21 

21 

22 

217 

20.7 

19.2 

17.2 

14.8 

12.1 

9.16 

6.85 

4.53 

2.55 

1.10 

0.22 

22 

23 

22.7 

21.7 

20.0 

18.0 

1.5.4 

12.6 

9.90 

7.16 

4.73 

2.67 

1.15 

0.23 

23 

24 

23.7 

22.6 

20.9 

18.7 

16.1 

13.2 

10.3 

7.47 

4.94 

2.78 

1.20 

0.24 

24 

25 

21.7 

23.5 

21.8 

19.5 

16.8 

13.7 

10 8 

7.78 

5.14 

2.90 

1.25 

0.25 

25 

26 

25.6 

21.5 

22.7 

20.3 

17.4 

14.3 

11.2 

8.10 

5.52 

3.01 

1.30 

0.26 

26 

27 

26.6 

25.4 

23.5 

21.1 

18.1 

14.9 

11.6 

8.41 

5.55 

3.13 

1.35 

0.27 

27 

28 

27.6 

26.4 

21.4 

21.9 

1H.8 

15.4 

12.0 

8.72 

5.76 

3.25 

1.40 

0.28 

28 

29 

28 6 

27.3 

25.3 

22.7 

19.1 

16.0 

12.5 

9.03 

5.96 

3.36 

1.45 

0 29 

29 

30 

29.6 

28.3 

26.2 

23.4 

20.1 

16.5 

12.9 

9.34 

6.17 

3.48 

1.50 

0.30 

so 

It. 

5.45 i 

5.15 | 

4.45 

4.15 

3.45 < 

3.15 

2.45 | 

2.15 

1.45 

1.15 

0.45 

0.15 

R. 

Rise 

Time of soundin 

g in hrs. and min. 

before or after that of low water. 

Rise 






























































670 


Astronomy. 


To Find at what Time the Sun Sets and liises. 

Let v denote the time angle from 6 o’clock to when the sun sets or rises, then— 

Sin.o = tan.? tan.D. 

Example. What time does the sun set and rise on the 21st ot June, in 60° lati¬ 
tude? 

The declination on this day is about 23° 27'. 

Sin.o = tan.60° X tan.23° 27' = 0.75131 = sin.3/t. 14/n. 48 s. 

The sun rises at 2 h. 45m. 12s., and sets at 9/t. 14/a. 48s. 

To Find the Length of Day and Night. 

Day.—D ouble the time of sunset, is the length of the day. 

Night.—D ouble the time of sunrise, is the length of the night. 

Amplitude. 

The angle or bearing from east or west to where any heavenly body sets or rises, 
is called the amplitude of that body, which, denoted by x } will be— 

Sin.x = sec.? siu.Z>. 


The amplitude is used for finding the variation of the coinpass. 

Example. The sun’s declination being 18° 25' south, required, his amplitude in 
latitude 48° 45' north ? 

Sin.a; = sec.48° 45' X sin,18° 25' = sin.28° 38' south, 
the amplitude required. 

Azimuth. 


1 = latitude, D = declination, and a = altitude. 
z — angle of azimuth, or bearing of the heavenly body from meridian to the 
polo above horizon. 

When the latitude and declination are of 


Same Names— 

l + a— J) + 90 

m =- 

2 


Different Names— 

/ + fl + D + 90 


m = - 


2 


n = ±m + ^90 — Dj‘ 


Subtract the smallest, and the 
remainder is n. 


Cos. £ 2 = |/cos.m cos.n s^c./ sec.o. 















Alphabets for Headings. 


G71 



al)cDefgi)tjitlmnopqrstttbtoxg|. 




' && 
>) OP&, 


aicbi\^i\Urano^(\x$invroxrjt, 


a®e©eg@$3«s9R3iog>Dgi@a:u®fflx 


§) 3 abcbcfg^ijflmno^qr§ftuwx9j 




&£6££F^IWW?W 

IpWXtjZ. 1234567890 

;(bcc)cf^)i|i{Sii()i}opc|!rsf(|btoxl)Z. 

B11AF All BHJM3B ABFIHABFf. 


Keep your hand horizontal for the letters a, g, h, r, s and t 
For j and z describe the letter with the finger in the air 
For x make a motion up and down with the index finger. 
























THE 


RELATIVE PROPORTIONS 

OF THE 

Steam-Engine: 

BEING 

A RATIONAL AND PRACTICAL DISCUSSION OF EVERY 
DETAIL OF THE STEAM-ENGINE. 


BY 

WILLIAM DENNIS MARKS, Ph.JL, C.E., 

WIIITNKY PROFESSOR OF DYNAMICAL ENGINEERING IN THE UNIVERSITY 
OF PENNSYLVANIA, AND CONSULTING MECHANICAL 
AND CIVIL ENGINEER. 


WITH NUMEROUS DIAGRAMS. 


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REFERENCE BOOKS. 


Worcester’s Unabridged Dictionary 

Of the English Language. Standard Royal Quarto Dictionary of the English 
Language. Profusely Illustrated. New Edition, with Supplement, containing 
12,500 New Words. Also a Vocabulary of Synonymes of Words in General 
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Lippincott’s Dictionary of Biography. 

A Complete Pronouncing Dictionary of Biography and Mythology, containing 
Notices of Eminent Personages of all Ages and Countries, with the Correct 
Pronunciation of their Names. Edited by J. Thomas, M.D., author of the 
System of Pronunciation in “ Lippincott’s Pronouncing Gazetteer of the World.” 
Royal 8vo. New Edition , Revised and Enlarged. 2550 pages. Library 
sheep. $12.00. Half Russia. $15.00. 

Lippincott’s Gazetteer of the World. 

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Tables showing the Population, etc., of the Principal Cities and Towns of the 
World. New Edition. One imperial octavo volume. 2680 pages. Library 
sheep. $12.00. 

Chambers’s Encyclopaedia. 

Revised Popular Edition. With many additional Full-page Illustrations and 
Maps. Complete in 10 vols. Library sheep. $30.00. 

Chambers’s Book of Days. 

A Miscellany of Popular Antiquities Connected with the Calendar, including 
Anecdote, Biography and History, Curiosities of Literature, Oddities of Life, 
etc. Edited by Robert Chambers. Profusely Illustrated. Two vols. Royal 
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Brewer’s Reader’s Handbook 

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Brewer’s Dictionary of Phrase and Fable. 

Giving the Derivation, Source, or Origin of about 20,000 Common Phrases, Illu¬ 
sions, and Words that have a Tale to Tell. New Edition. $2.50. 

Brewer’s Dictionary of Miracles, 

Imitative, Realistic, and Dogmatic. With Illustrations. $2.50. 

Edwards’s Words, Facts, and Phrases. 

A Dictionary of Curious, Quaint, and Out-of-the-Way Matters. $2.50. 

Allibone’s Quotations. 

Prose Quotations. Prose Quotations from Socrates to Macaulay. With In¬ 
dexes. Dictionary of Poetical Quotations. Covering the Entire Field 
of British and American Poetry from the Time of Chaucer to the Present Day. 
Great Authors of all Ages. Being Selections from the Prose Works of 
Eminent Writers from the Time of Pericles to the Present Day. By S. Austin 
Allibone.' Three vols. 8vo. Cloth. Per set, $9.00. 

English Synonymes. 

By Richard Soule. New Edition. A Dictionary of English Synonymes, and 
Synonymous or Parallel Expressions. Designed as a Practical Guide to Apt¬ 
ness and Variety of Phraseology. Large i2mo. Extra cloth. $2.00. 


Published by J. B, LIPPINCOTT COMPANY, Philadelphia. 

6 













-A- VALUABLE WO IR, IKI 

Now Offered at Greatly Reduced Prices. 




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***For sale by all Booksellers, or will be sent, transportation free, upon receipt 
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J. B. LIPPINCOTT COMPANY, Publishers, 

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7 












CHAMBERS, BROTHER & CO., 



\ 



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— 

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8 



























































































































